Doublewell Tunneling via the Feynman-Kac Formula Citation Hardarson, Askell (1988) Doublewell Tunneling via the Feynman-Kac Formula. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/34ks-xy63. https://resolver.caltech.edu/CaltechETD:etd-09062005-152643 Abstract We discuss asymptotics of the heat kernel [equation; see abstract in scanned thesis for details] and its x-derivatives when T, λ → ∞ and (T/λ) → 0 where H(λ) = - ((Δ/2) + λ²V) and where V is a double well. When the groundstate is localized in both wells for λ large we derive, by the Feynman-Kac formula, W.K.B. expansions of the groundstate, the first excited state and their gradients. As a consequence we get a general asymptotic formula for the splitting of the two lowest eigenvalues, E₀(λ) and E₁(λ). This formula allows us, in principle, always to go beyond the leading order given by [equation; see abstract in scanned thesis for details] where C is the action of a classical instanton. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Simon, Barry M. Thesis Committee: Simon, Barry M. (chair) Luxemburg, W. A. J. Katok, Anatole Koonin, Steven E. Defense Date: 14 September 1987 Funders: Funding Agency Grant Number Alfred P. Sloan Foundation UNSPECIFIED Record Number: CaltechETD:etd-09062005-152643 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09062005-152643 DOI: 10.7907/34ks-xy63 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3352 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 12 Sep 2005 Last Modified: 02 Jul 2025 23:20 Thesis Files Preview PDF - Final Version See Usage Policy. 12MB Repository Staff Only: item control page

DOUBLEWELL TUNNELING VIA THE FEYNMAN-KAC FORMULA Thesis by Askell Hardarson In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1988 (Submitted September 14, 1987) -11- Acknowledgements I am most grateful to my advisor, Barry Simon who suggested my thesis problem, for his encouragement, patience and great support. I would like to thank W.A.J. Luxemburg for his support and very pleasant discussions and I wish to thank him and Barry for their enjoyable courses in functional and asymptotic analysis and in mathematical physics. It is a pleasure to thank the IHES and David Ruelle for their hospitality during the months of the preparation of this thesis and to thank Tom Spencer and others for many helpful discussions over there. I also would like to thank the Alfred P. Sloan Foundation for awarding me Dissertation Fellowship, 1986-1987. Finally I would like to thank my friends, above all Matt Newlin, Pierre Baldi and Clemens Glaffig for keeping me company and to thank Lillian Chappelle, Valarie Steppes-Glisson and Nancy O’Connor for the expert typing of this thesis. -111- Abstract τH(Λ) We discuss asymptotics of the heat kernel e when T, λ → ∞ and -y → 0 where H(λ) = — ^ (x,y) and its x-derivatives + λ V and where V is a double well. When the groundstate is localized in both wells for λ large we derive, by the Feynman- Kac formula, W.K.B. expansions of the groundstate, the first excited state and their gradients. As a consequence we get a general asymptotic formula for the splitting of the two lowest eigenvalues, Eθ(λ) and E∣(λ). This formula allows us, in principle, always to go beyond the leading order log(E1(A)-Eθ(A)) given by — C where C is the action of a classical instanton. -IV- Table of Contents Acknowledgements.......................................................................................................................................................... ii Abstract................................................................................................................................................................................. iii Introduction.......................................................................................................................................................................... 1 The Agmon Distance and the MinimalAction.............................................................................................22 Proof of Theorem C..................................................................................................................................................... 41 A Proof of Theorem B............................................................................................................................................... 45 The Feynman-Kac Formula and LargeDeviations.................................................................................. 76 The Proof of Theorem A.......................................................................................................................................... 85 A Proof of Lemma 8................................................................................................................................................. 121 A Proof of Lemma 9................................................................................................................................................. 138 Differentiating the Heat Kernel......................................................................................................................... 144 A Proof of Lemma 10................................................................................................................................................151 A Proof of Lemma 7................................................................................................................................................. 165 The Proofs of Sublemma 4 and Proposition 1......................................................................................... 185 Some Basic Properties of Agmon Geodesics and Minimal Action Paths.............................. 200 Some Estimates (Sublemma 8).......................................................................................................................... 224 On the Solutions of the Jacobi Equations.................................................................................................. 243 Proof and Sketches of Proofs of Lemmas 2 to 6.................................................................................... 263 References 271 -1- 1. Introduction. §1.1. The statement of the problem. §1.2. Reduction to asymptotics of eigenfunctions and their gradients. §1.3. L2-asymptotics from Perturbation Theory, suffice near the bottom of the wells. §1.4. General upperbounds for eigenfunctions and their gradients. §1.5. Finer analysis near K the set of midpoints of Agmon geodesics between a and b. §1.6. Heat kernel asymptotics. §1.7. W.K.B.-expansions near K. §1.8. The asymptotics of Ej(A) — Eθ(λ) as A → ∞ . §1.9. The plan of the paper. -2- 1. Introduction. §1.1. The statement of the problem. Let H(λ) = + λ2V on Lj(Rd) d = 1, 2, 3, ... with λ > 0 where d a2 Δ = 53 ^t½5 *s the Laplacian and V is a double well defined by: i=l öxf (1.1) V ∈ Coo(Rd , R), V ≥ 0, V(x) = 0 «· X ε{a , b} where a ≠ b, the Hessian matrices V,,(a) and V7,(b) are strictly positive and ∣χ∣^moo V(x) > 0 As λ grows H(λ) looks more and more like that if V is replaced by the sum of its quadratic approximations at a and b. More precisely (Simon [1]): Fix an integer n. Then for λ large enough H(Λ) has at least (n + l) eigenvalues 0 < Eθ(A) ≤ E∣(λ) ≤ ∙∙∙ < En(λ) below its continuous spectrum and (1.2) E fλ) —- → en where 0 < eθ ≤ ej ≤ ∙∙∙ are the eigenvalues of h(a) φ h(b) with (1.3) h(c) := —+ 2 {V,,(c) X , x) on L2(Rd), for c ∈ {a , b}. For En(λ), as above, we let Ωn(x,λ) denote some corresponding eigenfunction with ∣∣Ωn∣∣2 = 1∙ -h θ(^ _1 2) as λ —► oo . The classical example of a double well is the symmetric double well in one 1 1 9 19 dimension: V(x) = rj(x+τ⅛) (x~¾) (e∙g∙, Landau-Lipschitz [1], Kac-Thompson [1], Reed-Simon [1] and Harrell [1]). Here V is even, the eigenfunctions f^’s are even and ¾n + l s are θdd, f°r n = θ, 1’ ■" ‘ M°re°ver> up to a sign, -3- (1.4) Ωθ(x,λ) = Ω1(x,λ) - 1 1 I 1 ⅛>0(λ2(x+i)) + Al 9p0(λ2(x-I)) + rθ(x , λ) I 1 y0(λ2(×+⅜)) + il ⅛⅛(λ2(x~⅛) + rl(x ’ λ) _1 where ∣∣r∣(∙ , λ)∣∣2 = 0(λ 9 ) as A → oo for i ∈ {0 , 1}. çJq is a normalized groundstate of h (a) = h(b), defined by (1.3) with a = — τ⅛ and b = | . Hence if we define, for some small ε > 0 and c ∈ {a , b}, (1.5) jc(x) : = 1 if ∣x-c∣ ≤ ε 0 otherwise we have ∣∣jaΩθ1∣2 ∣∣⅛ω0∣∣ = 5 + θ(λ z) as λ → 00 , which says the groundstate is living in both wells as λ —► 00 . 9 1 1 Similarly, in this L -biology, (Ωθ + Ω∣) lives near a — — ⅜ and Ωθ — Ω∣ near b — ∙ Applying the time evolution e ^(i) tθ (jj e -itH(λ) ,n n . k 7 (∏0+Ω1) = e u ) we ge⅛ (Ωθ+e -it(E1(λ)-E0(A)) υ Ω1), so after time t = ,r∕(E^(λ) — Eθ(λ)) a state living near a = — J⅛ has evolved to a state living near b = A . Hence the splitting Ej(λ) — Eθ(λ) tells how long it takes to tunnel through the classically forbidden region between the wells. Since “Every child knows that the amptitude for transmission obeys the W.K.B. formula” (Coleman [1], p. 806-807) we expect E^(A) — Eθ(A) to be of the -4- ×2(λ) ∕ ' ^'2,∖''∕ order exp(-λ ∫ V x1(λ) λk En(λ) ∖ 1 . 2(V(x)---- ⅛- ) dxl where a = -⅛ ≤ ×1(λ) < x9(λ) ≤ 2 = b ’ λ 2 Efl(λ) are the classical turning points at energy —-3— . This suggests A 00 which follows from: Theorem (Simon [2]). (1.7) lim A-→oo log(E1(λ)-E0(A)) _ = -p(a,b) where, for x, y ∈ (1.8) ||ja%ll2 HJbβO^ > θ then If , p(x , y) := inf {∫θ ^2V(γ(s)) ∣γ(s)∣dsι 7(O) = x and γ(l) = y∣ is the Agmon distance between x and y. Note if d = 1 and a < b then p(a,b) = ∫ 2V(x) dx . This theorem was originally proven using Feynman-Kac formula integrals and the proof allows a weaker hypothesis: (1.9) p∣jaΩo∣∣2 N-i l∣Jb^0"2 ≥ constant A for some m > 0 m > 0 and path -5- Assuming (1.9) we want to go beyond the leading order in (1.7) using path integrals. When there are finitely many Agmon geodesics between the wells, satisfying a nondegeneracy condition we get (see (1.56) below) Ej(λ) — Eθ(λ) = cθ 3 e ) (1 + y + 0(λ 3 2)) as λ→ oo . In this case Helffer and Sjöstrand [1] have expansions to all orders in . One can obtain this much more precise result by using our method, but now we are mainly concerned with error terms. We give a more general formula for E∣(λ) — Eθ(λ) (see (1.50) below) which also follows from the discussion before Theorem 6.6 in Helffer and Sjöstrand [1]. What « ∙ is new here is the method. We make asymptotic expansions of the heatkernel e (x,y) and its x-derivatives at T, λ → ∞, τH(A) ∖ λ < λ and we use those to get the asymptotics for Ωθ , Ω∣ and their derivatives that we will use, as we describe now. §1.2. Reduction to asymptotics of eigenfunctions and their gradients. Following Harrell [2] we use f ∂Ω∩ ∂Ω1λ ; («1 √ - ⅝ √) ds<×> (1.10) E1(λ) - Eθ(λ) = -------------------------------------2 ∫ Ω∩Ω1 dx nr '-' ∙*∙ where w is basically (explained later) {x : p(x,b) < p(x,a)} and here n is the outward normal. Now the problem reduces to find asymptotics of Ωθ , Ω∣ and their gradients as λ → ∞ . -6- §1.3. L 2 -asymptotics. from Perturbation theory, suffice near the bottom of the wells. To find asymptotics of ∫ Ω∩(x,A) Ω1(x,λ) dx as A → ∞ for any w, that w υ 1 contains a neighborhood of b and does not intersect some neighborhood of a, it suffices to have L -asymptotic expansions of Ω and from perturbation theory. From Simon [1], when V is polynomially bounded and more generally from Helffer and Sjöstrand [1] (see also Combes-Duclos and Seiler [1]) follows: There are functions a , β : (0,∞) → (0,1) with (1.11) α2(λ) + β2(λ) = 1 -I and α(A) = ∣∣jaΩθ∣∣2 (l+0(λ 2)) , β(λ) = ∣∣jbΩθ∣∣(l+0(λ _I 2)) so under assumption (1.9) we have (1.12) α(λ) β(λ) > c∣ A~m > 0 for A large. Moreover, for c ∈ {a,b} and any integer k > 1 there are polynomials P11 ,c , Po , ... , P.K,C , where P1 1 9C is odd, such that: (1) Ωθ(x,λ) = α(λ)(^ka(x,λ) + Γka(x,y)) + 0(λ)(≠k,b(×>λ) + rk,b^x,^)) and (2) Ωχ(x,λ) = -^(λ)(≠ka(x,λ) + rka(x,λ)) +α(λ)(≠kjb(×,λ) + rk,b(x,λ)) where -7- (1.13) ≠k,c(x-λ) = λ 3 i *,c(λ2(x-c)) Pk,c(^(×~c)) P1,c(A2(x-c)) ⅛ *’∙ + 1 + λz )■ A Here φc is a normalized (Gaussian) groundstate of h(c) = —+ A (V,,(c) x,x) for c ∈ {a,b} and the errors satisfy ∣∣rk c( ∙ , A)∣∣2 = θ(λ (k + l) 2 as A -→ ∞ . Those expansions imply the Rayleigh-Schrodinger series (Reed-Simon [1]) for Ej(λ) and Eθ(A) are the same and are asymptotic series (Simon [1]): There are constants {e^n∖°° such that for anyf > 0 0 n=0 x (1.M) (i-l) (!) ⅛W = .(θ) + _Q V + _ J. ... j__θ___ j. of—1 — ⅝(λ) A + + (i-l) + A By (1.2) eθ as A → ∞ = eθ = e∣ is the groundstate energy of h(a) and of h(b). If (1.13) holds with k = 2 we get, since P1,C 1 is odd, (1.15) ∫ Ωθ Ω∣ dx = a(λ) ∕7(A)ζl + constant + From now on we will assume: _3 2)) for any w as above. -8- There are functions a , β : (0,oo) —► (0>l) satisfying (1.11) and (1.12). With πι as in (1.12) there are polynomials P1,c , ... , P2m+2,c (*) for c ∈ {a,b}, where P∣ c is odd, so that (1.13) holds with k = 2m + 2 . Moreover (1.14) holds for (. = 2 . Now we can concentrate on finding asymptotics of κ∫ ” in (1.10). <9w §1.4. General upperbounds for eigenfunction and their gradients. Upper bounds for ∣Ωn(x,λ)∣ uniformly on compacts (u.o.c.) give upper bounds (Simon [6], Theorem C.2.5) for ∣VΩn(x,λ)∣ u.o.c., see also Remark (9.5). To get upper bounds for ∣Ωn(x,λ)∣ u.o.c. (or at infinity) one can use (Simon [2], Proposition 3.1) (1.16) τEn(A) τH(A) e λ ∣Ωn(x,λ)∣2 ≤ e λ (x,χ) . The Feynman-Kac formula (see §5.1) and large deviations (§5.2), yield: Theorem (Simon [2]). τH(A) (1.17) lim —- ------- τ----- = —A(x,y,T) -A-→OO ^ (1.18) A(x,y,T) = i∏f{∫θ (i -y2 + V(γ)) dt ∣γ(0) = x and γ(T) = y} u.o.c. in Rd x Rd × (0,oo) where is the classical action for —V. (1.17) together with (1.2) and (1.19) A(x,y,T) → min{p(x,a) + p(a,y), p(χ,b) + p(b,y)} u.o.c. in Rd × Rd as T →∞ gives the first part of -9- Theorem (Simon [2]). (1) (1.20) For each n lim ⅛l⅛⅛-A21 < -min{p(x,a), p(x,b)} and λ→oo λ lim foS∣^Ωn(×<-∖l < — min{p(x,a), ∕>(x,b)} λ→∞ ^ (2) (1.21) u.o.c. There is R > 0 s.t. for each n there are C2 and Cg > 0 with ∣Ωn(x,λ)∣ < Cβ e ^c2∣x∣ if ∣χ∣ > R a∏d λ > some λθ(n) . §1.5. Finer analysis near K. the set of midpoints of Agmon geodesics between and and b. Thinking of ∂w as {x : p(x,a) = p(x,b)} we see from (1.20) that the main contribution in “J ” in (1.10) is from a neighborhood of the set of midpoints of Agmon (9w geodesics between a and b, given by (1.22) K = {x : ∕>(x,a) = p(x,b) = . By (1.15) going beyond the leading order in (1.10) reduces to go beyond the leading order in (1.20) near K. To make sense out of this we need (Carmona-Simon [1]) (1.23) p(x,y) = rmfθ A(x,y,T) . For (1.23) to attain minimum we may have to take T —► 00 . If x £ {a,b} and c ∈ {a,b} then (Simon [2]) -10- (1.24) OO η Ο OO 1 Ο ∕>(x,c) = inf{Jθ (±√ + V(γ))dt : γ(0) = x,γ(oo) = c} and (1.25) p(a,b) = inf{J,.oo⅛T, We will denote by g + V(γ)dt) : γ(-oo) = a,γ(oo) = b} the minimizing paths of (1.24). paths for (1.25) are called instantons (Coleman [1]). The minimizing If g minimizes (1.25) then so do all time translates of g. For x ∈ K = {x : ρ(x,c) = p(x,b) = p⅜y--} we let gx be the instanton with gχ(0) = x. The gx,s are reparametrizations on [-oo,∞] of the Agmon geodesics between a and b minimizing (1.8), that are parametrized with respect to arclength. Moreover, Lemma L K is compact and has a neighborhood U such that Xc (1.27) For x ∈ U and c ∈ {a,b} that is a unique g ’ minimizing (1.24), the second variation (see (2.11)) of it is positive definite. Moreover (1.28) p(x,a) and p(x,b) are C°° on U and the gradient (p'(x,a) — p,(x,b)) ≠ 0 on U. For x ∈ K (p,(x,a)-pf(x,b)) = 2gx(0) = gχ(t)∣ „ ∙ v——U (Pr∞f in Chapter 13.) By (1.28) Z := {x : p(x,a) = p(x,b)} ∩ U defines a hypersurface containing K and for x ∈ K Z is transversal to gx , at x = gχ(0). -11- If w is a bounded open set with smooth boundary such that 3w ∩ U C Z and ∂w∖ U Ç {x : p(x,a) ≥ + εθ}, for some εθ > 0, then + εo ’ Xx>b) ≥ by (1.20) (130) +θ(< ∂Ω1 dS(X) "⅜f ~^ Ωθ ^⅛) dS(X) ~ iZ Ωθ 0n° -λ(p(a,b) + ε0) )∙ To obtain refinements of (1.20) near K we use te∕λ) (1.31) λ Ωi(x,λ) = e and (1.32) tH(Λ) ∫e (x,y) Ωi(y,λ) dy λ τΞii^ ≠-Ωi(x,λ) = e λ τH(Λ) ∫-2-0xj λ (x,y) Ω∣(y,λ) dy if i ∈ {0,l}, j ∈ {l,...,d} and make asymptotic expansions .H(Λ) -τϊ λ (x,y) and of of e τH(Λ) e (χ,y) (refinements of (1.17)): §1.6. Heat kernel asymptotics. Theorem A. Let V be a double well and assume ∣V7(x)∣ = 0(e about derivatives. δ > 0 and, If xθ ∈ K = {x : p(x,a) = p(x,b) = A∣x∣2 ) for statements then there exists -12- 1. There is 7 > 0 such that if δ ∈ (O,⅛] then τH(Λ) τH(A) ⅛ ⅛j e λ (x>y))l, e λ (χ,y) = (1.33) O(mln(e-'re°ii** e-'i,<x∙i) , e-τ¾<b> e→<x,b)} -Xjl uniformly for x ∈ B(xθ , δ) y g B(aJ) U B(bJ) as T, A → 00 and T/A → 0 Here eθ(c) trace,) V,,(c) for c ∈{a,b}. 2 2. Assume (1.34) V,'(b) = Ω2 where Ω = diag(w1 ,..., W(J) with 0 < ω1 ≤ ∙∙∙≤ ωj then there are positive constants ≤ y , ’ c4 ’ and T0 and functi°as aθ(×,y)>af (x,y)> a11j(×,y) on B(xθ , 6) × B(b∕) × [Tθ , ∞) for i ∈ {l,...,d} with aj(x,y), aT(χ,y), a^(x,y) ∈ C00(B(χθ , 6) × B(b,⅛)) for T ≥ Tθ and j ∈ {l,...,d} such that (1.35) H(λ) 1 -d g a -T — λ (×,y) = π 2 A ( ∏ ωi) ao(χ5y) exp(-AA(x,y,T)) e“Te°(b) ι=l aT(x,y) rr4 τ2 {1 + -½- + 0(⅛) + 0(T exp(~λ ~⅜))} and (1.36) 4⅛' τH(Λ) _d λ <«> = ’ 2 {⅛2 ÷ ⅛ 5 d § λ ( i-51 ωj) exp(-λ A(x,y,T)) + θts>+ 0⅛ £»} -13- uniformly for x ∈ B(χθ∕), y ∈ B(b,<5) as T, A → ∞ and T^ ≤ A where (1∙37) aJ(×>y) = ao(x) (1 + 0(e ““ è τ 1 + ∣y-b∣)) xb with aθ(x) = (det X ’ (0)) _1 2 > constant > 0 and xb X ’ as described in Theorem B below. w τ 1 ) (1.38) A(x,y,T) = ρ(x,b) + p(b,y) + 0(e (1∙39) P(b,y) = ⅛ ∑ ωi(yi-b∣)2 + 0(∣y-b∣3) (l∙40) 9A(x,y,T) _ <9p(x,b) ι n/ ~<5iι∖ -----+ θ<e ) and (1∙41) aT(χ,y) = 0(T) = aj,.(x,y) uniformly on B(xθ , 6) × B(b,<5) × [Tθ , ∞}. J- J The proof of this theorem, especially the uniformity for large T (see chapter 6), is the issue here. We use large deviations for T in a fixed time interval and then the semigroup property to get large T’s. Since the total time the instantons spend outside given neighborhoods of a and b is bounded from above, we can show it suffices to do finer analysis near a and near b. There V' is close to its quadratic approximations and we can do the analysis. We use (1.34), (1.20), and (1.21) to estimate the contribution in (1.31) and (1.32) from R^∖(B(a , ∣) U B(a , ^)) for 6 as Theorem A. Then we use (1.35) (resp. (1.36)) and the L^-asymptotic expansions in (*) when integrating -14- over B(a , U B(b , in (1-31) (resp (1.32)). There we use the method of stationary phase (Hörmander [1], Theorem 7.7.5) which requires improvements of (1.37)-(1.41) (see Chapter 4, lemmas 6 and 7). After using the asymptotics of E∣(λ) and Eθ(λ) in (*) we put T = constant log λ for some positive constant (idea from Breen [1]) and we get: §1.7. W.K.B. expansions near K. Theorem B. If ∣V,(x)∣ = 0(e A∣χ∣2 ) as ∣x∣ -→ ∞ and (*) holds then there is a neighborhood U of K = {x : ρ(x,a) = p(x,b) = f1ι,c and fη a∏d functions fc (see (1.44)) ∙ ∈ Coo(U,R) for c ∈ {a,b} and for j ∈ {l,...,d} such that if, for c ∈ {a,b} and j ∈ {l,...,d}, (1.42) Fc(x,λ)= = fc(×) + f ∙(χ,λ) 0p(x,c) fι,cj(x) 0X’ fc(x) + ’ λ then (compare with (1.13)) d _3 Ωθ(x,λ) = λ4 α(λ)(Fa(x,A) + (A _ e + λ5∕3(α)(Fb(χ,λ)+O(λ 2)) e_Ap(x’b) d _3 _ Ω1(x,λ) = -λ4 ^(λ)(Fa(x,λ) + 0(A 2)) e P(X,a) + λ4 α(λ)(Fb(x,λ) + 0(λ 2))e Ap(X'b) -15- (1.43) -A Ωθ(x,λ) = λ4 α(λ)(Faj(x,λ) + 0(λ 2)) e λp^ + λ4 ∕7(λ)(Fbj(x,λ) + 0(λ~2)) e-λXx'b) and -I d β Ω1(x,λ) = -λ4 ∕J(λ)(Faj(x,λ) + 0(λ^2)) e“W’a) d _3 _ + λ⅛ α(λ)(F∣j j(x,λ) + 0(Λ 2)) e x, uniformly for x ∈ U as λ → ∞ For x ∈ and c ∈ {a,b}. (1.44) _d _______ 1 _1 fc(x) = τr 4(det Jv"(c))2 (det Xx,c(0)) 2 Xc where X ’ is the unique nonsingular matrix solution on [0 , ∞) of the Jacobi equation: (1.45) X(t) = Vπ(gx,c)(t) X(t) satisfying (1.46) Xx,c(t) = (I + 0(e M)) e W as ∞ where (Proof in chapter 4 .) Remarks. (1) Equations (1.43) hold near any point xθ such that: If c ∈ (a,b) and p(xθ , c) = min{p(xθ , a), p(xθ , b)} > 0. -16- (1.47) . x0 , c then there is a unique Agmon geodesic g and its second variation is positive definite (holds for xθ ∈ K by (1.27). If for xθ with p(xθ , b) < p(xθ , a) say we put fa, fj a and f∣ all equal to zero. So we only need to worry about the Agmon geodesics to the closest well(s) (because the eigenfunctions are living in both wells). In particular (1.43) holds near the set of Agmon geodesics U {gχ(t) ∣ t ∈ [∞ , oo]}. x∈K We have, for example, for some δ > 0 (1.48) d f ζx) _3 Ωθ(x,A) = λ4 ∕7(λ)(fb(λ) + ⅛- + 0(λ 2)) e~λXx'b) uniformly for x ∈ B(b,5). Moreover we have f^(b) = π _d 4 ^detj V,,(b)) and p(x,b) = ^((V"(b))2 (x-b), (x-b)) + 0(∣x-b∣3). 1 Hence (1.13) width k = 2m + 2 looks like a Taylor-expansion in λ2 (x — b) of the result in (1.48). Without assumption (1.47) one can presumably often get expansion similar to d d those in (1.43). With new fc , f1 c and f1 cj ,s and the prefactor A4 replaced by λ4 (d-Ι)’ Pc(λ) where 1 ≤ Pc(λ) ≤ A 2 , e.g., Pc(λ) = λs(log λ)k . §1.8. The asymptotics of E^(A) — Eθ(A), as A → oo . Theorem B and lemma 1 yield (see chapter 3) -17- Let U be a neighborhood of K = {x : p(x,a) = p(x,b) = Theorem Ç. as in . A∣x∣2 Theorem B and lemma 1. Assume (*) and ∣V (x) ∣ = 0(e ) as ∣x∣ —+ ∞ ∙ Then there are functions Fθ(x) > 0 (see (1.52) below) and F1(x) ∈ C0°(U,R) such that d+2 E1(λ) - Eθ(λ) = -----------------λj------------ (1.50) 2α(λ)0(λ)(l+^+0(λ 5)) (Fq(x) + ¾⅛} + 0(λ ∫ 3 2))e'λ(p(x,a) + p(x,b))ds(x) Z={x,p(x,a)=p(x,b)} ∩U where 0(λ _3 9 ) is uniform on Z. If X ∈ K (1.51) mx := nullity of the Hessian of (p(a,x) + p(x,b)) along Z, at x = dim{f ∈ L2(R,Rd) : f(t) = V,,( gx(t)) f(t) and <gx(0) ,f(0)> = 0} = dim{f ∈ L2(R,Rd) : f(t) = V,,(gχ(t)) f(t)} - 1. (1.52) Fθ(x) = ∣p,(x,a) - p,(x,b)∣ fa(×) fb(×) > 0 for x ∈ U where fa(x) and fb(x) are defined in (1.36). Remarks. (1) Observe the simple dependence of a(λ) and ∕J(λ). Recall α(λ) = ∣∣ja Ωθ∣∣2 (l + θ(^⅛)) and ∕7(λ) = ∣∣jb Ωθ∣∣9(l + θPi)) λ'2 -18- (2) For X ∈ K (1.53) ∣√(x,a)-√(x,b)∣ = 2,∣2V(x) see lemma 2 below. (3) Note p(x,a) -(- p(χ,b) ≥ p(a,b) with equality only on K which makes the nullity of 9 the Hessian of p(x,a) -)- p(x,b) along Z at x ∈ K important, gx is L -solution of the Jacobi equations. (1.54) f(t) = V"(gx(t)f, t ∈ R and (1∙51) is saying we should only look at the subspace of solutions of (1∙54) orthogonal to gx . Now we look at a few cases influenced from reading papers on short time asymptotic of diffusion processes on a R,iemannian manifold (see Molchanov [1] and C. Bellaiche [1]). Case χ. K = {xθ} and mxθ = 0. Then there are coordinates x = (xj , ..., Xj) near xθ such that x^ = 0 defines Z and positive constants a∣ , ..., (1.55) such that p(a,x) + p(x,b) = p(a,b) + ∑ o∙ x? + 0(∣x∣3) for x ∈ Z. i=l 1 Hence (Erdelyi [1]) there are constants cθ and c∣ such that (1.56) C 3 _3 E1(λ) - Eθ(λ) = 2^(⅛) λ2 (1 + T + θ(λ 5 ))e~Ap(a’b) as λ → ∞ . -19- Remark. (1.57) Taking straight line segments as trial paths for the Agmon distance, we get: There exist 6 > 0 and C > 0 such that p(y,×θ) ≤ C∣y-xθ∣ for all y ∈ B(xθ , δ). Hence by (1.55) and (1.28) we have in case 1; (1.58) There is a unique Agmon geodesic gxθ between a and b and there is a hypersurface Z that intersects gxθ at xθ transversally and a neighborhood U of xθ and a constant ~c such that p(a,x) + p(b,x) ≥ p(a,b) + = p(x,xθ), for all x ∈ U ∩ Z. This nondegeneracy assumption was made by Helffer and bjöstrand [1] in a multiwell situation when they proved, as mentioned earlier, (1.59) E1(λ) - Eθ(λ) = cθ λ2 e Ma’b) (1 + ½ + ... + ⅛ + 0(λ (n + 1))) for all n, as λ → ∞ Case 2. K = {xθ}, mxθ = 1 (so d > 2) and in some coordinates near xθ p(a,x) + p(x,b) = ∕>(a,b) + x 2p 1 d-1 ο + ∑ x^ if x ∈ Z, for some p ≥ 2. Then there are «=2 e constants c∣ ··■ Cβp-∣ so that (Erdelyi [1] or 01ver [1]) E1(λ) - Eθ(λ) = "0 2o(λ)∕J(λ) as λ —+ ∞ . ∕2'⅛) vι + ·■■ + (1 + l2p c3p-l (3-Aτ ι 2 2p× _3 — λp(a,b) + 0(λ 2))e -20- K = {xθ} with mx Case à- = 2 (so d > 3) and in some coordinates near xθ d-1 p(a,x) + p(x,b) = p(a,b) + xχ + x^ x^ + x^p + ∑ x^ with p ≥ 2. The Laplace 1 12 2 £ integrals needed are in C. Bellaiche [1] and we have: If p = 2 co λ2 e~λXaΛ) (1 -(- o(l)) as λ → ∞ and if p > 2 2α(λ)∕J(λ) - ----2o(λ)∕3(λ) log λ e (J + o(l)) ! as λ → oo . m.. V _l_I Λ.1 U∙∙-r en **_:'* V <'', rl —***X — U. e=k+2 E∣(λ) — Eθ(λ) = K is a submanifold and k = dim K(≤ d — 1) and for x ∈ K mx = k (resp. Case 4. ∑1 E^(λ) — Eθ(λ) and in s∩mp Cnr'Γdi∏HtβS — in χ2 (resp. ρ(a,x) + p(x,b) = p(a,b) + x^p k+i e ∙∙∙ are constants cθ , (1 + y + 2∖∖ -λP(a,b) ))e 0(λ + Λ fa.hl — rλΓ4-, rnfχ.H^ v--7- z — Γ ∖-T~∕ + ∑ É=k+1 x^ , for P ≥ 2) then there e co such that E^(λ) — Eθ(λ) = 2α(A)ff(A) as λ → ∞ (resp. E^(λ) — Eθ(A) = ,3-f-k+l____ 1∖ 3λλ e -λp(a,b) ? 2 2p' (1 + -⅛ +'"+-7Γ⅛ + θ(λ~2^ as A —► oo χ2p .'∙2 2p× Estimating p(x,a) -I- p(x,b) > p(a,b) on Z gives a general upper bound E1(λ) - Eθ(λ) = (d+2) A 2 —λp(a,b) (λ)/?(λ) β as A -→ oo . (See Hellfer and Sjöstrand [1], Theorem 6.6). "0 (λ)/?(λ) -21- §1.9. The plan of the paper. The main difficulty here is the uniformity for large T in Theorem A. Our method is to use large derivations on an interval [Tθ , 2Tθ] where Tθ is a fixed (large!) number. Then we use the semigroup property of the heatkernel for T = (n + 1) T∣ where T^ ∈ [Tθ , 2Tθ). In both cases and in proving Theorem B using (1.31) and (1.32) we need to analyze the minimal action, the Agmon distance, their paths, the related Jacobi fields and to compare those. We state these results in chapters 2 and 4 but their proofs or sketches of proofs are in chapters 11 to 16. We proof Theorem C (given Theorem B) in chapter 3, Theorem B (given Theorem A) in chapter 4 and in chapters 5 to 10 we prove Theorem A, the central part of the proof is in chapter 6. -22- 2. The Agmon Distance and the Minimal Action. §2.1. An Introduction - Proposition 1. §2.2. The positivity of the second variation and the solutions of the Jacobi equations. §2.3. Asymptotics of the Agmon geodesics, minimal action paths and the solutions of releated Jacobi equations. X V §2.4. The positivity of the second variation of grp . §2.5. The Greens matrix Gx,^,^(t,s) = (ξj4p- + V,,(gψ^)) Gt (t,s); some asymptotics ,χ,y,τ §2.6. Finer asymptotics of G ’ ’ (t,s) when y is exponentially close to b. -23- §2.1. An Introduction - Proposition L The Agmon distance defined in (1.8) by p(x,y) = inf{∫θ^2V(γ(s)) ∣∙γ(x)∣ ds : 7(O) = x, 7(l) = y} is the geodesic distance in the metric 2V(x)dx 9 which (and for more general V’s) is called the “Agmon metric” since the studies of Agmon [1] of exponential decay n of L -solutions of (-Δ + v) u = 0, at 00 . For V > constant > 0, at 00 , as we consider, it was earlier used by Lithner [1] for the same purpose and (1.21) follows from his work. Carmona and Simon [1] proved Agmon’s upper bound was a lower bound for the ground state, at ∞ . They used Feynman-Kac formula (see Simon [5]) and (1.23) that says ρ(x,y) = rinfθ A(x,y,T), where A(x,y,T) is the classical action given by (1.18) A(x,y,T) = inf {JT(a 72 + V(7)) dt: 7(O) = x, 7(T) = y} 7 0 z In tunneling problems the Agmon distance was introduced by Simon [2] when proving (1.7) by reducing to the upper bound in (1.20) and (1.21) and (2.1) log Ω∩(x,λ) lim -------- Ö------- = —min{p(x,a) , p(x,b)}. A→∞ ^ Helffer and Sjöstrand [1], [2], and [3] (see also Sjöstrand [1]) use the Agmon distance and perturbations of it extensively their studies of multiwells, and Dirichlet’s problems that they reduce to. -24- In the analysis of e τH(λ) i λ (χ,y) and Λ e j- H(λ) \ ^ (x,y) (see (1.17), (1.19 and Theorem A) the Agmon distance enters mainly through Proposition 1. (2.2) lim A(x,y,T) = min{p(x,a) + p(a,y), p(x,b) + p(b,y)} + 0(e~aT) 1 →∞ uniformly on compacts on x , for some a > 0 (depending on the compact) as T → ∞ . (A Proof in Chapter 12.) We will study p(x,c) and p(a,b) given by (1.24) OO 1 Q p(x,c) = inf{Jθ (A √ + V(γ))dt : γ(0) - x, p(oo) = c} if X £ (a,b) and c ∈ {a,b} and (1.25) oo 1 9 p(a,b) = infiJ-σo(⅛ 7 + v(7))dt ∙ 7(-∞) = a, γ(∞) = b} which implies that minimizing paths gψ,y for A(χ,y∕Γ), gX,C for p(x,c) in (1.24) and gx for p(a,b) in (1.25), if x ∈ K, all satisfy the same Euler-Lagrange equations. (2.3) γ(t) = V,(γ(t)). xb That is convenient when we compare g ’ (t) to g^1,y(t) on [0 , ι] and gy,b(t) to g^,y(T - t) on [0 , ∑] if all have those paths are unique and, say, -25- (2.4) ρ(x,b) + p(b,y) < p(x,a) + p(a,y). Note that to avoid the potential term in A(x,y,T) = ∫θ ⅛⅛τ,y)2 + V(g⅜y)) dt to grow as T → ∞ the minimal action paths have to run into the wells and stay there most of the time. Conditions like (2.4) make the g^>,y ,s for large T, go near b and not near a. §2.2. The second variation and the solutions of Jacobi equations. In the rest of this chapter we look at xθ satisfying (recall lemma 1) (2.5) p(xθ,b) < p(x∩0 , a) + p(a,b) (2.6) There is a unique Agmon geodesic g x∩,b minimizing p(xθ , b) in (1.24) and (2.7) χ∩,b the second variation of g υ is positive definite (see (2.11) below). It follows (see lemma 2 below) that these conditions are satisfied for x near xθ . X V Moreover for x near xθ , y near b and large T there is a unique grp variation is positive definite. and its second Moreover (2.5) implies (2.4) for x near xθ and y near b XV x b Vb and so we know we should compare gy' with g ’ and g ’ . To describe what (2.7) means, let J = [0,∞) or J = [0,T] and Q(t) a symmetric continuous (d × d)-matrix on J then -26- Definition (2.81. The system (2.9) — f(t) -∣- Q(t) f(t) = 0 on J is nonconjugate if no nonzero solution vanishes twice on J. This is equivalent to (Hartman [1], Coppel [1]) . For any subinterval [t^ , tg] C J, t2 ((0,0) + (Qrhr)Y) dt ≥ 0 , I(j7jt1,t2) := ∫ (2.10) tl Vη ∈ Dθ(tpt2) := {rç ∈ AC([tpt2], Rc^) : ⅛ ∈ L^(t∣,t2), ∏(t^) = = 0 = ^(t2)} and I(τ∕,tpt2) = 0 only if η = 0. Definitions. The second variation of (2.11) gX’b is positive iff (2.10) holds with J = [0,oo) and Q(t) = V,,(gx,^(t)). The second variation of (2.12) *s positive definite iff (2.10) holds with J = [0,T] and Q(t) = V"(gτ,y(t))∙ ,, x∩,b Hence (2.7) is telling us the Jacobi system (1.4) — f(t) + V',(g (t)) f(t) = 0 is nonconjugate on (0,∞) meaning that (1.54) has no nonzero solution vanishing twice on [0,oo). Remark (2.13). (See A. Bellaiche (1] and Kobayashi [1]). On a Riemannian manifold m of class Cr for r > 2 one defines the cutlocus of m, by c(m) := {(x,y) ∈ m × in there are at least 2 minimal geodesics between x and y, or x and y are conjugate along a minimal geodesic (second variation is not positive definite)}. Moreover one defines -27- the cutlocus of x ∈ m by c(x) := {y : (x,y) ∈ c(m)}. Then c(m) is a closed subset of m X m, c(x) is a closed subset of measure zero in m and the energy E = Cr~1 on m × m∖c(m). 1 2 d (x,y) is Moreover if g is a minimal geodesic between x and y then x and any point z on g before y are in c(m). Note the metric tensor gjj(x) — 2V(x) 6y , for the Agmon distance vanishes at -T H(λ) a and b. Therefore we cannot quote those results and therefore e behaves as d described in Theorem A and not like the free one: (g^p) exP(----- We xb Vb mainly make statements about p(x,b), ρ(b,y), g ’ and g ’ for x near xθ and y near b and not about p(x,y) and Agmon geodesics between x and y. §2.3. Asymptotics of Agmon geodesics, minimal action paths and solutions of related Jacobi equations. We will assume (1.33) V,,(b) = Ω^ where Ω = diag(ωj , ..., ω^) with θ < ωι ≤ ω2 ≤-"≤ ωd ' If (2.5), (2.6) and (2.7) hold then Lemma 2. There exists <5θ , 6∣ and Tθ > 0 such that xb If x ∈ B(xθ , 5q) then there is a unique Agmon geodesic g ’ . It’s second variation is positive definite and it depends Coo-ly on x ∈ B(xθ , όθ). For each 1. α∈N<} (2.14) (√∙b(t) - b) = Ο«,-“1') = ⅛! g×∙b(.) uniformly on B(xθ , έθ) × [0,∞) . -28- (2.15) For y ∈ B(b,δθ) and i ∈ {l,...,d} (gy,b(t)-b)i = (y~b)i e 1 + 9 — min(ω∙ ,2ω1)t 0(∣y-b∣2) e 1 1 vk -<1(gy, (t))i∙ 2 i_ For X ∈ B(×q,⅛q) the Jacobi system — -i= U (t) + Vπ(gx, (t)) U(t) dtz X L· Xb t ∈ [0,∞) has matrix solutions X ’ (t) and Y ’ (t) satisfying (2.16) Xx,b(t) = (I + 0(e ^lt)) e~βt = -Xx,b(t) Ω^1 γx,b(t) = (I + 0(e 0l⅛ efit = Yx,b(t) Ω -1 uniform on B(xθ , 5θ) × [0,∞) which are infinitely differentiable in B(xθ , <5θ) with For each a ∈ Nθ ∣α∣ ≥ 1 (2.17) α1∣ ∙xb . Jαl xb -M o÷ Λ1∣“ —61t ⅜-7j XX’b(t) = 0(e 1 ) e-fit = ∣-δ Xxφ(t) θx ∂×l ∣y Yx'b(t) = 0(e ^)βΩί = |!ί Yx,b(t) uniformly on B(xθ , <5θ) × [0,∞) and (2.18) det(Xx, (t)) > constant e ~t(∑ ωi) '-> 0 uniformly on B(xθ , 5θ) × [0,∞) . Moreover 0 for -29- (2.19) ⅛ (⅛)i gX’b(t) = ((⅛)i Xx,b(t))(Xx,b(O))-1 ej if i ∈ {O,1} and j ∈ {l,2,...,d} where βj = (0∣j)d-1 and 0jj is the Kronecker delta and = -(gx⅛)j for j = 1, ..., d (2.20) 2. If (x,y,T) ∈ B(xθ , 0q) × B(b,6θ) x [Tθ , ∞) there is a unique minimal action path XV xy ∙xy gψ and it has positive second variation (as defined in (1.12)), grp (t) , grp’ (t) , A(x,y,T) are infinitely differentiable with respect to (x,y) ∈ B(xθ , όθ) x B(bθ , 6θ) (see (2.26) below). The Jacobi system (2.21) (--a∑ + v"(gx,y)) U(t) = 0 for t ∈ [0,T] dtz has solutions χx,^,^ and γx,y,^ satisfying (2.22) XX’y’T(t) = (I + 0(e + ∣y-b∣)) e“fit = XX,y,T(t) Ω^1 Yx,y,τ(t) = (I + 0(e-*lt + ∣y-b∣)) efit = YX,y,T(t) Ω~1 uniformly (x,y,T,t) ∈ B(xθ , 6θ) × B(b,5θ) x [Tθ , ∞) x [0,T] and (2.23) d -t( ∑ ωi) det XX’y’T(t) > (const) e i=1 >0 uniformly (x,y,T,t) ∈ B(xθ , δθ) × B(b,<5θ) × [Tθ , ∞) x [0,T] -30- The X x,y,T *x,y,T γx,y,T and Ÿ χ,y,τ ’s are infinitely differentiable and satisfy (2.24) For each a , β ∈ N' 0 |a| + |/?| rp ∣α∣ + ∣∕3∣ . χ y <p 2-------- χx,y,1(t) = 0(1) e-βt = 2-------- Y X x,y,i ,j, (t) <9xa<9y ∂-χa∂γ^ ∣a∣ + ∣∕3∣ y ∣α∣ + ∣β∣ .χy<j> 2-------- 3- Yx,y,1(t) = 0(1) ei2t = 2-------- Yx,y,1(t) 9xa5y ∂xα0y uniformly for (x,y,T,t) ∈ B(xθ , 6) x B(b,δθ) × [Tθ , ∞) x [0,T]. With XX’b and Yx,b as in (2.15) we have (2.25) (XX’y’T(t) - Xx,b(t)) efit = = 0(e Ul2 + ∣y-b∣) = (XX’y’T(t) - Xx,b(t)) eβt and (YX’y’T(t) - Yx,y(t)) e~m = 0(e Wl2 + ∣y-b∣) = (YX’y’T(t) - Yx,b(t)) e~m uniformly for (x,y,T,t) ∈ B(xθ , δθ) × B(b,δθ) × [Tθ , ∞] × [0,T]. Moreover, with g = gψ,y , X = χx,y,^ and γ = γx>y>^ we have -31- A(x,y,T) = (g(T))i (2.26) i∆!⅛l> = _(έ(0)). ⅛S = 1 te(T)2 + V(y) <⅛>'i ⅛ «(«) = ⅛ <⅛>j s<*> = <⅜>l (x(*)-Y(*) (Y(T)-1X(T)) (x(O)-Y(O) (Y(T))-1X(T))-1 ei W ⅛ «“> = ⅛ <⅛>i = = <⅛>i (Y(t)-X(>)(X(0))-1Y(0)) (Y(T)-X(T) <X(0))-1Y(0))-1 ei for j ∈ {0,l} and i ∈ {l,...,d} and finally If a ∈ Nq then (2.27) ∣-δ V(g^,y(t)) = 0(1) uniformly for (x,y,T,t) ∈ B(xθ , 6θ) × B(b,5θ) × [Tθ , ∞) × [0,T] Remark. By the Euler-Lagrange equation (2.3) gψ,y(t) = VV(g^,,y(t)) if t ∈ [0,T] we X V have constant energy along gψ, (t) given by (2.28) Eτ(x,y) := ±(⅛y(0))2 - V(g*'y(0)) = i(g⅜,y(t))2 - V(g*,y(t)) for all t ∈ [0,T]. -32- Xb Similar 0 energy along g ’ by (2.3) and (2.14) (2.29) ±(gx,b(t))2 - V(gx,b(t)) = Γ(gx,b(s)2 - VV(gx'b(s)) gX’b(s) ds = 0 . Using(2.26) and(2.28) we get the Hamilton-Jacobi equation (Gelfand-Fomin [1]) i(Vx A(x,y,T))2 + V(x) = i(Vx A(x,y,T))2 + V(y) = --⅛-) ∙ x>y §2.4. The positivity of the second variation of gψ In the next lemma we describe in terms of inequalities what “the second variation of g^-,y is positive definite” means. That is useful when we go beyond the leading order in τH(λ) ----- ------ (x,y) = —A(x,y,T) as A → oo . Some information about the Agmon distance, the minimal action and the comparison of different minimal action paths, that we need later will follow. Lemma 2∙ With xθ , <5θ , <5^ , and Tθ as in lemma 2 X V 1. There exists a constant k1 > 0 such that if ∣∣γ — grp, ∣∣ ≤ δ∩ then L∞(0,T) (2.30) where ∫τ(⅛2 + (V"(y)η,η)) dt > k1∣∣r,∣∣2 for all η ∈ Dθ(0,T) 0 1 Lσo[0,T] -33- (2.31) Dθ(O,T) := {η ∈ AC([O,T],Rd) : η(O) = Ο = ∣7(T) and ⅛ ∈L2[0,T]} with (2.32) IMI L∞[O,T] sup O≤t<T ( Σ »?i (t)) i=l 1 Also (2.33) ∫τ(±(⅛y + ⅛)2 + V((g*’y + η))dt - ∫τ(⅛(gζjy)2 + V(g⅞y))dt ozτ 0 > k1 minf^Q , ∣∣τj∣∣2 1 for all η ∈ Dq(0,T), k U L∞[0,T]' U uniformly for (x,y,T) ∈ B(xθ , όθ) × B(b , 6∩) × [T∩ , ∞) . 2. Uniformly on compacts in (2.34) {(x,y) ∈ Rd × Rd : ρ(x,b) + p(b,y) < ∕>(x,a) + p(a,y)} ⅛Ty(t)∣ ’ ∣⅛yW - bl = 0(e 1 + min(∣y-b∣ , <5θ)e Eτ(*,y) = ⅛g⅞y(t))2 - V(g×>y(t)) = 0(e (2.35) w T 1 ) (see (2.28)) . A(x,y,T) = p(x,b) + p(b,y) + 0(e WlT) where (2.36) p(y,b) = ∣ ∑ ωi(yi - bj)2 + 0(∣y-b∣3) uniformly on B(b , 5θ) . 3∙ If (xn , y∏ , Tn) -→ (x,y,T) ∈ B(xθ , δθ) × B(b , δθ) × [Tθ , ∞) then -34- (2.37) ,∣g×n,yn ( ±sι _ ffx,y∣∣ ®T "l 0° L∞[O,T] rn 0 as η → ∞ 4. There exists k2 > 1 such that (2.38) ∣gτ1,yi(t) - gT2’y2(t)|, ∣gτ1'yi(t) - gT2’y2(t)| ≤ k2(∣χ1-χ2∣e — ω1t — ω1(T-1) + ∣y1-y2∣e ) for t ∈ [0,T], (χ1 , χ2) ∈ (B(χθ , 6q))2, (y1 , y2) ∈ (B(b,6θ))2 and T ∈[Tθ ,∞), (2.39) g⅜,y(t) ∈ B(b,⅞) if t ∈ ⅛ , and (2.40) χb χb — Wι(T-1) —ω1T ∣grp, (t) — g ’ (t)∣ ≤ k2 e e 1 if t ∈ [0,T] for all (χ,y,T) ∈ B(χθ , 5θ) × B(b,6θ) x [Tθ , ∞). Finally if xθ = b (2.41) ∣gτ1'yi(t) - gτ2'y2∣ ≤ maχ(∣χ1-χ2∣, ∣yχ-y2∣). (A proof in chapter 16.) Note (2.38) and (2.40) imply (2.42) χv χb —ωηT —ω1(T-1) ∣g⅜y(t) - gX’b(t)| < k2(∣y-b∣ + e 1 )e 1 -35- §2.5. The Green’s matrix Gx,y,^it.sl = When analyzing the heat kernel e + -T (t, s); some asymptotics. H(λ) we will change a measure on the path space and instead of using the Brownian bridge measure whose covariance operator (see Simon [5], chapter 2) is ( — -S-- on [0,T] with Dirichlet’s boundary dt2 conditions)-1 we use the measure associated with the mean zero whose covariance operator is (( — 2 Gaussian process + V,,(gψ^(t)) on [0,T] with Dirichlets boundary dtz conditions) -1 We need to analyze its integral kernel which we call the Green’s matrix Gx,y,T(s,t) . Observe if x = y = b then (2.43) Gb'b'τ(s,t)= [e^ni'-"l - (e2«T - I)-l {eii<s+,> - e"'-*’ -eΩ(l-s) + sΩ(2T-s—1)}) (2Ω)-1 where Ω = diag(wj,...,ωtj) = Jv,,(b) and we have a d-dimensional oscillator process tied xyT down at 0 and T. For certain error terms we need small perturbation of G ’ ’ (t,s). Lemma 4. With xθ , 6θ , and Tθ as in lemma 2 let W^(t) for t ∈ [0,T] if (r,T) ∈ [0 , Γj] × [Tq , ∞), r∣ > 0, be continuous symmetric (d × d)-matrices with -36- (2.44) T ∫ ∣Wj-(t)∣ dt = 0(r) as r J. o uniformly for T ∈ [Τθ , ο©]. 1. Then there exists rθ such that (2.45) Ü = (V'⅛y) + W!p(t) U has solutions X*,y,T(t) and γ×>y,T on [0,T] for (x,y,T,r) ∈ B(xθ , <5θ) × B(b,0θ) × [Tθ , oo) × [0 rθ] with (2.46) Xr’y’T(t) - XX’y’T(t) = t — <51(t-s) „ T ∩4, 0(∫ e 1 ∣W⅛(s)∣ds + ∫ ∣WJj,(s)∣ ds) e~i2t = (X?’y’T(r) - XX’y,T) and (2.47) YX’y’T(t) - YX’y’T(t) = 0(f 0 ∣W⅜(s)∣ ds + JT ∣W⅞(s)∣ds) efit = ( Yx'y'τ(t) - YX’y’T(t)) t uniformly for those (x,y,T,r),s (2.48) XX’y’T(t) = X?’y’T(t)|r_0 and YX’y’T(t) = Y×’Υ,Τ(1)|Γ=θ are those in (2.22). 2 2. If (—^i⅛ ^*^ V,z(gτ,y(t)) + Wγ(t)) is the Dirichlet operator on [0,T] there dι constants kg and k^ such that -37- (2.49) (—+ V'⅛y) + W⅜(t)) ≥ ⅛ > 0 (recall (2.30)) on Dθ(0,T) (defined , ∙1∙ dt in (2.31)) uniformly for (x,y,T,r) ∈ B(xθ , 5θ) × B(t>1 , 5θ) × [Tθ , ∞) × X yiT [O,rθ] and its Green’s matrix is given by width X = Xr, and Y = Yf,y,T , (see Heimes [1]) (2.50) Gr,y,τ(t,s) = [X(t)-(Y(t)-X(t)X~1(0)Y(0))(Y(T)-X(T)X-1(0)Y(0))~1X(T)]χ-1(s) (Y(s)-X(s)X- 1(0)Y(0))(Y(s)-X(s)X~ 1(s)Y(s))-1 if 0 ≤ s ≤ t ≤ T and r,χ>y>τrt cλ _ '-*r — (Y(t)-X(t)X~1(0)Y(0)) {I-(Y(T)-X(T)X-1(0)Y(0))-1 X(T)X~1(s)(Y(s)-X(s)X-1(0)Y(0))}(Y(s)-X(s)X-1(s)Y(s))-1 if 0 ≤ t ≤ s ≤ T. The Green’s matrix satisfies Y T7 T -U>1 It —sl (2.51) ∣Gr,y, (t,s)∣ ≤ k4 e and (2.52) ∣Gr,y,τ(t1,s) - G?’y,T(t2,s)l < k4 e 1 if 0 ≤ t, s < T uniformly for those (x,y,t,r) ,s . 3. For GX’y’T(t,s) = Gr’y’T(t,s)|r_0 we have: For each a ∈ Nθ 1' 1 2' if 0 ≤ t1 , t2 , s ≤ T -38- (2.53) Λ∣α∣ xvT Λ∣α∣ xvT — ω1∣t-s∣ ∣∣-δ Gx,y, (t,s)∣ , ∣∣^δ Gx,y, (t,s)∣ = 0(e 1 ) if 0 ≤ t, s ≤ uniformly for (x,y,T) ∈ B(xθ , 5θ) x B(b,5θ) x [Tθ , ∞). [A sketch of a proof in chapter 16.] J 4⅛ »»XV (2.52) shows the process with covariance operator (—-a-75 + V (gτ' (t)) dt2 + wF(t))"1 will have continuous sample paths (see Simon [5], Theorem 5.1) and (2.51) shows the paths won’t go far away for large T (by Donsker-Varadhan, see Stroock [1], Cor. (3.50)). §2.6. Finer asymptotics of G xvT (t,s) when χ is exponentially close to b. Finally we need a little bit finer analysis when y = y satisfying (2.54) ÿ = y(x,T) = b + 0(e — ω1 T i ) uniformly for (x,t) ∈ B(xθ , 5θ) x [Tθ , ∞) . Lemma 5. With xθ , jθ , and Tθ as in lemma 2 and y satisfying (2.54) we have >d2 1. For each a ∈ Nθ there are diagonal matrices A(α, ∙ ), B(α, ∙ ) ∈ Cβ°([0,∞),Ru ) (i.e., each derivative is uniformly bounded on R), independent of x and T, such that (2.55) -Ω(T-v) ⅛ XX’y’T(T—x)|y_y = (A(α,v) + 0(e -5i(T-v) )) e ∂y, — - and ∂' ' d χx>y>τf⅛)∣ _ ω- 1 ∂ya dtx y=y y t=T-v -39- Yx∙y'τ(T-v)∣y =y = (B(α,v) + 0(e il<T V>)) eß(T V> (2.56) = ⅛ γ*'y,τ<t>∣ y=y j ω^1 t≈=T-v uniformly for (x,T) ∈ B(xθ , δθ) × [Tθ , ∞). 2. GX’yT(t,s) = Gj,b(t,s) + Gj(t,s) + Rx,τ(t,s) where (XX,b and YX,b as in (2∙16)) (2.57) Gx,b(t,s) = Xx,b(t)(Xx,b(s))-1(Yx,b(s)-Xx,b(s)(Xx,b(0))~1 Yx,b(0)) (Yx,b(s)-Xx,b(s)(Xx,b(s))-1 Y^ x,b(s))-1 if 0 ≤ s ≤ t ≤ T and Gx,b(t,s) = (Yx,b(t)-Xx,b(t)(Xx,b(0))-1 Yx,b(0)) (Yx,b(s)-Xx,b(s)(Xx,b(s))-1 Yx,b(s))-1 ifO<t<s<T and satisfies : Vα ∈ Nθ (2.58) (Gf∙b(t,a) - e-iι∣t-sl(2Ω)-1) = 0(e~ilmin<s∙t3 e-.l*-l uniformly for (x,t,s) ∈ B(xθ , <5θ) × (0,∞) x (0,oo) (2.59) Gj(t,s) = —e xT and R ’ (t,s) satisfies Ω^2Τ 8 (2Ω)-1 ) -40- (2.60) ∫τ ∫τ ∣Rx,τ(t,s)∣ dtds = 0(e 6lT) = |RX,T(t,s)| if 0 ≤ s, t ≤ T Ο Ο uniformly for (χ,Τ) ∈ B(xθ , όθ) × [Τθ , ∞). Moreover with the same uniformity XyT (2.61) Q⅜÷--.. (T—ν,Τ—u)|y=y = {A(ei,v) - A(epu) + B(ei,u)}(2Ω)~1 e — Ω(u-v) -{B(ei,v) - B(ei,u) - B(βj,O) + A(ep0) - A(ei,u)}(2Ω)~1 e — Ω(u+v) + 0(e — ⅛l(T-u) — ω∣(u-v) )ifO≤v≤u≤T and ∂G χ,y,τ (T-v,T-u)|y=y = B(ei,v)(2Ω)"1 e —Ω(v-u) -{B(ei,v) - B(e∣,u) - B(e∣,0) + A(e∣,0) - A(βj,u)} 1 -∩(u-∣-v') —δ1(T-v) —ω1(v-u) (2Ω)-1 e l , + 0(e 1 e 1 )ifO≤u≤v≤T -41- 3. Proof of Theorem Ç Recall (1.10) ∂Ω∩ ∂Ω1 ∏ω1√ - ⅝ √) ∙*S(×) ei<a> - V) j⅞ ⅞ c1 dχ------------------w where we take W to be an open bounded set containing a neighborh∞d of b and not intersecting some neighborhood of a. Moreover with a smooth boundary 9W such that for an open neighborhood V of K = {x: p(×,a) = p(x5b) = ^}, as described in Proposition 1 and and Theorem B ΛλΛ7 ∩ 5» Ù ----Λ^V ττ — — ∕v ιx ∙∙ r∖^∙>'^j — ΓΊ TT --- ~ — and 9W∖U Ç {x: p(x,a) > (3.1) ≥ + eθ, p(x.b) + εθ} , for some εθ > 0. By Lemma 1.2 defines a smooth hypersurface and for x ∈ K, Z intersects gx transversally at x = gx (0), i.e. gx (0) is a normal of Z of x ∈ K. By hypothesis (*) we have (1.15). <9Ω∩ By (1.20) we can bound Ωθ, ∣-g-y∣ ? ∣Ω∣ ∣ and 9Ω-1 ∣^3∣ z -λ(min(p(x,a), p(x,b)) -εθX 0(e I uniformly on ∂ W, since W is compact. Thus by (3.1) we have (1.31): <9Ωf 5Ω1 X<ωi-5Γ - ¾ √> dsf×' = Z σW λ→ ∞. <9Ωr 0Ω1 - ⅝ √) ds<×> + « —λ(p(a,b) + fθ) by -42- Now we use the formulas for Ωθ, Ω∣ and their gradients is Theorem B. Since α2(λ) + β2(X') = 1, Fa(x) Fb(x), Fftj(x) and Fbyx) are 0(1) and p(x,a) = A>(× b) on Z, we get: 9Ω∩ 0Ω1 n .^-0 - ⅝ √ = A m {(⅛y - ⅛i) Z W>)f , f ta fb fl,a θp(x,a) fb<×> ⅛ ⅛ i1,b + il,bj ⅛ il,aj ib, + + 0(A 5 ]1 e-'<','x'a> + i,<x'b>> for X ∈ Z. W contains point with ρ(a,x) — p(b,x) > 0 near Z and the outward normal is p,(x,b) - √(x,a) ∣√(x,b) - p'(x,a)∣ Therefore ∂Ωθ (Ωi ∂n 9Ω 1) = A Ω, 0 <9n (d + 2) F1(x) (F0(x) d----- χ------1- 0(λ )) e — λ(p(x,a) + p(x,b)) uniformly on Z where Fθ(x) = ∣ pz(x,b) — p,(x,a)∣ fa(x) fb(x) and Fθ and F∣ are Co° on Z by Theorem B and lemma 1. -43- This proves (1.50) of Theorem C, (1.52) follows from above and (1.53) from (2.20) and (2.29). To prove (1.51) of Theorem C we note if f (t) = V"(gx(t)) f(t) on R (3.1) and ∫ then, since : (3.2) ∣f∣2 dt < ∞ x,b _ . ~ gxl [0,∞) ⅛ ∞) is a bounded solution of γ(t) = V” (gx,b(t)) γ(t) on (0,∞). Xb Recall from Theorem B that X ’ (t) is a nonsingular matrix solution of (3.2) xb -M -J V,'(b) t with X ’ (t) = (I + 0(e ) e as t→oo. Any bounded solution of (3.2) (see Coppel [1] Chapter 2, proposition 1) is of the form γ(t) = Xx,b(t) (Xx,b(0)) (other solutions grow exponentially at ∞) . Therefore f as in (3.1) is given by ⅛,∞)ω = χxb<t) (χx,b(θ))~1 f(θ) (3∙3) ∙f∣(-θo,0] (~t> = χx,a(t) (χx,a(°)^1 f(θ) ¼(0) -44- Hence if f satisfies (3.1) and h ∈ CcJ' (R, R^) then (3.4) ∫°° <(-f + V" (gx) f), h) dt = ∫°° ((f, h) + (V"(gx) f, h)) dt = — oo —∞ = (∫θ + ∫°°) = -((Xa,x(0) (Xx,a(0))^1 + Xx,b(0) (Xx,b(O))-1)f(O), h(0)) -∞ o by (3.1) and (3.3) = ((p"(x,a) + p"(x,b)) f(0), h(0)) by (2.20) and (2.19). Λ ∙Λ This shows the number of independent Lz- solution of f = V (gx) f is equal the nullity of the Hessian of p(x,a) + p(x,b). If we restrict ourselves to solutions f of (3.1) with (gx(0), f(0)) = 0 so f(0) is tangent to Z at x ∈ K and h with (gx(0), h(0)) = 0 we get by (3.4) the first part of (1.51). The second part follows from gx(t) = V,(gx(t)) and gx is in L^(R) by (2.14). -45- 4. A proof of Theorem B. §4.1. The setup. §4.2. The contribution to (1.31) and (1.32) from the complement of neighborhoods of a and b are small. §4.3. The use of the L 2 -asymptotics of the eigenfunctions. §4.4. A stationary phase theorem from Hörmander [1] and lemmas 6 and 7. §4.5. Asymptotics of terms not containing the polynomials. §4.6. The terms with the polynomials. §4.7. The Remainder. §4.8. Cancellations and taking T = constant log λ completes the proof. -46- 4. A proof of Theorem 13. §4.1. The setup. As mentioned earlier, we use (1.33), (1.20) and (1.21) to estimate the contribution in (1.31) and (1.32) from Rd∖(B(a,∣) U B(b,∣)), for δ as in Theorem A. Then we use (1.35) (resp. (1.36)) and the L^-asymptotic expansions in (*), when integrating over B(a,^) U B(b,φ in (1-31) (resp. (1.32)). In these asymptotic expansions and estimates we have “the same kind” ττ ∕ ∖ ∖ -T ∖ Λ "*^ ∖ of formulas for e (x,y) and e (x,y), and for Ωθ and Ω∣ . Therefore we only do the calculations for one of Ωθ , Ω^ , Ωθ and , Ωθ say, and we claim the others are similar §4.2. The contribution to (1.31) and (1.32) from the complement of neighborhoods of a and b1 are small. If xθ ∈ K and with δ > 0 as in Theorem A we write (4.1) Ωθ(x,λ) = e τE0(A) τ∏(Λ) λ ∫e λ (x,y) Ωθ(y,λ) dy ,E0(A) J + J + B(b,^) τH(A) λ (χ,y) Ωθ(y,λ) dy J 1∖(B(a,φ U B(b,φ)' -47- By (1.33) in Theorem A, (1.14), and the assumption (*) in §1.3 we have teq(λ) (4-2) e ^d∖(Bj(a,∣) U B(bφ) Kd∖(B(a,∣) U B(b,∣)) uniformly for x ∈ B(xθ , 0) when T, λ → ∞ and I → 0. A By (1.20) and (1.21) ∫ %(y,A) dy = Rd∖(B(a,f) U B(b,f)) ) fio(y^) dy = Rd∖B(0,R) e-λ{min(p(y,a),p(y,b)- ε} Rα∖B(0,R) for large R and for any ε > 0. If y we get B(a,φ U B(b,φ then p(y,a), p(y,b) ≥ some εθ > 0 . After taking ε = -48- &ο “” λ λ λco R \ e 2 + e 3 ) = 0(1), { Rd\(B(a,|)UB(b‡) which is all we need. Now we have, by (4.2) τ⅞(λ) τH(A) e λ (χ,y) Ωθ(y,λ) dy e λ r . , x jRd∖(B(aφuB(b,j)) (4∙3) <?2 . -λ(mιn(p(x,a),p(x,b)) + γ¾-). = θ(e ) rΓ uniformly for x ∈ B(xθ , 6) as T, λ → ∞ and y → 0 . 9 §4.3. The use of the L--asvmptotics of the eigenfunctions. If Xq ∈ K then, by Theorem A, there is a 6 > 0 such that we can τH(A) expand e (x,y) as in (1-35) for x ∈ B(xθ , 6) and y ∈ B(a,δ) U B(b,6). Now we look at ∫ e -τφ λ (x,y) Ω∩(y,A) dy, say. After change of B(b,⅛) variables we have (1.34) V,,(b) = Ωz . By assumption (*), Ωθ(x,A) = α(λ)^2m+2,a(χ,λ) + r2m+2√x'λ)) + ^λX≠2m+2,√x^ + r2m+2,√x'λ)) -49- with ^2m+2 g I z p1 √λ2(χ-c)) , p2m+2√λ (χ-0)∖ c(x,λ) = λ2 ⅞oc(λ2(x-c))(l + —≈---- j---------- + ∙∙∙ + ) -(m+∣) and llr2m+2( ' ’ ^)∣∣2 = θ^Α ) ' Here φc is the Gaussian groundstate of h(c) = + i(Vπ(c)x,x) on L2(Rd), l,c ’ "∙ ’ p2m + 2 c are Pu*yuυιl^*ials5 Pj c is odd, α2 + /?2 = 1 and α(λ)∕3(λ) ≥c^λ m>0asλ→∞ Now we write this as (4.4) d 1 Ωθ(x,λ) = ∕J(λ){λ4^b(λ2(x-b)) (1 + pi,√λ 2(x~b))λ2 + p2,b<λ2(χ-b))λ 1 + f2,b(x'λ))} 1 —- —i/O ∖ where y>b(x) = (det Ω)4 π 4 e ’ “·* ∣∣⅛b( ∙ . λ)∣∣l2(B(b>i)) = »(A '2) as Λ ∞ where we used the L2(B(b,6))-norm of λ4 ∕J(λ) φb(λ5(x~b)) Pi,b^A (x~b)) _I 0(«Λ,Λ *) = _3 O(^Λ)Λ-5, fo, i = 3.......2n, + 2, .g λ5 τhe L2(B(M)).nom of -50- ßW r2m+2,t∕x'λ) is θ(^(λ)λ ~(m+∣) 2 ) (^(λ)λ 2) = _3 is exponentially small, so it is 0(∕3(λ) λ <*(λ) r2m+2,a(x'λ) ιs θ(α(λ) λ -(m+0) 2 ) α(λ) < 1 and λ~m ≤ α(λ) β(λ) 9 ). = θ(λ ≤ β(X) trace(Vπ(b))2 and hence by (1.25) e eo 2^ and A of ⅜sa(λ2(x-b)) ' O Finally, the L (B(b,6))-norm of -(m+s) 2 ) = θ(∕3(λ)λ where > 0. (x,y) — π _d d 1 2 λ2(det Ω)2 aθ∖x,y) τH(Λ) By (*) we know -λA(x,y,T) -Teθ I 1 (x,y) + 0(Ij) τ∙4 + 0⅛ m2 exp(-A -⅛)) c, O + a ⅛±2 Therefore by (4.4) with Pθ ∣j(y) = 1, -T (4∙5) ∫ e B(b⅛ 7Γ H(λ) (χ,y) %(y,λ) dy -⅛ -Te0 ÿ I ∕ 2 ∕I0 i(χ,T,λ) Ili(x,T,λ)> 4 e 0 β(λ) λ 4 (det Ω)4 ∑ -.------ + ----- ■.— ∖i=θV λ⅛ λ^1+^ ' + Rj(x,T,λ) + R,2(χ5T,λ) where -s 2), since -51- (4∙6) I0j(x,T,Λ) = J e-A(A(X-y'T)+^«(y-b).(>"-b)>) aJ(x,y) B(b,∣) pi,t>(λ (y-b)) dy for i ∈ {0,l,2} (4.7) Ili(x,T,A)= ∫ e B(b,⅛ --A(A(x,y,T)+I<i2(y _b),(y-b)>) 1 aJ(χ,y) aj,(x,y) Pi b(λ2(y-b)) dy (4.8) R1(x,T,λ) = β{X) ∫ e B(b‡ for i e {0,l>2} -τS(λ) λ (χ,y) f2,b<y’A) dy and (4.9) ~ — Te f λ R2(x,T,λ) = λ4 e θ ∕J(λ)Jθ(g) + 0 4- exp — λ- £ I0,j(x’T’A)> i=0 For a fixed (x,T) we integrate over y in B(b,⅛) with help from: §4.4. A stationary phase theorem from Hörmander [1] and lemmas 6 and We will frequently quote the following; Theorem (Hörmander [1], Theorem 7.7.5). Let K C R be a compact set, X an open neighborhood of K and k a positive integer. If u ∈ Cθ^(K), f ∈ Cθk^*^l(χ) and Imf ≥ 0 in X, Imf(xθ) = 0, f,(xθ) = 0, del f"(xθ) ≠ 0, i, ≠ 0 in K∖{xθ} then (4.10) ∣∫u(x)e* dx — e θ (det(^f*,(xθ)∕2πi)) LjU∣ ω J ≤ Cu~k 53 sup∣Dau∣, ∣o∣≤2k (4.11) u > 0. Here C is bounded when f stays in a bounded set in C 3k+l (X) and ∣x-xθ∣∕∣f,(x)∣ h as a uniform bound. With Sxθ(x) = f(×) - f(×0) - <f"(xθ)(χ-χθ^ (x ~ x0>V2 which vanishes of third order at xθ we have L.u= ∑ j v-p=j ∑ i"j 2-v(f,,(xθ)-1D,D)v(gχ u)(xθ)∕μ! v! . 2v≥3μ 0 This is a differential operator of order 2j acting on u at xθ . rational homogeneous functions of degree —j in f,,(xθ), The coefficients are ..., ∖xq) with denominator (det f,,(xθ)) . In every term the total number of derivatives of u and of fi, is at most 2j. Remark. Here i = — 1 and the integration in (4.10) is over K. We will need a continuation of Theorem A: -53- Lemma 6. In part 2 of Theorem A we have: For each (x,T) G B(xθ,6) x [Tθ , ∞) the function (4.12) B(b,6) 3 y ∣→ A(x,y,T) + ^(Ω(y-b)(y-b)) [y ∣→ A(x,y,T)] attains its absolute minimum value at a unique point y = y(x,T) [resp. y =y(x,T)] satisfying (4.13) y — b = 0(e Wl^) [resp. y — b = 0(e Wl^)] uniformly for (x,T) ∈ B(xθ , <5) × [Τθ , oo). (Proof in Chapter 16.) For (x,T) ∈ B(xq , é) x [Tθ , ∞) we write (here i = 4 — 1) (4.14) A(x,y,T) + A(Ω(y-b), (y-b)) =: A(x,y,T) + i(Ω(y-b), (ÿ-b)) - i fχ,τ(y) and (4.15) A(x,y,T) =: A(x,7,T) - i f χ,τ(y) rw which defines fχ rp and fχφon B(b,<5). Then lm fx,T(y) ≥ θ, lm fx,τ(y) = 0 only if y = y(x,T) and fi X, <τ1 ,(y) = 0, i.e., y(x,T) is the only point in B(b,<5) __ ∕×√ where fχ y is real and stationary and similar y for f χ ιp ∙ To use Hörmander’s stationary phase theorem with uniformity (see (4.11)) for (x,T) ∈ B(xθ , 5) x [Tq , ∞) we state -54- Lemma 7. In Theorem A.2 with same notation as there and y = y(x,T) as in lemma 6 there exist positive constants cθ β for β ∈ Nθ with |/?| ≤ 2, c^ β for β ∈ Nθ for ∣∕3∣ ≥ 3, cs , Cgj for j ∈ {l,...,d}, c7 and C00(B(xθ , 6)) - functions a∙^(x) and a1j(x) for j ∈ {l,...,d} such that uniformly for (x,y,T) ∈ B(xθ , δ) × B(b,6θ) x [Tθ , ∞) and for (x,T) ∈ B(xθ , 6) x [Tθ , ∞) when y — ÿ, we have 1. For each a ∈ Nθ -r ao (χ,y) = θ(1) (4∙l6) and for each β ∈ Nθ with ∣β∣ < 2 (4.17) J/?l T aθ (x,y,T)|y_y = cθ β aθ(x) + 0(e -<5iτ 1 ) ∂y where cθ θ = 1. 2. For each a ∈ Nθ (4.18) Jα∣ ,, A(x,y,T) = 0(1) and A"y(x,y,T) = = Ω + 0(e (4.19) and 1 + ,y—b∣) > constant > 0, Jβ∣ — 61T ⅛ A(x,y,T)∣y=y = c4 β + 0(e 1 ) if ∖β∖ > 3 ∂y -55- A(x,y,T) = p(x,b) + 0(e 2“ιΤ) = A(x,y,T). (4.20) 3. For each a ∈ Nθ (4.21) ∂l 1 xτ' ar (×,y) ∂ya 11 (4.22) O rp aT(x,y,T)|y=y = ⅛ T + aχ(x) + 0(Te^ 1 ). 0(T) and 4. For each (αj) ∈ Nθ × {l,...,d} ÛII (4.23) Γp afj(x,y) = 0(T) and -6 T aT,(x,y) = Cθj T + a1j(x) + 0(Te 1 ) (4.24) 5. With fχ ψ(y) and f χ ψ(y) defined in (4.14) and (4.15) we have -⅛—^-L < c7 < ∞ and Jr—< cτ (recall (4.11)). ∣f,x,τ(y)l - 7 ∣fx,τ(y)∣ ^ 7 (4.25) (A proof in Chapter 11.) Now we take X = B(b,δ) and K = B(b,^) in Hömander’s stationary phase theorem. -56- §4.5. Asymptotics of terms not containing the polynomials. Recall from (4.14) we defined for each (x,T) ∈ B(xθ , δ) x [Tθ , ∞) fx,Ι,(y) = *(A(x,y>T) + i(Ω(y-b), (y-b)) - (A(x,y,T) + ⅜(Ω(7-b),(7- b)))) for y ∈ B(b,δ), where y = y(x,T) is the unique minimum point of y ∙→ A(x,y,T) + ^(Ω(y-b), (y —b)) and is the only point where fχ ψ is real and stationary. Now using the method of stationary phase we get τ . m ., -λ(Ax,y,T)+i<Ω(y-b),(y-b)) iλfχ τ(y) r 1o,(∕x,τ,λ)= ∙f ∣y-b∣≤∣ e -λ(A(x,y,T)+i<Ω(7-b),(y-b))) = e [λfχ'τ(7)] (det (LθaJ(x,y))(y) + ao (χ,y) dy e (L1 aj(x,y))(7) (∣ ∑ λ ) 27ri (Lk_ 1 ao(χ>y))(y) √k→) suP ∣^aJ(χ,y)l ) ∣αf∣≤2k (x,y,T)∈B(x0,⅜)xB(b,∣)×[T0,∞) + 0 λk for any integer k ≥ 1 and we take (4.26) rd+4 k ∈ l 2 1 2 d+4 2- + 1)∙ By (4.25), ∣y-y ∕∣f^τ(y)∣ ≤ c7 < 00 + -57- nl“l and (4.28) —~ A(x,y,T) = 0(1) uniformly for ∣α∣ ≤ 3k + 1 ∂y and (x,y,T) ∈ B(xθ , 6) × B(M) × [Tθ , ∞) and thus the 0( ∙ ) is uniform for (x,T) ∈ B(xθ , × [Tθ , oo). By (4.19) -∣ λf"~fvΙ ^x, ι v∙, , 1 Kω⅛FJ) =( ∖ tc∖r∖ det t λ∕ _ -⅛1J τ ∖ 2π d ) = λ (det Ω) _1 d — rp 2 jr^(l + O(e ^ )) uniformly for (x,T) ∈ B(x, 6) x [Tθ , ∞). By (4.17) Lθ(aJ(x,y))(y) = aθ(x,y) = aθ(x)(l+ 0(e -61T 1 )), rp rp (L∣aθ (x,y))(y ) is a differential operator of order two acting on aθ (x, ∙ ) at y with coefficients that are rational homogeneous functions of degree —1 in f^ψ(y) — 2Ω + θ(e"^δiτ), ¾(7) = A^(xy,T)|y=y and f^(y) = A⅛>yy(x,y,T)ly=y ∙ By (4.17) ∣a∣ aθ1(x,y)∣y_y = Cθ β aθ(x) where Cθ β are constants and aθ(x) is Cσo on B(xθ,6) by Theorem A and by (4.19) Jβ∖ A(x,y,T))y_y = C4^ + 0(e — 61T ) where c4 is -58- T — —ό1Τ constant. Hence L∣(aθ (x, ∙ )(y) = aθ ∣(x) + 0(e 1 ) uniformly for (x,T) ∈ B(xθ > 6) × [Tθ , ∞) where aθ ^(x) is in Cσo(B(xθ , 6)). Since we have uniform bound for any fixed number of derivatives of aθT (x, ∙ ) and A(x, ∙ ,T) in (4.16) and (4.18) and the form of each Lj , j = 2, ..., k where k ∈ [—, (4.27) -O + 1) we have, by above Iθ,θ(x,T,λ) = λ 2(det Ω) 6 T d9 ( —*1T 1 (x)(l-f~θ(θ ) 1 Ί 2 7r {aθ(x)(l+0(e 1 )) + -½----------------------- + "*+θ^-(tι)V λ 1 9 .(<L+4) ) + 0(λ d λ5(det Ω) 2 ) > exp(-λ(A(x,y, T) + ∣(Ω(y-b),(y-b)))) = Id —2w1T 2 τr2 aθ(x)(l + 0(e~5lT)) e-λ(p(χJ>) + θ(e )) ; ⅞ βx) 1 d + + 0φ) uniformly for (x,T) ∈ B(xθ , 5) × [Tθ , ∞) where aθ is as in Theorem A and a0,1 = aO 1 -g⅛- ∈ Cσo(B(xθ , 6)). Above we used aθ(x) > const. > 0 by Theorem A, A(x,y,T) = p(x,b) + 0(e -2ω1T -w-∣T ) by (4.20) and y = y(x,T) = b + 0(e 1 ) by (4.13). estimates being uniform in those x and T’s. All -59- In a similar way using also (4.21): For each a ∈ Nθ , a∣^(x,y)∣ — θ(T) uniformly for (x,y,T) ∈ B(xθ , 6) × B(b,6) × [Tθ , oo) and (4.22) a^(x,y) = c5 ' T ÷ a∣(x) + 0(Te - 61T ) uniformly, we get by estimating (LjaJ(x,y ) a^(x,y))(y) - 0(T) for j = 1, -, k - 1 and ( ∑ sup ∣0y(aT(x,y) aT(x,y))∣) ≈ 0(T) ∣o∣≤2k (x,T,y) uniformly for (x,T) ∈ B(xθ , 6) x [Tθ , ∞), with k as in (4.26) above. — λ(A(x,y,T)+i(Ω(y-b),(y-b))) Ilθ(x,y,T)= (4.28) ∫ ∣y-b∣≤∣ θ (x,y) aj (x,y) dy = = λ 2(det Ω) 1 d -i1τ. 2 2 _2 π2 aθ(x)(l + 0(e )) (⅛ ∙ τ + alι0(x) + 0(Te-iiτ))e-λ<',(×'b)+°" —2ω1T. ”(1 + 0(I)) uniformly for (x,T) ∈ B(xθ , S') × [Tθ , ∞). §4.6. The terms with the polynomials. If hx,T<y) e c°°(B(M)) fθr (x,T) ∈ B(xθ , 6) × [Tθ , ∞] and Pα(y-b) = (y-b)α where a = (α1 ,..., αrf) ∈ Nθ is multi index with ∣α∣ = ∑ ∣αi ∣ = n then if we take k ∈ [n+θ^t^θ , π+θ+^) we have . 1 1 2 2 -60- (4.29) J λ2 ∫ Ptt(y~b) hχ τ(y) e ∣y-b∣<∣ n = λ2[λ d 4(det Ω) -λ(A(x,y,T)+i(Ω(y-b),(y-b))) 2 dy Id —δηT 2 π2(l + 0(e 1 ))(Lθ(hχτ Pα))(y) , L1() ι , Lk-i(hx,TP«)(y\ 1 + λ + ∙∙∙ + .(k-l) } + Λ ( ∑ sup ∣9yh τ(y)∣) 0 V ∣,∣<2t (x,y.T)----- l eip(_A(Xx>b) + uniformly for (x,T) ∈ B(xθ , 6) × [Tθ , ∞) and for λ > 0 (as in the proof of (4.27), this uniformity follows from (4.18) and (4.25)). ___ ___ Since L· is a differential operator of order 2j and y = y(x,T) = b + 0(e — (λ7,∣ rp ) uniformly for (x,T) ∈ B(xθ , 6) × [Tθ , ∞) we have Lj(hχ,τ(y) Pα(y))(7) = ~ — w1(n-2∙)T 0(e ) for integers j ≤ ~ and Lj(hx,T(y) pα(y))(7) = 0(l) fθr J ≥ ∣ where ~ is to remind us on the dependence of h X« rAr(y) and its y-derivatives of order <2j. If n is odd we write (4.29) as -βί­ (4.30) f ∫ ο λ2 -λ(A(x,y,T) + i(Ω(y-b),(y-b))) Pα(y~b) hχ j√y) e 2 dy ∣y-b∣≤∣ = λ2[λ {(0(e 2 π2(l + 0(e 6lT)) 2(det Ω) -ω,T -ω1nT ~ -ω1(n-2)T 1 ) + Ö(- λ 2 ljn+l(hx,τ(y)pa(y)×y ) + 0(-⅛ + ∙∙∙+0(- + vv 2 rn+3λ Λ 2 < ∕ (n+d+4) )} 1 —2Uf T + °( (n-∣1d+6)^ eχp(-λ(∕,(x'b) + θ(e 1 ))) = λ 2(det Ω) 2 π2(l + 0(e 6ΐΤ)) uniformly for (χ,Τ,λ) ∈ B(xθ , ό) × [Τθ , ∞) × (0,oo) when e constant and (4.31) 0( ∙ ) = 0(∙ ∑ sup ∣∕J∣≤n + d + 8 (x,y,T) |ôy h τ(y)∣). t — ω1T ton- 1 (Ln+l(hx,T(y)P«(y))(y) _wT 1 v , , . ,∩, -2ω1Τη <J=--------- ----------------- + o(⅛ + 0(e 1 λ2)) e~λ(p(χ,b)+O(e )) λ sume -62- Similar if ∣α∣ = n is even we have for Pα(y-b) (4.32) λ2 ∫ Pα(y - b) hχ τ(y) e (y-b)α À(A(X,y,T>+3<n(y-b),(y--b)>) dy I∣y-b∣≤∣ » ι -, Λ - λ 2(det Ω) + 0(i) + 0(e 1 d -⅛,T 2 τr2(l + 0(e 1 ))(Ln(hχ,τ(y)pα(y))(y) WlT λ2)) uniformly for (x,T,Λ) ∈ B(xθ , δ) X [Tθ , ∞) X (0,oo) as long as e ωl m λ I 2 ≤ some constant and with 0( ∙ ) as in (4.31). Write each m. Pi,b(y) = Σ ( Σ i0(y-b)α + ∑ ⅛∖y-b)*) ’ k=l 'lα∣=2k ∣α∣=2k + l 7 m∙ = Σ ( Σ co' pα(y-b) + ∑ c√ Pα(y-b)) k=l ja∣=2k ∣α∣=2k+l 7 then with hχ ψ(y) = aθ,(x,y), and hence 0( ■ ) = 0( ∙ ) by (4.16), we have by (4.30) and (4.32) -63- — λ(A(x,y,T) + ^(Ω(y-b),(y-b))) (4.33) Iθi(x,T,λ)= ∫ ∣y-b∣≤< 1 aθ (x,y) Pijb(A¾-b)) dy = = ( m« Σ ( i°λ 2(det Ω) Σ 2 τr2(l + 0(e 6lT)) ∖k=l v∣α∣=2k (Lk(aJ(x,y) Pα(y))(y) + O(i) + 0(e + mi ∑ _ ω rp A 1 λ2}) ∕∙∖ _d —I d _â {G« A 2(det Ω) 2 τr2(l + 0(e 1)) ∣α∣=2k-± (⅛(,0⅝,y)P,t(y))(y) + ο(Λ) + o(e_„lT λ5 Λ exp(-λ(p(x,b) + 0(e , —2ωη Τ ))) with uniformity as in (4.30). Note the first contribution is from the even part and the second is from the odd part of the polynomial. rr -T If 2i = ∣α∣ then L∣(aθ(x, ∙ ) Pα( ∙ ))(y) = constant aθ(x) + 0(e ) uniformly for those (x,T),s, since L∣ is a differential operator of order 2i acting on aθ (x, ∙ ) Pα( ∙ ) at y = y, with coefficients, being rational functions of y-derivatives -64- of A(x,y,T) or order 2,3, ..., 2i + 2 at y = y, of the form (constant + 0(e Unless all derivatives act on Pα we get 0(e — ω∙< T -61T i )). _ ) = 0(∣y — b∣). When all derivatives act on Pa , we get τ -<51T -i1T aθ (x,y )(constant + 0(e 1 )) = (aθ(x) + 0(e )) (constant + 0(e -01T -61T 1 )) = constant (aθ(x) + 0(e 1 )). Similarly, if 2i — 1 = ∣a∣ rn _ — -δ U-∣1T ± (Li(aθ (x,y) Pα(y)) (y) = constant aθ(x) + 0(e i ), since by (4.17) J0∣ τ a^(x,y)|y_y = cθ aθ(x) + 0(e -61T ) uniformly |/?| ≤ 2 ∂yp and we get 0(e — Wι T ) unless one derivative acts on aθ and differentiation of order 2i — 1 on Pa . In that case we get Z) τ —*ιτ aθ(x,y)∣y~y (constant + 0(e )) = constant δ1T (aθ(x) + 0(e 1 ). Recall P∙^ jj is odd and we get by above from (4.33) -65- (4.34) m, ,(0jλ∖ ⅛,ι=.∑ ∑ (⅛ k=l, ∣α∣=2k-l ((ck a0(x)÷0(e *iτ)) exp(-λ(p(x,b) + O(e / + θ(/ -2ω1T 1 )) = λ -<51T 1 )) (aθ 1(x) + O(e u>∙l_______________________ 1 λ5 exp(-λ(p(x,b)+O(e — 2wι Τ ωιτ ⅛ 1 λ2))} 2(det Ω) _3 T, nf∖ 1 d -i1T 2π2(l + 0(e 1 )) 2z(detΩ) 2 τ 1 i"l1 \Ζ _ ; τ, nz_ 2∖ -⅛ ä -61T 2 π (1 + O(e 1 )) ))) _ —Τ uniformly for (χ,Τ) ∈ B(xθ , 6) × [Τθ , ∞) when λx e 1 ≤ some constant where aθ j(x) = constant aθ(x) ∈ C0°(B(xq , 6)). Similarly (4.35) Iθ2(x,T,λ) = λ _d _1 d _ β rp 2(det Ω) 2 π2(l + 0(e 1 )) τ {aθ,2(x) + 0(e 1 ) + 0(λ + 0(e —2w∣T ))) ΓTΛ ) + 0(e Wl λ2 )} exp ( — λ(ρ(x,b) -66- with same uniformity as in (4.34) where aθ 2(x) = const, aθ(x) ∈ Coo(B(xθ , 6) Adding (4.27), (4.34) and (4.35) gives (4.36) 2 I0i(x,T,A) -ä _1 d _,T £ -.------ = λ 2(det Ω) 2 π2(l + 0(e 1 )) i=0 λ∣ exp(-λ(p(x,b)+0(e 1 T )) an 1 (x) 1 1 [aθ(x)(l + -⅛- + 0(i)) + -⅛ λ ∕(a0,ι(×)+θ(e i1T 1 )) 1 λ'2 + ⅜(a0,2(x) + θ(e = A 2 (det Ω) 2) + 0(e WlT A2)) I — C√-ι mi J. ∏ -i1T 1 ) + θ(λ + 0(e 1 λ2))] — d9 — o-1 ∣T X >*∩ I (x) — -<5 2 τr2 (1 + 0(e 1 )) aθ(x)(l + ---- -1- 0(λ „ 1 —9 exp(-λ(p(x,b) + 0(e CCICM — <5 + 0(A -2ω1 T i ))) uniformly for (x,T,λ) ∈ B(xθ , 5) × [Tθ , ∞) × (0,∞) when e ai θ(χ) ∈ C∞(B(xθ , 5)). — ω-∣ T 1 λ2 < a constant and where -67- T T Now take hχ,'p(y) = aθ (x,y) aj (x.y ). By (4.21) ^-q af(x,y) = 0(T) uniformly and so 0( ∙ ) in (4.31) is 0(T) From (4.3) and (4.31) we get x (4.37) l x Ili(x,T,A)= r -λ(A(x,y,T)+∣<Ω(y-b),(y-b)>) τ Je 2 aj(x,y) ∣y-b∣≤f 1 aT(x,y) Pi,b(A2(y-b)) dy m∣ _d _1 d = ( ∑ ( ∑ (⅛ A 2 (det Ω) 2 π2(l + 0(e^6,τ)) 'k=l a:|a|=2k (Lk(aJ(x, y)a[ (x,y)Po(A)(y) + 0(J) + 0(Te + Σ (c α λ ∣α∣=2k-l _u τ A 1 λ2)) _d _1 <1 __β rp 2(det Ω) 2 τr2(l + 0(e~ 1 )) (Lk(ao (χ,y )af(χ,y )Po(y))(y) 1 A5 rp —ω1T ο -2ω1T + 0(⅛) + 0(Te 1 λ2)) exp(-λ(p(x,b) + 0(e 1 ))) λ5 for (x,T,λ) ∈ B(xθ , δ) × [Tq , ∞) × (0,∞) with uniformity when A constant. IΛ e -- ζtJ Jτ ≤ -68- When i = 1 we have the contribution from the even part in (4.37) vanishing since P∣,jj is odd and hence _Q (4.38) I1 1(x,T,λ) = λ __ω rp rp 2(1 + 0(e 1 ))(0(⅞ + 0(Te 1 1 λ2)) λ^2 ’ exp(-λ(p(x,b) + 0(e —2ω,T 1 ))) with uniformity as in (4.36). Generally we have __ UJ Ili(x,T,λ) = λ exp(-λ(p(x,b) + 0(e — 2cjη T __ β rp 2(1 + 0(e 1 ))(0(T) + 0(Te __ rp JL 1 1 λ2)) 1 ∙ 0 )) with the usual uniformity, which we use with ι = together with (4.37) and (4.38) and get (4.39) 2 (l-¼-) _— __— — __$ T ∑ L .(x,T,Λ)∕λ 2 = λ 2(det Ω) 2 π2 aθ(x)(l + 0(e 1 )) i=0 1,1 [l(c5 . T + al θ(x) + 0(Te β rp 1 ))(1 + 0(J)) ι τ —w-∣ T q.. + ⅜(θ(π) + θ(τe λ )) λ5 λ'2 1 —ω1T i — 2ω1T + p(0(T) + 0(Te λ2))]exp(-λ(p(x,b) + 0(e 1 ))) = λ _d _1 d _ β rp 2 (det Ω) 2 π2 aθ(x) (1 + 0(e 1 )) -69- -6iT c5τ+al,θ(x)+θ(τe exp(-λ(p(x,b) + O(e )) Te-2ωιτ + o(⅛) + o(V )] — 2ω-∣ T i ))) uniformly for (x,T,λ) ∈ B(xθ , 6) ∈ B(x0 , δ) × [Tθ , ∞) × (0,∞) with e 4 T λ2 < constant where a∣ θ ∈ Coo(B(xq , 6)). Now we finally worry about §4.7. The Remainder. By (4.36) _i 2 (x,T,λ)∕λ 2 Σ I 0,i i=0 exp(-λ(p(x,b) + 0(e —2u>∙t T ))) with the usual uniformity and by (4.9) (4.40) — — Te 4 2 2 R2(x,T,A) = λ 4 e θ ∕J(λ)(O¾) + 0(¾- exp(-λ c1 -⅛) À T? 2 ( Σ ⅛ i(x,T,λ)∕λ2) i=l υ,1 = 0(λ4 ∕7(λ)e^τeθ(0(^) + (^ exp(-λ cχ ^))) exp(-λ(p(x,b) + 0(e -2ω1T 1 )))). -70- By (4.8) R1(x,T,λ) = ∕3(λ) ∫ e τ∏(Λ) λ (x,y) ~2 b(y,λ) dy B(b,∣) and by (1.35) e l'5W d _Te λ (x,y) = 0(λ2 e θ -λA(x,y,T) ) uniformly for (x,y,T) ∈ B(xθ , δ) × B(b,0) × [Tθ , ∞) when < λ and T, λ → oo. We use the Cauchy-Schwartz inequality (4.41) τH(Λ) R1(x,T,λ) ≤ ∕J(λ)∣∣e λ (x, ∙ )∣∣2 |[?2?b( ■ ,λ)∣∣2 with II ∙ ∣∣2 = the L2(B(b,2))-norm. _3 By (4.4) ∣∣r 2,b( ∙ ,λ)∣∣2 = 0(λ 2). ifχ,τ(y)∙ Now write as in (4.15) A(x,y,T) = A(x,y,T) — -71- 'γh(a) d t ∣∣e^τT- (x, . )∣∣2 = 0p e-τe∙>( J λ,.z mχ 1 e-2iA'-y-T>dy)5) B(b⅛ = θ(λ2 e^τ'0 e--'λ'×S.'1')(j .2i"×∙τ,yidy)2) = J .^τ'° .- <*∙7∙t> 0(Λ^h aa uniformly for (x,T) ∈ B(xθ , όθ) × [Tθ , ∞), by (4.25), (4.18), (4.19) Hörmander’s stationary phase theorem. Now (4.20) of lemma 7 and (4.41) imply (4.42) R1(x,T,λ) = 0(e d —*Be∩ a U λ4 A 3 f) —2w1 T 2 exp(-λ(p(x,b) + 0(e 1 ))) uniformly for (x,T,λ) ∈ B(xθ , δ) × [Tθ , ∞) × (0,∞) when T4 ≤ A. §4.8. Cancellations and taking T = constant log A completes the proof. From (4.5), (4.36), (4.39), (4.40), and (4.42) we have (4-42) ∫ e τH(A) d λ (x,y) Ωθ(y,λ) dy = λ4 ^(λ) fb(x) e ∣y-b∣<∣ — 2ω,T — <5ηT — Te∩ (l + 0(λe 2 1 ))(l+0(e 1 )) e 0 λp(x,b) and (c5'τ+fl b(×) + θ(τe τ4 [1 + -2--------½°-- ------------------- + 0(2ς) + O(A 2) ,o c + 0(-ζ- exp(-λ —τ))] uniformly for x ∈ B(x∩ , δ) as T, λ -→ ∞ and ≤ λ ≤ e 1 , where f^(x) = » ^^det^J V,,(b) aθ(x) and aθ as in Theorem A. Here f1 ∣j(x) = a1 θ(x) + aθ 1(x) ∈ Coo(B(xθ , 0)) with a1 θ(x) from (4.38) and aθ 1 from (4.36), eθ = tracejv"(b) -------- 2-------- trace ω ---- 2---- , _ = .J smallest eigenvalue of V,,(b) and 0 < 6∣ ≤ ωl = c^(b), , _ - ,p ωιfb) _ — = c^(b) > 0 are constants and with ∕J(λ) as in assumption (*). We get similar contribution from {y : |y—a∣ ≤ and a small one from Rt^∖(B(a,^) U B(b,∣)) by (4.3), which adds up to (1) T(eθ + -TΓ+ θ<⅛)) (4.43) Ωθ(x,A) = e (1 + 0(λe λ *i « — Te -2ω1(a)T —6,(a)T s lk ))(1 + 0(e ll > )) (c5(a)T+f1 a(x) + 0(Te ⅛ , , x °{A1 „(A) f,(x) .-λi,'x∙i> 6l(a)T)) [1 + —------------------- r----------------------- + -73- + λ4 β(∖) fb(x) e Ä/,(X’b) (1+ O(λe ^^»(l+OPe 0l(b)T)) (c5(b)T+f1 b(x) + 0(Te 0lT)) -3 4 [1 + -2----------- ù°-_-------------------- + 0(λ 2) + O(⅞) λ /Τ2 ∕ C4(k) -j- O(-^- exp(-À ^ ))jj unιiormiy lor x ∈ n(xθ , ∂) when i . with Take < λ ≤ min(e ωι (a)τ u1(b)T e ) <5θ = min{6-^(a), 0^(a),<5^(b),ω∣(a),ω^(b)} > 0 (l) τ(e0+-⅜- θ⅛)) and write e Te θ(l + ejjυ ∙ J + 0(⅛) as T, λ λ ∞ , y → 0 tc∣ 6∩T —2ωη(c)T —0∩T ο rr If λ≤ e υ then (1 + 0(λe i )) = (1 + 0(e u ) and if T8 ≤ λ then £ λ I rrt2 0(£§) and e -λSι^ rT14 = 0(e ) and we write (4.43) as -74- (4.44) Ωθ(x,λ) = λ4{o(A) fa(x) e Ap(x’b) (1 (c1(a) + e∞)T + flb(x) + 0(Te + 0(e -0∩τ 0 )) -6θτ υ )) 1 + _3 O(λ 2) nT + λ4 ∕J(λ) fb(x) e MX’b) (1+O(e -5 "θ ’)) (c1(b) + e<lj)T + f1 √x) + 0(Te 1 + —6 T θ )) +O(Λ -2 2) 6 T uniformly for x ∈ B(xθ , 6) when T, A → ∞ and Tθ ≤ λ ≤ e θ To show c∣(a) + eθl) = 0 take x ∈ B(xθ , S') with p(x,a) < p(x,b). Put T = Ä log λ in the R.H.S. of (4.44) and then T = 00 A log A in the R.H.S. and subtract. 00 That gives 0 = Ωθ(x,λ) - Ωθ(x,λ) = λ4 α(λ) fa(x) e_AXx’a) (1 + 0(λ^4)) Z (cι(a) + eθυ) (^- log A - ∙^- log A) + 0(log A) _3 . ( ----------------------- Q-------- r-Q---------------------- — + 0(λ 2) ∖ -75- the prefactors are not zero and we get (c1(a) + eθ ) = 0(λ 2) as λ ∞ which implies c∣(a) + eθ^ = 0. By taking x ∈ B(xθ , δ) with p(x,b) < p(x,a) we get Cj(b) + eθl) = 0. Now we go back to (4.44) with T = log λ and use c^(a) + eθ(1) = 0 = Cj(b) + eθl) and we get (4.45) Ωθ(x,λ) = λ4α(λ) fa(x) e -λp(χ,a)α + θfλ-4n ι + (¾,a(x)÷ 0(A log Λ)) + θ(λ-2} + 0(λ)fb(x)e Ap(x’b) (1 + 0(λ~4)) 1 + (f1 b(x) + 0(λ-4 log λ)) ---------------------------- + 0(λ * 2) d 3 = λ4{α(λ) fa(x) e -λp(x,a) (1 + ⅛x) + θ(λ-2)} + β(λ) fb(x) e Mx’b) (1 + ⅛≤ + 0(λ uniformly for x ∈ B(xθ , 6) as λ ∞. 2))} -76- 5. The Feynman-Kac Formula and Large Deviations. §5.1. A use of the Cameron-Martin formula §5.2. Large deviations, Varadhan’s theorem and Schilder’s theorem. -77- 5. The Fevnman-Kac formula and large deviations. §5.1 A use of the Cameron-Martin formula. References for this section are Simon [2] and [5], poor Barry! Given x,y εR^ and t > 0, let Cx,y([0,t]) be the Banach space of continuous functions Z:[o,t] → R*^ with Z(0) = x and Z(t) = y and with the supremum norm d "z∣∣L∞[o,t] = O<us≤t (i∑1 2i'8>) <5∙1> Let Pχ t be the Gaussian measure on Cxy([o,t]) with mean m(s) = Eχ,yjt(Z(s))= (1— s)x + syifθ≤s<t and covariance. (5.2) Eχ y.t((Zi(s) - mi(s) (Zj(u) - iΩj(u)) = s(l - u∕t) if 0 ≤ s≤ u ≤ t and for ij ∈ {1, ,d}. When x= 0 = y, {Z,(s)}θ<s<t is called the Brownian bridge or tied down Brownian motion. We will denote Eθθ.t( ∙ ) by E*( ∙ ). , Z The conditional Wiener measure is given by ‰,y√ ∙ ) = <27rt) 2 exp(-⅛4 px,y;t< ∙ ) If V ≥ 0 and continuous and H = ~~ Δ + V the Feynman-Kac formula says (5.3) as e ___ ‡TT t (x,y) = ∫exp(-∫ V(ω(s)) ds) dμ∩ f (ω) which can be written ∩ u,x,y ,r -78- (5.4) e-t∏(x,y) = (2πt) _u 2 2 exp(-∣x 2^-) E∣(exp(-∫θ V((l-∣) X + ∣y + <t Z(∣)) ds)). In the double well case, we are staring at, we take t = T/λ and H = H(λ) and we have (5.5) e τH(A) T λ (x,y) = ∫ exp(-λ2∫θ V(ω(s)) ds) dμθ,χ,y.τ∕λ (ω) d 2 J = (o⅛,)2 eχpf- λ⅛τy' ) e7∙ (eχp(-j v((!-⅛)x + ⅛y + Jτ∕λ Z(⅛)) ds)) . Writing formally ∫ dμθχy.t (ω) = “J” d0°ω exp(-± ∫θ û2(s) ds) 6(ω(0)-x) 6(ω(t)-y) gives e tH^\x,y) = “J” do° ω0(ω(0)-x) 0(ω(t)-y) eχp(-⅛ ∫ ω2(s) ds -λ2 ∫ V(ω(s)) ds). 2 0 0 rΓ With t = y and γ(s) -- ω(s∕λ) we have e τH(A) λ (x,y) = “f d∞ 76(γ(0)-x) 6(γ(T)-y) exp(-λ∫τ (U2 + ν(γ)) ds) 0 2 TH(A) which reminds on (1.17) ⅛----- ------ ⅛ = -A(χ,y,T) u.o.c. in τ Rd X Rd X 0,∞), where A(x,y,T) = inf {∫ (1 γ2 + V(γ))dt : γ(0) = x, γ(T) = y}. (1.17) was proven using large deviations that we say a few words about below. Now we prepare for going beyond the leading order in (1.17), when there is a unique -79- minimal action path g^y and the second variation is positive definite. We want to estimate as a small error term the contribution in (5.5) from the paths in the complement of some neighborhood of grj√ ( ∙ λ). the integrand around g^y Then we make Taylor expansions of in this neighborhood (see Schilder [1]) to get finer asymptotics of the contribution from this neighborhood. Our first step is to use Cameron-Martin formula (Cameron-Martin [1], Freidlin [1]) that implies: If f ∈ Cθθ([0,t]) and ∫ t .9 fz(s) ds < ∞ then μθ θ (f + Z) is absolutely continuous with respect to μθ θ θ.t (Z) and the Radon-Nikodym derivative is given by (5.6) exp(-A ∫t (f(s))2 ds) exp(-∫ f(s) dZ(s)) z 0 0 t . where ∫ f(s) dZ(s) is ItO stochastic integral, to obtain 0 Sublemma 1. If gj^,y is a miminal action path and denotes expectation w.r.t. Brownian bridge on [0,T] then (See also Davies and Truman [1]) (5.7) e - TH(A) J λ (x,y) = ∫ exp(- λ2 ∫ (ω(s)) dμ τ (ω) = 0 0,x:y,† d τ _1 = (⅛)2 exp(-ÄA(x>y/r)) Eξ(eχp(-∫θ <γ(⅛yCt) + λ 2 z(t)) - - V(g*y(t)) -λ 2 VV(gxy)) ∙ Z(t)) dt)) and if r > 0 then -80- τ .0 J (5.8) F(x,y,T,λ,r) := ∫ exp(-A^∫θ ∖'(-'(s)) ds) XV,( ■ A) ∣∣σo ≤ r) χ(u ∈ Cχfy([0,^]) : ∣∣tv∙ — g,j' du — rp £ (ω) = (27,r)2 cxp(-AA(x,y,T)) eJ (exp(-∫ (V(g*y(t) + θ5×>y* ∖ 0 _i + A _i 2 Z(t)) - V(gy,(t)) - A 2 VV(g*y) ∙ Z(t)))dt)) χ(Z ∈ Cθθ([0,T]) (∣∣Z∣∣00 ≤ rA^)) Proof. We prove (5.8) then (5.7) follows by taking r —► 00. If {ω(s)}θ<s<t is the conditional Wiener process on [0,t] starting at x and ending at y and {α(s)}θ<s<ι is the Brownian bridge on [0,1] then (see Simon [5], chapter 2) ω(s) = (1- ∣)× + ∣y + 4"t α(∣) and α(∑) — t _1 2 α(s) where = means the distribution on both sides are the same. (5.9) Therefore F(x,y,T,λ,r) = ∫ exp ( —∫ V(ω(s)) ds) 0 ×Y, χ(ω ∈ Cχjy([0,I]) : ∣w(s) - g^ j(s ∙ λ)∣ ≤ r) d/Z0,x,y;ï = (2⅛)2 eXp( Ej[exp(-λ∫ V((l-A) x+ ‡y + A ∣2 2Ιr-) 2 α(t)) dt) -81- χ(z ∈ Cθθ([O,T}) : ∣(1-1) X + ∣y + λ 1 22 a (t) - g×y(t)∣ ≤ r for t ∈ {0,T])] . 1 Now a → a + f =: Z where f(t) = -λ2(g*y(t) - (1— ‡) x + ‡y) (so (1-∣) X + 4y + λ _i _i 2"(t) = gψr(t) + A 2 Z(t)) is a bijection Cθθ([0,T]) Cθθ([0,T]), ∫ T . 2 ∣f r dt < ∞ and so from (5.6) we have the Radon-Nikodym derivative is given by dn (q1 T T — T τ,⅛. = exp(-∫ (f(t), dZ(t)) - A∫ ∣f(t)∣2 dt) = exp(-λ2 ∫ < 0μrjΛzJ n0 z ο o0 0 λ rτ∕Λ∙χy ι2 VV(gψ(t)) ∙ Z(t) > dt) exp( — —^ (g^p )2 dt) exp(A(×-y)' ^y )) where we integrate by .. X V ∙ A parts the ItÔ-integral since f is absolutely continuous (Nelson [1]) and used gr^ (t) = VV(g⅞y) and where dμτ(Z) = (2πT)2 dμ0 θ,0.τ(Z) is the Brownian bridge measure. By (5.9) F(x,y,T,λ,r) = (⅛)2 exp(-^ Jθ (g⅞y(t)2 dt) Ey(exp(-Aιfτ(V(giy+λ δ 0 _1 _1 2 Z) - λ 2V'(g⅞y)Z) dt) 1 J ∕Q x[Z ∈ C00([0,T]) : ∣∣Z∣∣ Εοο[θ/Γ] ≤ rλ2]) = (⅛) exp (- λ A(x,yT)) e£( exp (-λJ∙τ(V⅛y(t) + λ _I ' -i 2 Z(t)) - V⅛y(t)) - λ 2 1 V'(g⅞y(t)) ∙ Z(t)) dt) x (Z : l|Z||Loo[0/r] ≤ r A2)]· which finishes the proof. -82- To use this (see lemmas 8, 9 and 10 below) we need: §5.2 Large deviations x Varadhan’s Theorem and Schilder’s Theorem. References for this section are Varadhan [1], [2]. Stroock [1] and Jain [1]. Let X be a separable complete metric space, 58 = the Borel σ- field over X and Pε be a family of probability measures. We have in mind X = Cθθ([0,T]) and with ε = 5.10) μψ( ∙ ) = (2πT) d n 1 , Pε(A) = ∕irp (λ2A) where AiQjQjQ∙,j'( • ) is the Brownian bridge measure, τ so formally Pε(A) = “J ” exp(-^ J z2(s) ds) d∞ Z .A 0 and we want results like Laplace method (Erdelyi [1]), when _x? o 2ε X = R and Pε(A) = ∫ -g----- i dx . A (2τrε)5 (5.11) Definition (Varadhan [1]). We say {Pε} obeys the large deviation principle with rate function I( ) if there exists a function I( ∙ ) from X into [R] such that (i) 0 ≤ I (x) ≤ ∞ for all X ∈ X (ii) I( ∙ ) is lower-semicontinuous (iii) For each £ < ∞ the set {x : I (x) ≤ £} is compact subset in X (iv) For each closed set C C X lim ε log Pε(C) ≤ — inf I(x) ε→0 x∈C (V) For each open set G C X ε log Pε(G) > — inf^ I(x)∙ -83- Then we have Varadhan’s Theorem (Varadhan [2], [3]). Let Pε satisfy the large deviation principle with a rate function I( ∙ ). Then for any F : X → R bounded and continuous we have (5.12) For any closed set C in X (i) lim e log ∫ exp(-¾^) dPε(x) ≤ - inf (F(x) + I(x)) ε-→0 c x∈C For every open set G in X (ii) (5.13) exP(-i-r~') dpε(χ) ≥ ~ inf ε lθS I (f(x) + Kx)) In particular if G = X = C then (5.14) lim e log ∫ X s~4U exp(-¾^) dPε(x) = -inf (F(x) + I(x)). x∈x Moreover (5.15) If F : X -→ R U {oo} is lower semicontinuous and bounded from below then (5.12) holds. Now we let X = C([0,oo), Rd) with the uniform convergence at compacts, and M the Borel field over X. The d-dimensional Brownian-motion {bj(s)}s>θ l<j<d ^at are mean zero Gaussian random variables with covariance E(bj(t) bjf(s)) = <5^ min (t,s) _1 give a probability measure P on (X,M). Let Ρε(Α): = P(e 9 A). XT: = Cθ([o,T],Rd) = {f ∈ C([0,T], Rd) : f(0) = 0} and Iτ(f) = I ∫ τ ∣f∣2(t) dt for f ∈ Xτ. -84- Then: Theorem (Schilder). The Pε∣χ ,s satisfy the large deviation principle with rate function Irp. τ Schilder [1] showed if F : C([0,T]) -→ R is continuous and F(Z) + i ∫ ∣Z∣2(t) z 0 dt has a unique minimum at some Zθ with some additonal hypotheses among which [-(l-ε)D2F(Z∩ + τj)Z2] was: There are ε > 0 and δ > 0 such that E(exp---------------- 2---- ----------- ) ≤ constant < ∞ if ∣∣η∣∣ oo r ≤ ⅛∙ (the second variation positive definite) then L {O,TJ (n+l) 1 E(exp(-ε-1F(ε2Z)) = (Γθ + Γ1ε.2z + Γ2ε +∙∙∙+ Γn εz + 0(ε i )) exp(-ε 1(F(Zθ) + ± JT ∣Z∣2 (t) dt))) For further results see for instance: Azëncott [1], Chevet [1], Donsker- Varadhan [1] Ellis-Rosen [1], [2] and [3], Freidlin-Wentzell [1] Pincus [1], Simon [5] and Stroock [1]. -85- 6. The Proof of Theorem A. §6.1. Some lemmas and the idea of the proof. §6.2 Going from (R^)n to (B(b,iθ))n in (6.10). §6.3. The main contribution in (6.10) is from §6.4. The conclusion of the proof of (1.35). §6.5. A Proof of (1.36). §6.6. A Proof of Theorem A.l. X B(gip^(iT1 ), ⅛) -86- §6.1. Some lemmas and the idea of the proof. We make the same assumptions as in lemmas 2-7 on xθ and b: (1.33) V,,(b) = Ω2 where Ω = diag(ω∣ ,..., ωfj) with 0 < Wj ≤ u>2 ≤∙∙∙≤ ωfj (2.5) p(xθ , b) < p(x0 , a) -(- p(a,b). (2.6) There is a unique Agmon geodesic g x∩,b .... minimizing p(x0 , b) = inf{∫ζ°(i γ2 + V(γ))dt∣γ(0) = xθ , γ(∞) = b} and (2.7) Xn,b The second variation of g υ is positive definite. Lemma 8. rΓ t With <Sθ , Tθ as in lemma 2, rθ in lemma 4, aθ (x,y) and a∣ (x,y) as described in Theorem A, we have 1. For some k5κ > 0 (6.1) rp F(x,y,T,λ,⅛) = ∫exp(-λ2 ∫λ V(ω(s))ds) 0 x({~ ∈ Cx,y([0 £]) : ∣∣w - g*>y( ∙ λ)∣∣oo ≤ ⅜})dpθ χ y 1(ω) d ,j, = exp(-λA(x,y,T))(A)2 bj(x,y){l + Mpy) ,τ4 τ2 —λ kcΓ ÷ θ(^2^) ÷ θ(ξξ" eχP(------ —))} uniformly for λ T (x,y,T,r,Λ) ∈ (B(xθ , 6θ) u B(b,δθ)) x B(b,5θ) × × [T0 , ∞) × (0,rθ] x (0,∞) -87- where (6-2) bo∖χ,y) = 22(i∏1ωi)2 ao∖x,y) e -Tefl trace , V,,(b) θ and eo = 2. If X = wθ ∈ B(xθ , δθ) U B(b,6θ) z = wm + 1 ∈ B(b,6θ) and Tj ∈ [Tθ , ∞) then m inf { ∑ a(wj,Wj+1 ,..., T1)} = A(x,z,(m + l)T1) w1,-⅛∈r° i=0 and is attained if and onlv if x,z i = g(m+l)T1(ιτP f°r i 6 i1’-’“}· Moreover <6∙3> m I m (i50 <A>2 ∙ bo1(>"i ∙ ∙i-i'l (B(b,⅛))m 1-θ exp(-λA(wj , wi+1 , T1)) dw ,..., dwm x S (m + l)T, 1 τrι = ⅛P2 b0 1 (x,∙z) exP(-AA(x,z,(m + 1)T1) (1 + 0(∣))m where 0(^) is uniform for (xtz,T∣,λ,m) ∈ (B(xθ,6θ) U B(b,6θ)) X B(b,6θ) X [Tθ , ∞) X (0,∞) X N. (A Proof in Chapter 7.) -88- Observe that (6.1) gives the lower bound in (1.35) of Theorem A and part 2 says the leading order of the approximation in part 1 satisfies the semigroup property. Lemma 9. There are positive constants kg and λθ such that (6.4) Hfλ'> 4 — λkβr^ exp(-T—y^)(x,y) = F(x,y,T,λ,r) + 0(λ2 e ) exp(-λA(x,y,T)) = F(x,y,T,λ,r)(l + 0(e — Λk r2 θ )) uniformly for (x,y,T,r,λ) ∈ (B(xθ,i0) U B(b,i)) x B(b,⅛θ) × [⅞ 5 2Tθ] × [0,rθ] x [λθ , ∞). (A Proof in Chapter 8.) Note that (6.1) and (6.4) yield (see (6.19) below) (6∙5) exp(-T¾^)(x,y) = d aT∕ x ⅛p2 b0 (x>y∏1 + a--λx,y + O(i)}exp(-λA(x,y,T)) λ uniformly for (x,y,T) ∈ B(xθ , δθ) × B(b,6θ) x [Tθ , 2Tθ] as λ —+ oo (see also Davies and Truman [1]) t. Lemma 10. With ∣V,(x)∣ = 0(e I I2 ) for statement about derivatives, there exist β ∈ (0,1), kγ , kg , and λθ positive such that -89- (i) If xθ ∈ K = {x : p(x,a) = p(x,b) = f⅛⅛} X ∈ B(xθ , δθ), w 0 B(a,6θ) U B(b,δθ), T ∈ [Tθ , θo), T1 ∈ [T0 , 2Tθ], λ ∈ [λθ , ∞) and i ∈ {1, ..., d} then (6.6) exp(-T1 —Ç—^)(x,w) λ Ί _Tl^ ∣i -g- e i kγexp(-λ(min { sup {p(x,c) + Xc>y)} + ks}) c∈{a,b}y∈B(c,^60) ( J ≤ 1 λ (x,w)∣ k7 exp(-λ(A(x,y,T) + kg)) ∀y∈B(b,/?£()) UB(a,^θ) (ii) If X ∈ B(b,δθ), y e B(b,∕J0θ), w £ B(b,<5θ) T ∈ [Tθ , ∞), Tχ ∈ [Tθ , 2Tθ], λ ∈ [λθ , ∞) and i ∈ {l,...,d} then (6.7) exp(-T1⅛^(x,w) ∣Λ -25x∙ e" H _T 2W 1 Λ (x,w)∣ k7 exp(-λ( sup {p(x,b) + P(b>y)}+ko)I y∈B(b,^θ) k7 exp(-A(A(x>y>T)+k8)) and similar when (x,y) ∈ B(a,δθ) × B(a,∕2δθ) (iii) With everything as in lemma 8.1 and ∕ a∣x∣2 ∣V (x)∣ = 0(e ) for some A < ∞ then uniformly for (x,y,T) ∈ B(xθ , δ) X B(b,δθ) X [Tθ , 2T0] and i ∈ {l,...,d} -90- (6'8) ~5 ⅛ e τH(A) a λ (X,y) = ⅛π^2 ∕0A(x,y,T) τ τ ∂b^(x,y)> σ° 0 WΛ {2a^q ψ,√ 8x∣ ⅛t-W ⅜J+0^,}, as λ -→ ∞ A Proof in Chapter 10. Now we want to get the upper bound in (1.35) by showing the main contribution in (6.9) τH(A) T e λ (x,y) = ∫exp(-λ2 ∫ V(ω(s))ds) dμ τ (w) 0 0,x,y,j is from {ω : ∣ω(s) — gψ,y(sλ)∣ ≤ ψ} and use lemma 8.1. with T = (n + 1) If T ∈ [2Tθ , ∞) we write ∈ [Tθ , 2Tθ) and for (x,y) ∈ B(xθ > × B(b,6) where δ ∈ (O,∕36θ] we write wθ := x, wnψj : = y and _TH(A) (6.10) e λ -T1¾^ n (x,y) = ∫ .∏ e (wj , wj + 1) dwχ , ..., dwn . (Rd)n First we show the main contribution in (6.10) is from (B(b,⅛Q))n . In the final formula for the splitting this step puts the contributions from paths going between a and b more than once, among them multi-instantons (see Coleman [1] and Zinn-Justin [1]), into error terms. The second step is to show that w∣ ∈ B(g^,,y(iT∣), con^atll) js what really matters. In step 3 we bound ∫ wi∈B(g⅞y(iT1),εen⅛ant) τ1∕w .∏ e 1~u _T h(λ) 1 λ (wi , wi+1) dw1 , ..., dwn -91- from above by ∫ exp(-λ2 ∫ 0 ∣ω-g*,y( ∙ A)∣∣ L ∖,(-,(s)ds) y({ω ∈ Cx y([0, ∙⅞∙]) Λ τ < ^}) ci∕ι τ (ω) (1 + “something small”) [U, » j u,x,y, x which we know by lemma (8.1) and finishes the proof of the upper bound in (1.35). To get (1.36) we differentiate (6.10) under the integral sign (6∙11) ⅛ e i τH(Λ) _T H(A) _nT H(A) λ (x,y) = ∫ ⅛ e 1 λ <x'wl> e 1 λ (wl ’ y) dwl pd i and we only need the analysis of Q e τ hw ∖ -- J∙ I (x,w∣) in lemma 10, (9.3) of Proposition 2 that says ___ (6.12) τH(A) 1°ε⅛e λ (χ>y)∣ îîîn -------- i------ τ--------------Λ~→∞ λ ≤ -A(x,y,T) u.o.c. in Rd × Rd X (0,∞) and T ∈ [Tθ , ∞)-analysis of e that we have then finished. mH(A) j- ∖ (w^ , y) -92- Now with kg as in lemma 10 we pick 6 ∈ (O,∕J0θ] such that kς sup ∣p(x,a)-p(x,b)∣ < -⅛x∈B(xθ,6) (6.13) sup {p(y,b)} ≤ sup {p(y,a)} y∈B(b,6) y∈B(a,∕J0θ) and we do the first step of section 6.2. §6.2. Going from fRd)n to fBfb.⅛0Ωn in (6.101 In this section we will prove that if δ ∈ (O,∕76θ] satisfies (6.13) and T∣∈[Tq,2Tq) then uniformly for x = wθ ∈ B(xθ , 6), y — wn+1 ∈ B(b,6), T = (n + 1) Tj ∈ [2Tθ , ∞) and λ ∈ [λθ , oo) we have (6.14) exp(-τ5^)(x,y) = ∫ (B(b,6θ)) n n ∏ e i=0 (wi , wi + ι) dwl - dwn + 0(exp(-λ(p(x,b) + sup y∈B(b,6) , n ∫ .∏ e (B(b,<5θ))n 1 = θ (wj , wj+1) dw1 ... dwn d = ⅛P2 b0 (x,y) exp(-λA(x,y,T)) (1 + 0(J)). -93- Proof. For X ∈ B(xθ 5 Æ), y ∈ B(b,j30θ), T = (n + 1) T∣ with ∈ [Tθ , 2Tθ) we write (6.15) e -T H(Λ) τ H(Λ) λ (x,y) = ∫.∏θe 1 λ (wj , wj+1) dw1 ... dwn i B(a,60) e -T + J B(b,60) + J Rd∖(B(a,6θ)UB(b∕θ))' H(λ) H(λ) — nT λ (x,w1) e 1 λ (wj , y) dw1 = : I∣ + HIj + Vj which defines , IIIj , and Vj (in the same order). For j = 2, ..., n put (6-16) H(λ) n -T 1 λ ∏ e I: = ∫ (wi , wi + ι) dwl — dw∏ i=0 wk∈B(a^0) for k l,...j and (6-17) ∏j = ∫ wk∈B(a,5θ) WjgB(a,0o) n ∏ i=0 for k = l,...j-1 , H(λ) 1 λ (wj , Wj+1) dw1 ... dwn -94- For £ ∈ {1, ..., η — 1} h = Wjt∈B(b,6θ), 1 k = l,...,f τ «(λ) n ~ 11 ï ∏ e Λ (wi , wi+1) dw1 ... dwn i=0 η ∫ wk∈B(b,6θ), k = l,...,£+l „ + J wk∈B(a,6θ) ∏ e i=0 ~~ -l i H(λ) τ- (wi , wi+1) dw1 ... dwn H(λ) -τι~r- , η e '∙ IWj ,wi + 1jow1 ...αwn ∙-∏ for k1Ξl,...,fi wg+l^β(a⅞) - ⅞ + l + Π£ + 1 and so (6.18) I1 = I2 + II2 = (I3 + II3) + II2 = In + ∑ ∏j J =2 Before continuing we note by (6.4) of lemma 9 exp(-T1 S<Ξ)) (u,v) = F(u,v,T1,λ,^) + 0(λ2 2 e — λA(x,y,T∣) — λkθ rθ uniformly for (u,v,Tj,λ) ∈ (B(xθ , <5θ) U B(b,<5θ)) × B(b,5) × [Tθ , 2Tθ] × [λθ , ∞) Hence by lemma 8.1 -95- (6.19) exP(-(u,v) = ⅛)2 b0 1 (u>v) exp(-λA(u,v,T1)) {1 + ⅛ + θ(½) + θ(⅛ exp(- ^⅞))} (l+0(λ2e — λk λkθrθ)) = λ η τ1 aF1(u,v) 1 = ((<τ=r bn 1(u,v-) expf — λA(u,v,Tπ )) (1 -1--- 1—5--------1- O(-⅛)) 2r υ ' ∙ ±" ' λ 'λz" = ((⅛)2 bO 1(u<v) exp(-λA(u,v,Tι)) (1 + O(i)) uniformly for (u , v , T∣ A) ∈ (B(xθ , <5θ) U B(b,6θ)) × B(b,6θ) × [Tθ , 2Tθ] × [λθ , ∞) and similarly we have for (u , V, T1 ,λ) ∈ (B(xθ , δθ) U B(a,0θ)) × B(a,6θ) × [Tθ , 2Tθ] × [A∣ , ∞) For j > 2 we write -96- (6.20) IL = ∫ ( ∫ .∏ e B(a,δθ) w2,...Wj-1∈B(a,δθ)1-° -τ Η(Λ) 1 λ (wi , wi 1) dw1 , dwn) Wj⅛B(a,δθ) nH(λ) = (x,w1) IIj(w1 , y) dw1 J e B(a,δθ) By the semigroup property and e -tυ H(λ) ∖ ∖ o λlx-v∏ (x,y) ≤ (?^.) exP(---- 9t2t ) if t > 0 we have π∕wι-y> = j-ι -τ1¾^ > ∏ e (wi , wi+1) J w2,... Wj-ι∈B(a,δθ) WjgB(a,δθ) v=o Wj+1,.∙.wn∈Rd τ H(λ) n - i 1 λ (wi , wi+1) 1 dw1 , ..., dwn ∏ e A i=j j_l _T SW .∏ιe 1 λ (wi.wi+1) J w2,...,Wj-1∈B(a,δ0) WjgB(a,δθ) -(n-i)T1⅞υ (Wj , y) dw2 , ..., dWj -97- ≤ ( sup ( Wj^B(a,5θ)x J w1,...,Wj-1∈B(a,0θ) ∫e j-ι -τ1^^ .. .∏ιe (wi,wi+1)dw2,...,dwj-iη , .yr H(λ) — (n—j)T1 — (wj,y)dwj j—ι -τ1iiW wjΓB(M0),w1.∙..,wi-1∈B(M0) i=l ' <wi∙"i+l> d"2 ■ ' - dwi-l, _T where if j = 2 we mean sup (e w2gB(a,6θ) H(A) i λ (w∣,w2)) which by (6.7) is less than kg exp(-λ(A(w∣ , z , S) + kg)) for any z ∈ B(a,∕J0θ) and S ∈ [Tθ , ∞) so we have (6.21) ∏2(w∙^ , y) < kg exp(-Λ(A(w-^ ,z , S) + kθ)) for any such z and S. If j ≥ 3 then by (6.19), (6.3) and (6.7) (6.22) ∏≈(w1,y)≤ sup ( ∫ j Wj^B(a,6θ) w2,..., { ⅛ια⅛2 bo‰wi+ι) -T ∙∙∙ ∫ Wj-1∈B(a,6θ) -λA(w.,wj ,1,T1) 1 1+1 1 (i + oφ) H(λ) 1-Γ (wj-l ’ wj)} dw1 , ..., dwj-1 -98- < ; . g (j-2)T1 -λA(w1,w∙1,(j-2)T1) <⅛>2 ⅛ 1(wι*wj-j)e 1 j 1 w. j-l∈B(a,6θ) k7 exp(-λ(A(wj-1 , z , S) + kg)) dwj-1(l + 0(∣))θ for any z ∈ B(a,∕76θ) and s ∈ [Tθ , ∞). If X ∈ B(xθ,<5θ) U B(a,6θ),y ∈ B(a,6θ) (and similar for b) and T,S∈[Tθ,∞) then min -fAfx,w,Tl + A(w,y,S)} = A(x,y,T + S) and is attained at the unique point w∈B(a,6θ)" w = w(x,y,T,S) = gτ + s(τ) € β(a, ^∑), by lemma 2∙ Now we will use Hörmander’s stationary phase theorem. To get uniformity, as in (4.11), we want (6.23) |w—w∣ < constant <∞ for all ∣Aw(x,w,T) + A(v(w,y,S)∣ (x,y,T,s) ∈ (B(xq,6q) U B(a,6θ) x B(a,0θ) × [Tθ , ∞) × [Tθ , ∞). That follows from A(v(x,w,T) + A(v(w,y,S) = = (Aw(x,w,T) + A(v(w,y,S)) - (Aζv(x,w,T) + Aw(w,y,S))∣w=w = ∫ (Aww(x,w + t(w-w),T) + A‰(w + t(w-w),y,S))(w — w) dt 0 1 -<5,T -i1S = Jθ(2Ω + 0(e 1 + e 1 + ∣w + t(w-w) — a∣))(w — w) dt . -99- In the last step we used (4.18). With k = integer part of (f∣ + 2) =: [^ + 2] we get from (4.10) (6.24) J (≠)^ bj(u.w) e-AA<X-W'T> .-jA('>'>y∙S) dw B(a,Sθ) = 0(√τ'°'*, e~ W,y,T + S) (1 + 0(l,) uniformly for (x,y,T,S,λ) U (B(xθ , 0θ)) × [Tθ , ∞) × [Tθ , oo) × (0,∞). Since, by a statement similar to (6.2) near a, instead of near b, we have ∣3y ⅛W)I = e θ( ) 0(∣9y aj(x,y)∣) where eθ(a) = trace J V,,(a) . The uniformity in (6.24) follows from (6.23) together with (4.18) and (4.16) that say ∣0y A(x,y,T)∣ for ∣a∣ ≤ 3[^ + 2] + 1 and ∣9y aj(x,y)∣ for ∣∕3∣ ≤ 2[^ + 2] are uniformly bounded on (β(x0 , 6θ) U B(a,6θ)) × B(a,0θ) × [Tθ , ∞). So (6.21), (6.22), and (6.24) gives If j ≥ 2 then -100- ~ — (j— 2)Tηe∩(a) πj(wl , y) = θ(e exp(-λ[A(w1 , z , (j-2)T1+ S) + kg])) (6.25) (1 + 0(^))θ 1) uniformly for any z ∈ B(a,β6θ) and S ∈ [Τθ , ∞). Put (6.19) and (6.24) into (6.20) and use (6.23) and you get: If j ∈ {2, ..., n} then IIj = 0(e —(j —l)T1e∩(a) —λ{A(x,z,(j-1)T1)+ S)+ks) 1 θ e 1 8 (1 + o⅛))j) (6.26) IL = 0(e j —jTη e∩(a) 1 υ exp(-λ{p(x,a) + sup (p(y,a))+kg}))(l + 0( y∈B(a,∕J6θ) Using j ≤ n = 0(T) in (6.26) we get (6.27) ∑ IL = 0(exp(-λ{p(x,a) + sup (ρ(y,a)) + kg})(l + 0(y))) j=2 j y∈B(a,∕Jiθ) λ uniformly for (x,y,T,λ) ∈ B(xθ,<5θ) × B(b,∕3<5θ) × [Tθ , ∞) × [λθ , ∞). In In — ∫ wk∈B(a,<5θ), k = l,...,n τ »(A) n - 11' λ ^ ∏ e A (wi , wi+l) dwl ’ —’ dw∏ i=0 li— » Taking S —+ oo (use Proposition 1) and then inf over z ∈ B(a,∕76θ) gives -101- _τ ∏W we use (6.19) for e (wj , *'j-∣) for i = 0, 1, ..., n — 1 and (1.17) for i = n which implies e _T HW 1 λ (wn , y) < Cε exp(-λ[A(wn , y , T1) - ε]) uniformly on B(a,6θ) × B(b,∕30θ) × [T,2Tθ], and we have, since Ci V- 4_■ ∩fh) -vA/J n = -1 +■ (VÏÏ -×az '»S p J (B(a,oθ)) n-l∕ , H T1 -λA(wi,wi + 1,T1) ) i∏0(⅛> b0 <»■ ∙wi+l)β c^-Λ[A(wn,y,T1)-t] dwj ........ dwn fl+θ(Tn By (6.3) and (6.2) we get I„ = 0(Cε e e — nT1e∩(a) π 1 θ λ2 j B(a,6θ) -λ[A(x , wn , nT1) + A(wn , y , T1)-e] ,~√Γ"2λ dwn (1 + 0(†)) ) since A(wn , y , T1) ≥ p(wn , y) and A(x , wn , nT1) = p(x,a) + p(a,wn) + 0(e uniformly for (x,wn) ∈ B(xθ , 0) × B(a,6θ) we get — w∣(a)nT∣ 1) -102- rτ, Z X d -λt ⅛f t Jp(x,a) + p(a,w) + p(w,y)}+0(e — Te∣(a) 2 w∈B(a,⅛θ) In = 0 Cf∙e 1 λ^e —ωη(a)T )-ε] Since y is in B(b,∕36θ), In is smaller than the R.H.S. of (6.27) and so by (6.18) (6.28) L = θ(exp(-λ[p(x,a) + sup [p(y,a)] + kg) (1 + 0(^)) V y∈B(a,0δθ) 07 λ uniformly for (x,y,T,λ) ∈ B(xθ , όθ) × B(b,∕3δθ) × [2Tθ , oo) × [λθ , oo) Next write ιπ1 = I∏2 + IV2 = ∙∙∙ = IIIn ÷ £ IVj J=2 where IL is defined by replacing a by b, the definition of L , and in the same way we get IVj from ∏j We can replace a by b in the estimates for ∏j and copy estimates for IVj . We get from (6.27) (6.29) H(λ) n -T 1 λ (wi , wi=1)) dw1 ... dwn III1 = ∫ (.∏ e (B(b,<5θ))n 1-θ -λ[p(x,b) + + 0(e where by (6.19) and (6.3) sup (p(b,y))+kg] y∈b(b,β<5∩) τ θ7 (l+0φ) -103- (6.30) H(λ) n -T 1~T (wi , wi + 1)) dw1 , ..., dwn = ∫ i∙5∩e (B(b,6θ))n 1-θ - (^)2 bj(x,y) exp(-AA(x,y,T))(l + 0(^)) uniformly for (x,y,T,λ) ∈ B(xθ , όθ) × B(b,∕30θ) x [2Tθ , ∞) × [λθ , ∞). Finally by (6.6) of lemma 10.1 , nlA>∣ ι^ΙΓ J (6.31) -nT (x>wl) i(λ7 ι^^Γ (w1 , y) dw1 Rd∖(B(a,6θ)UB(b,0θ)) ≤ ( sup ''w1gB(a,δ0)UB(b,δ0) e _T H(A) -nT BD 1 λ (x,w1))(∫ e 1 λ (w1,y) dwj ≤ k7 exp(-λ( min [ sup {p(x,c) + p(c,y)}] + kR)) c∈{a,b} y∈B(c,∕3<5θ) and so τH(A) e λ (x,y) = I1 + I∏1 + V1 - = 0(e + λ[p(x,a) + sup (p(a,y)) + ks] y∈B(a,^6θ) ∙Γ f∙πne (B(b,iθ))nkl = 0 -τ 1 λ (1 + 0(J))) (wi,wi , i)) dwχ , ..., dwn + 11+17 -104- - λ[p(x,b) + + 0(e sup (p(b,y))+k8] y∈B(b,∕‰) τ, θ? (1 + 0(2))) + O(exp(-A( min [ sup {p(x,c) + p(c,y)}] + kg)) c∈{a,b} y∈B(c,∕J0θ) η - J (B(b,6θ)) η (∙ι π=∩0 e τ H(λ) "" 11 —∖ (wi > wi + l)) dwl > ···’ dwn + 0(exp(-A[p(x,b) + sup {ρ(y,b)} + -⅛))(1 + 0(∣)) y∈B(b,δ) 2 λ uniformly for (x,y,T,λ) ∈ B(xθ , δ) × B(b,0) χ [2Τθ , οο] χ [Αθ , οο]. Here we used (6.15) in the first step, (6.28), (6.29) and (6.31) in the second and (6.13) in the third. (6.14) follows now by (6.30). §6.3 The main contribution in (6.10) is from - xθ B(gψ^(iTj, ψ). We want to go beyond the 0(y)-term in the leading term of (6.14) given by (6.32) 1 , -τ1Hφ n (i *0 e <wi’ wi+ι))dwι ∙ ■ ∙dwn = (B(b16θ)) (Â)2 bo (x’y) e*p(-AA(*,y,T) (1 + θ(J)) -105- As a step towards reducing the problem to (6.1) in Lemma 8.1 we have: With kj as (2.33) we have (6.33) n -⅞H⅛ n _ 1 λ (wi, wi + 1))dw1 ... dwn f jr e (B(b10θ))n i=θ -τιιφ n ∫ wi∈β(gτy(iτι), ⅜) π (wi,wi + 1))dw1 ...dwn e T∙.> — -Λ Te∩(b) k1T exp(-λ(A(x,y,ι ) + ~∣y))) + 0((l + 0(†)) Tλ'* e uniformly for (x,y,T,λ,r) ∈ B(xθ,6) × B(b,0) × [2Tθ, ∞) × [λθ, ∞) × (0, rθ]. Remarks 1. If we put (6.19) into (6.32) we get 53 A(w∙ , w∙ , , , T∙∣) in the exponent i=0 ^,^ with wθ = x and wnψj , T1)} = A(x,y,T) min w^ ∙ ∙ ∙v n is attained if and only if w∣ = gτy(iτi) fθr i ∈ i1» n)∙ 2. Using the uniformity in (6.33) and taking r ∣ 0 we get n -T1 7Γ e J (B(b10θ))n i = 0 H(λ) wi + l) dwl ∙∙∙ dwn = -106- = 0((l + 0(∑)) Tλde Te°(b) exp(-λ A(x,y,T)) d which T ∙ λ 9 times the behaviour in (6.32). Proof of (6.33): Put (6.34) VIθ = n -τ1¾-^ π e (w; , w∙-^1) dw1 ... dwn 1-θ ∫ (B(b16θ))n for r ∈ (0, Γθ] and for j ∈ {1, ..., n} (6.35) VIj = ∫ wke⅛τy (^τι),⅜) -T n π e i=0 k=l,...j H(λ) (wi » wi+l) dwl ∙∙∙ dw∏ w. + 1,...,wn∈B(b,5θ) and (6.36) VIL = ∫ π e wk∈B(g*y(kT1)4) k=l,...j-1 1=0 -T H(λ) 1~T (wj, wi + 1) dw1 dwn wj∈B(b,δθ)∖B(g*y(jT1), ‡) wk∈B(b,6θ) k=j + l,...,n We see VIj = ^j-∣-l ^*^ (6.37) vι0 = VIn + ∑ v∏i. j=l J By (6.30) and (6.3) we get from (6.36): ^or j = θ> n~ 1 and so -107 S τ «(λ) e^Tl “ VII. ≤ J i=o j wk∈B(b,0o) for k∈{l,...,n}∖{j} Wj∈B(b,6θ)∖B(grpy(jTχ), jp) (6.38) iλλ2 bjτι (χ,wj)e <2τrJ b0 (wj, wi+1)dw1∙∙ ∙dwn λA(x,w.jT,) j (l+0(i)) Wj∈B(b,0Q)∖B(grp^(jTχ), ψ) x 4 (∏ + l-j)T1 -λA(w.,y,(n + l-j)τι) 1 2(n + l-j) l(w.,y))e J (l + 0(1)) ( ⅛)2bθ dWj Now we use (2.33) that says: ∫J ⅛2 + V(7))dt ≥ A(x,y,T) + k1 min(⅛ ∣∣γ - g⅜7 l∣2∞[0,T] for all 7 such that 7 — g*y ∈ Dθ(0,T), uniformly for (x,y,T) ∈ B(xθ,6θ) × B(b,<5θ) x [⅞> ∞). Hence inf XV {A(x,w∙jT1) + A(w.,y, (n + l)Tχ)} wjeB(b’c0)\B(gTy(jT1),‡) >x Vo (b2 + vMdt} ≥ A<x’y’T> + kι .inf 7=l∣7-gT ∣∣l∞10jι≥⅜ ⅜ By (6.2) -(n + l)Teθ(b) Jτi (n+u-j)Tχ suP (b∩ 1(x,w.) b (wj,y)) = 0 (e ) y,Wj∈B(b,6θ) 0 J x∈B(xθ,<50) -108- = 1 + 0(^) and since (1 + (1 + oφ) uniformly for (x,y,T,A,rj) ∈ B(xθ,5) × B(b,6) x [Tθ, ∞) × [λθ, ∞) × (0, rθ] × {1, ..., n} Now (6.33) follows from (6.37) and definitions (6.34) and (6.35). §θ·4 The conclusion of the proof of (1.351 We start by; If then ∏-∣ H(λ) (wi,wi + 1))dw1 ... dwn (6.40) wj∈B(gτy0τi)⅛) for j = l,2,...,n )) + 0 2 rz exp(------⅜-))} exp(-λA(x,y,T)) uniformly for (x,y,T,λ) ∈ B(xθ,6) x B(b,6) × [2Tθ,oo) × [λθ, oo). -109- Proof. Put Vj = gτy(jT1) for j ∈ {0,l,...,n,n + l} then vθ = x = wθ and vn_|_i — y = wn-∣-i an^ vj,vj÷ι xy I (t) — Sτ (t+jTj)∣^θr^ ] by tke un⅛ueness >n lemma 2.2. wj ∈ B(Sτr(j^l)'⅜) f°r J = 1> ■■■■> n then Sτ^ If is again unique and by (2.38) wPwi + 1 vi,vi + l, -ω,t -ω 1∣T1-t∣ lsψl (t)~6τ1 I ≤ k2(∣wj ~ vj∣e + ∣wj + l ~ vj+l∣e } - 2k 2ψ , uniformly. Hence if w∙w∙ 1 1 + l (sλ)∣ ≤ I for s ∈ [0, i] <*j(s) - gτJjι Jjτ then ∣αj(s) - gτy(jT1 + sλ)∣ = ∣αj(s) - gv τl1v'2∕(sλ)∣ W∙W∙ ∣ - W∙W∙ V∙V∙ 1 ≤ l«j(s) - gτjι j+ (sA)∣ + ∣gτjι j+1(sA) - gτjιj+1(sA)∣ ≤ ‡ + 2krjΓ = (1 + 2k2) X = r∣-∣ by (6.39). So we have a string of (n + l)-sausages inside a long and fat one: (6.41) A: = U {α ∈ Cx y([0,y]) : α(j-j4) = w-for j ≈ {0,lV∙∙,n +1} wi∈B(g*y(iT1)X) λ λ j i = l,...,n T1 wiw*-Li T and ∣α(j-ψs)-gτ^ j (sA)∣≤X on [0,= [0,for j∈{0, ..., n}} Ç {α ∈ Cx,y([0, ^]) : ∣q(s) — g^y(sλ)∣ ≤ ψ} = : B for r given in (6.39). -110- By definition T F'(×O',Tiλ,e) = ∫ exp(-λ2 ∫θ V(α(s)) ds) × ({α € CX(y([0, J]): »“ - 8T (∙>)∣∣lco[0 T, ≤ ≈>) ⅛0wi(α) Since μ∩ u y t is the restriction of the Wiener-measure to Cu5v([θ>t]) we get: (θ-42) ∫ wk∈B⅛⅜y(kT1ψ)k=l....... „ J . °n r(wi, wi+i, T1, λ, ‡) dw1, ..., dwn 1=θ p [Jexp(-λ2Γ λ V(α(s))ds)χ({α∈Cw.w ΧΥ/, rr, X r∖f=θ wk∈B(g-r>(kT1),⅜) k = l,...,n 1 0 w. w ∣(α(s)-gτ1 1+1(sλ)∣ ≤ ψ})dμ ([0,¾)∙. 1+1 ^ τ (α)dw1,...,dwn θ'wPwi+1,√ T = ∫ exp(-λ2 ∫ V(α(s)ds)dμ τ(a) A 0 0,x,y:± T ≤ ∫ exp(-Λ2 ∫λV(a(s)ds)dμ τ(α) = F(x,y,T,Λ, ⅜) B 0 0,x,y:± T' that we know how to expand, by (6.1). Recall Lemma 9 that says eχp(-τi 5W) (wi , wi + 1) = F(wi , wi+1, T1, λ, ‡) (1 + 0(eχp(-A½g-))) -111- uniformly. Using (η + 1) = 0(T), (6.41) and (6.1) with r as in (6.39), we conclude: (6.43) -T H(λ) l~λλ (wi,wi+l))dwl, dwn J (» e J ( * r(wi, wi+l, Tl’ λ, ψ)) dwl,∙ ∙dwn wk∈B(g^y(kT1),^) for k=l---- n wk∈B(gjy(kT1),τp for k=l,...,n -λkβr^ -λkβr2 (1 + 0(T exp -^f-))) < F(x,y,T,λ,ψ) (1 + 0(T exp( ζd* ))) = exp (- -AA(x,y,T)) bj(x,y) (l + 1 + θ (⅞ ∕rr2 —Aki-r^ χ — λkfir^ x + θ(⅜exp (—f- ))+ θ(τ eχp(—))) uniformly which proves (6.40). Now we get: ∕r λ .λ (6-44) -T e (6.14) H(λ) λ t ∖ (θ∙lθ) r ∕ n (x,y) = ∫ I . 7r , —.r1 - n Ηθ (Rd)n J ( " -T H(λ) 1 λ (wi-wi+l))dwp∙∙∙,dwn H(λ) 1 A (wpwi+l)) (B(b,6θ))n kl-θ 0(exp(-λ(p(x,b) + sup {p(b,y)} + -^))) (1 + 0(I)) Y∈B(b,0) 2 A -112- (6.33) ί .Xθ(B(g×y(iT1)4)) Τ' (i=oe _Τ Η(Λ) 1 λ (wpwi+ι))dwυ∙∙∙>dwn rl - Tin(b) + O((l + 0(J)) Tλde + 0(exp[-λ(p(x,b) + 0 k1Γ2 exp(-λ(A(x,y,T) + -⅛))) sup {p(b,y)} + -^)]) (1 + 0(I)) y∈B(M) 4 λ (6.40) / \ τ blFfx.vl -τ4 = (⅛∕ ⅛<×-''∙> ο + -uγl2 + 0⅛> + θ(^ exp(-Λ ⅛)) + 0(T exp(-λ ⅛)) + 0(Tλ2 exp(-λ-i^-))} exp(α-λA(x,y,T)) + 0(exp[-λ(p(x,b) + ko sup {p(b,y)} + -^)]) y∈B(b,⅛) 2 = ⅛Γ b0'>-'∙) <, + ⅛-, + °(ÿ + °⅛ exp(-λ ⅛-))} kcr2 exp(-λA(x,y,T)) Uniformly (x,y) ∈ B(xθ,6) × B(b, δ) as T, λ → ∞ and T4 ≤ λ. Proposition 1 in the last step. Where we used -113- §6.5. Proof of (1.36) Similar to (6.15) we write H(λ) .1 JL e τ λ (x v) = λ 5xi t ,y, Γ _i _a_ e j λ ∂χ.. , H(λ) 1 λ . vi,wi-∣-l)) dwl’ ··’ dwn _T 5iD 0 λ (x,wl)(.⅛ιe when T = Tθ ÷ nT^ ∈ [2Tθ, ∞). The reason for this “Τθ” is to make the large T dependence in formula (6.48) below simpler (Tθ is fixed but T∣ depends on T). When we obtained (6.14) from (6.15) we only used the estimates in (6.6), the upperbound in (1.17). We have (6.6) and (6.7) again H(λ) for∣i-0-e T λ (x,y)∣ and we replace the upperbound in (1.17) by (9.3) below that ∣A 0xi e (6.7) and says τH(A) lim λ→∞ ι°gl⅜e λ (χ,y)∣ λ ≤ -A(x,y,T) uniformly on compacts in Rd × Rd × (0, ∞). So similar to (6.14) we have τH(A) <6-45> -I ⅛ » (×,y) = ∫ (B(b,0θ))n n ( J -⅛e i -T1Φ e (wi, wj+1)) dwp...,dwn + _T H(λ) ° λ (χ>wι) -114- + O(exp{-λ(p(x,b) + k ,«⅛ sup (p(b,y)) + -^)} (1 + O(ι)) y∈B(b,δ) 2 λ uniformly for x ∈ B(xθ, δ), y = wn+ι ∈ B(b, δ), T = Tθ + nT∣ where T∣ ∈ [Tθ, 2Tθ) and n ∈ N, when δ satisfies (6.13). For w1 ∈ B(b,δθ) and y = wn+j ∈ B(b,δ) Ç B(b,∕Jδθ) we obtain an expansion of ∫( ⅛ e j i=l (6.46) _T 5W 1 (wj,Wj^^))dw2, ..., dwn as in (6.43) which along with (6.8) yield J (B(b,δθ)) Γ n H(A) _T H(A) -τ 1 A (wi,wi+1))dw1,...,dwn ⅛^ ° λ <*·»!> (i*1 fA)5 pA(xiw1,T0) ^ b(4,> ^2"i 1 s×ι τ θ ι /«a T∩ T∩ 9b∩υ ∖ (f^(χ,wl,Tθ) bθ0(x,w1) a10(x,w1) (x,w1)) Ί -i--------------------------------- 1----------------------- i------------- ÷ °⅛>J exp(-AA(x,w1,Tθ)) d ,p_ ψ bθ θ(w1,y) a<τ~τθ)fw X (l + ħ------- a lwl,y2 + e(τ,λ)) exp(-AA(w1,y,T-Tθ)) dw∣ uniformly where e(T,A) = θfe') + O⅛β×p(-⅛ -115- XV XV By the uniqueness of gψ in Lemma 2, w: = grp (Tθ) is the only point where the inf in inf {A(x,w1,Tθ) + A(w1,y,T-T1)} = A(x,y,T)and gi?W=g^y . Hence by (2.26) wl 1 υ x 1 10 '[o,τθ] -(⅛τT<θ)i = -(⅛°>i - aAix'y'T’ ∂x∙ 0 0 By sublemma 2 d l________ -Ϊ.Σ“· bθ∖x,y) = 2^^det JVπ(b)) 5 e * aθ∖x,y). The bounds on the derivatives of aθ(x,y), a∣ (x,y) and A(x,y,T), Ayy(x,y,T) ≥ constant > 0, (4.25) in Lemma 7 and Hörmander’s stationary phase theorem imply: (6∙47) ,Tq) Tθ (±i It 9A(x‰,w-----bθ υ(x,w) + JrA) 2π B(b,6θ) (⅛ a(x>w,T0) bθ 0(x,w) a1 θ(x,w) - exp(-λA(x,w,Tθ)) T-Tr ∕ ∖ \ö T — T∩ , a1 (w,y)x fc) b0 °(w>y) (1 + ~---- Â--------- ) exP(-λA(w,y,T-Tθ ))) bo (x>y) + i Λ’y^ + 0Φ) eχP(-λA(x,y,T)) dw = (⅛)2 1 A uniformly for (x,y,T,λ) ∈ B(yθ,6θ) × B(b,<5) × [2Tθ,oo) × (0, ∞), where -116- (6.48) cjr(χ,y) = bj(x,y) (a1θ(x,w) + a1 θ(w,y) ÷ (det((Aww(x,w,Tθ) + A((,w (w,y,T-Tθ))∣ _)) lw-w z0A(x,w,T∩) T∩ T — T∩ a Tn T—T∩ I ° bo o(χ<w)bo 0(w>y)) - ⅛boθ(χ>w)bo 0(w*y)]∣ [l1(—gχ. __ w=w. rp Q d____________1 -J ∑ wι = : 22(det Jv,,(b)) 2e i= 1 aT(x,y). Now (1.36) follows from (6.45), (6.46), and (6.47). The estimates in (1.37), (1.38), (1.39), and (1.41) are included in Lemma 7, in (7.8) below and Lemma 3.2. (1.40) follows from βA⅛√2 _ ⅛>⅛b) = ,0⅛J-(o)-g×-b(O))i .te×y(T, _ -.∖T),i + j2 (s×y(5)-8χbw,.ds . 0(e “12, — -ω τ + ∫θ (V'(g⅜y(s)) - V,(gx,b(s))i ds = 0(e "l2) by (2.26), (2.20), (2.14), (2.31), and (2.42). -117- §6.6. A Proof of Theorem A.l. ,H(λ) We only do (1∙34) for e ,H(λ) (x,y). The proof for follows since the estimates (6.6), (6.7) and upperbound in (1.17) of e τ H(λ) I ο — -l 1 T I that we use below, are also valid for l^- e (x,w∣) . (x,y)∣ then e -T H(λ) 1 λ (x>wl)> (See (6.6), (6.7) and (6.12).). Let δ ∈ (0, 6]. For x ∈ B(xθ, 0) y £ B(a, 6) U B(b, 6) and T = (n + 1)T^ with Tf ∈ [Tθ, 2Tθ] we write (6.15): -T ( J B(a,iθ) + I B(b,60) H(λ) H(λ) -T λ (χ,y) = ∫ (*θ e λ (wi,wi+1))dw1,...,dwn + I Rd∖(B(a,5θ))UB(a,^6θ) I∣ + I∏ι ÷Vj which defines I∣ II∣andVj )e -T n As before in (6.18) we write I1 = In + 53 j=2 j where -T 11j = H(λ) H(λ) — nT 1 λ (χ,w1)e 1 λ (w1,y)dwχ I wk∈B(a,0θ) for k = l,...j-l 'i=0 WjgB(a,<5θ) H(A) 1 λ (wi, wi+ι)) dw1,...dwn -118- and H(λ) -T η 1 λ (wi, wi+1)) dw1,...dwn. ιr e I ⅛ — i = 0 wk∈B(a,60)k = l,...,n 1 Also III-∣ = HI∏ + ∑ ΓV∙ with j=2 j ∏ιn = J (.%s0e wk∈B(b,6θ) for k = l,...,n 1 and . (iZ∏e IV∙ = ∫ wk∈B(b,<S0) for k=l,...j 1-θ WjgB(b,0o) _T 1 λ (wp wi+l)) dwp ∙ dwn _T ÏÎÜ) 1 λ (Wi’ wi+l^) dwl,∙∙∙dw∏ We estimate ∏j, IVj and V∣ as before and we get: (6.49) n Σ ∏j = 0(exp(j=2 >(x,a) + (6.50) n Σ ivi =: 0(exp( j=2 p(x,b) + sup {p(z,a)} + kg])) (1 ■+ z∈B(a,06θ) sup {p(z,b)} + kg])) (1 z∈B(b,∕75θ) and V1 < k7 exp(-λ( min [ sup {p(×,c) + p(c,z)} + kg])) i ‘ c∈{a⅛b} z∈B(c,∕36θ) If y £ B(a, 6q) then using (6.19) lemma 8.2 and (6.6) gives 1∏≤k7 / wk∈B(a,δ0) C⅛01((⅛)2 b01<wP wi+ι)e ( *’ 1+υ 1^1 + θΦ)) 1~u e -λ(A(wn,z,s)+kg) dwp. ..,dwn -119- - λA(x,wn,nT1) —λ(A(wn,z,s)+⅛)tjwjl(l÷ J nT1 =kτ J, « JA)2b° 1'xιw",c wn∈B(a,5θ) Οίψ» A z -τe∩(a) -λA(x,z,nT1 +s) τ = 0(e θ e 1 ')(l + 0φ) for any z ∈ B(a, /?5θ) and s > Tθ. Take s -→ θo and inf over z and you get: (6.51) In = 0(the R.H.S. of (6.49)), if y £ B(a, 6θ). If y ∈ B(a,6θ)∖B(a,∕J6θ) we expand: , H(λ) 1 λ = (⅛p bJ 1(wn,y) e-λA(wn,y,T1)cι + θφ) uniformly for (wn,y,Tj) ∈ B(a,δθ) × B(b,6θ) x [τθ? 2τθ] and then τ _ ∩f ~T0e0(a) -λA(χ,y T) τ 1n - 0(e e ’y’ 1(1 + θφ^ Since (see (1.38) and (Ï.39)) A(x,y,T) = p(x,a) + p(a,y) + o(e~Wl^^T) and p(a,y) = 1 .£ u,.(a)(yi - bj)2 + 0(∣y-b∣3) in coordinates such V"(a) is given by a diagonal matrix: V,,(a) = Ω2 where Ω = dιag(ω1(a),...,ωd(a)) with 0 < u,1(a) ≤ ...≤ ωd(a) -120- we have ln = 0(e-τe0<≈)e-A(p(x,a)+7S2) (1 + θφ) uniformly for x ∈ B(xθ,6) y ∈ B(b,6θ)∖B(b,6) as T,λ → ∞. Now we get by (6.49) γ nz -τeθ(a) p-λ(p(x,a)+γ52) In = 0(e e ) uniformly for x ∈ B(xθ,δ) y ‡. B(a,δ) as T,λ → ∞ and → 0. Similarly, we get ιπn = 0(e^ Te»(b) e-Ve(χ,b)+τJ2)) uniformly for x ∈ B(xθ,6) y £ B(b,6) as T, λ —► oo and y → ∞. Collecting terms we get: H(λ) -T λ ∕ ∖ n I ∙ r -τe0<a) -λp(x,a) -Teθ(b) -λp(x,b) —Λγ<52. λ (χ,y) = 0 (min{e υ e rv , e υ e v j} e ' ) uniformly for x ∈ B(xθ,6) y £ B(a, 6) U B(b, 6) as T, λ -→ ∞ and y → 0. -121- 7. A Proof of Lemma £. §7.1. Asymptotics for a Gaussian path integral. Our goal is to prove the asymptotics of F(x,y,T,λ,r) in (6.1). By sublemma 1 in Chapter 5 (7.1) d F(x,y,T,λ,ψ) = (27ψ)2 exp (-λA(x,y,T)) τ _I _I E⅛p(-λ∫o(v(g×y(t) + λ 2z(t))_v(gxy(t))_A 2v/(gxy(t))z(t))dt) χ(Z ∈ Cθθ [0, T] : λ _1 2 ∣∣Z∣∣ ≤ ‡)] for r ∈ [0, 6θ]. Now Taylor’s formula implies the exponent at the integrand is given by (7-2) 1∫ <V"(g*y(t) + Sλ 2 Z(t) Z(t), Z(t)) ds = ⅛ (v"(g×y(t) + λ 2 η(tγ> z(t) ∙ z(t)) 1 where ∣τ7(t)∣ ≤ ∣Z(t)∣ for all t ∈ [0,T]. When λ '2z ∣∣Z∣∣ < A each matrix lL _1 1 oo∣0,T] - T element at V,,(g^y + λ 2 rf) — V,,(g^y ) is 0(‡) uniformly. Hence the expression in (7.2) can be estimated from below by (7.3) ((V,,(gψy) + W!p) Z ∙ Z) where Wψ is a symmetric continuous matrix function on [0 ∙ T] with Γ T 0 ∣Wiλ(t)∣dt = 0(r) uniformly. 1 -122- By Lemma 4 there is a rθ ∈ (0, δθ] such that the Dirichlet’s operator + V"(g⅞y(t)) + W⅜(t)) > 0 on Dθ(0,T) (7.4) dt for all (x,y,T,r) ∈ B(xθ , όθ) × B(b,6θ) x [Tq , ∞) x [O,rθ]. We will now evaluate the Gaussian integrals Ej(exp (-i ∫θ <(V"(g×y) + W⅜) Z,Z) dt)). Recall we assume (1.34) V,,(b) = Ω2. Sublemma 2. if (x,y,T,r) ∈ B(xθ , όθ) x B(b,iθ) x [Tθ , oo) x [0,rθ] so that (7.4) holds, then _d rpι 2 e£ (exp (-^ ∫ <V,,(g*y) + Wij-,) Z,Z) dt)) = b^(x,y) where (7.5) T (7.6) bT(x,y) = det(Cr,y,τ(0))) _1 2 and C(t) = C?’y,T(t) is the matrix solution of { (7-7) C(t) = (V',(g*y(t)) + w⅜(t)) C(t) with C(T) = 0 and C(T) = -I Moreover (7.8) - —— trace Ω b7(x,y) = 22 (det Ω)2 e 2 a^(x, y) and a7(xy) = (det Xx,b(0)) 1 2 (1 + 0(e →1T + ∣y _ b∣ + r)) = = ag,(x,y)(l + 0(r)) uniformly. -123- Here Xxb and 6∣ are as in Lemma 2 so det (X X ’ b (0)) > constant > 0 on B(xθ , 6θ). 2. If z = g^y(tθ) for tθ ∈ [0,T] then Remark. We obtain (7.5) by reducing to a similar problem for the Brownian motion, which is solved in Truman [1]. See also Montroll [1] who proves (7.5) in one dimension. (7.9) necessarily holds if our expansions of the heat kernel satisfies the semigroup property. Similar enters for short time asymptotics of diffusion processes on a Riemannian manifold, see Molchanov [1] proposition 10.4 and A. Bellaiche [1] proposition on 8.12. Proof of Sublemma 2.1. Let E( ∙ ) denote the expectation with respect to Brownian motion then (Truman [1], section 3): If (7.10) A(t) = (V"(g×y) + W⅝) A(t) on [0,T] with A(T) = I and A(T) = 0 has a nonsingular matrix solution and for b ∈ Cθ[0, T] (7.11) then (7.12) E(m(b)) = (det A(0)) 1 2. -124- With X = (7.13) and Y = Y*y^ as in (2.45) of Lemma 4, Chapter 4. A(t): = (Y(t) - X(t)(X(t))- 1Y(T)) (Y(T) - X(T)(X(T))^1Y(T))~1 is the unique solution of (7.10) and one can see from (2.22) and (2.46) that it is nonsingular. Write (7.14) E(m(b)) = ∫ E(m(b)∣b(T) = c) dc Rd and let B(t) be the unique (recall (7.4))) solution of (7.15) B(t) = (V"(g*y) + Wir) B(t) on [0,T] with B(0) = 0 and B(T) = I. For c ∈ Rd let fc(t) = B(t)c then fc(0) = 0,fc(T) = c, fc ∈ L2 and b ∏∙ b + fc gives a bijection Cθ θ [0,T] → Cθ c [0,T]. The Radon-Nikodym is given by (see (5.6)) J(b) := exp(-i JT ∣fc(t)∣2 dt) 2 0 e×P(∫τ fc(t) db(t)) 0 Using E( ∙ ∣b(T) = c) = (2τT) (det A(0)) _d 2 eJ( ), (7.12), (7.14) and (7.15) gives _1 2 = E(m(b)) = ∫ E(m(b)∣b(T) = c) dc Rd = ∫ E(J(b) M(fc + b) I b(T) = 0) dc = (2πT) _d 2 ∫ exp(-i(B(T) c,c)) Ej'(exp(-i ∫θ ((V"(gτy) + W⅝) z,z))) dc -125- = T _d _1 2 (det B(T)) 5 τ EJ (exp(-i ∫θ ((V"(g*y) + Wrp) z,z) dt)) or The L.H.S. of (7.5) = (det [A(0)(B(T))^1])~2. (7.16) From (7.13) and (7.15) one can show det [A(0)(B(T))^^1] = det C(0) where C(t) := Cf,y,τ(t) = -(Y(t)-X(t)X~1(T)Y(T)) (Y(T)-X(T)X-1(T)Y(T))-1 which satisfies (7.7). To show (7.8) one uses (2.22), (2.23), (2.46), (2.47) and (2.18) but we leave out the details. Proof of Sublemma 212. χ,y,t + t0>∙ If i = ⅛⅝> then g×⅛) = ⅛y(t>∣[0,t0) “d ⅛-⅛ « = ⅛" 5 where 0i - V,,(g-p,y)Cι C∣(tθ) — 0 and 1, bθθ(×^) = (det cW) By par^ X = X×<'-τ and V = γx'y'τ ci<0> = -(Y(0) - x<°>x^1 go with c1(t0) iγ(⅛) - *<*o> Wo»'1 Y(‘o)) (tθ) -1 . Similarly y(t0)) T —1∩ b∩ θ(z,y) (det C2(O)) 1 2 -126- wberβ C2(0) = -(V(ιθ) - X(tθ) X 1(T) Y(T)) (Ÿ(T) - X(T)X"1(T) Y(T))-1 „x in Lemma 2 we see ∕2∙2β) *7j∙θtf1 'k _1 _1 l z ι t∩)∣z=z C⅝)-*⅝><χ(°>"lγ^ (γ(to)-χ<to) (χ(°)) Y(o)) Azz(*’ ’ .τ0)∣z=z= -(*(V-γ⅝)(γ(τ))'lχ∞> (W-γMγ∞> a∏i — 1 YZ'TλA~l X(T)) J(,(*' Azz l y,' Ι,lje c of (7∙θ) = (det N) where ,H∙s' n ≈ (Y(0)-X(0)X-1(tθ)Y(tθ))(Y(tθ)-X(tθ)(X(tθ))-1Y(tθ))~^1 [(Y(tθ)-X(tθ)(X(0)-1 Y(0)) (Y(tθ)-X(tθ)(X(0))-1Y(0)Γ1 ^(X(to)-γ(to)(γ(τ))~lχ(τ)) (X(to)-γ(to)(γ(τ))-lχ∞)'lj -lx (Y(tθ)-X(tθ)(X(T))-1 Y(T)) (Y(T)-X(T)(X(T))-1Y(T)) and we need to show det N = det[-(Y(0)-X(0)X^ 1(T)Y(T))(Y(T)-X(T)X~ 1(T) Y(T))'1]∙ Actually N = -(Y(0)-X(0)X~1(T)Y(T)) (Y(T)-X(T)X~1(T)Y(T))-1 which can be proved by straightforward calculations. -127- §7.2 More convenient path space measures. 2 By (7.4) + v"(g×y) + W⅜) > 0 on Dθ(0,T) for all (x,y,T,r) ∈ B(xθ , 6q) × B(b,5θ) × [Tθ , ∞) × [0,rθ] and recall we studied it’s Green’s matrix G*,y,T jn Lemma 5 in Chapter 2. Now we have Sublemma 3. If (x,y,T,r) ∈ B(xθ , 6θ) x B(b,5θ) × [Tθ , oo) × [O,rθ] then (7.17) Ej,(zi(s) Zj(t) exp(-i ∫θ ((V,,(g^y) + W⅛) z,z) du)) = = (Gr,y,τ(s,t)).^ Ej,(exp(-i ∫θ ((V,,(g^y) + W⅛) z,z) du)) hence by sublemma 2 (7.18) T _d τ 2 EJ [zi(s) zj(t) (exp(-i ∫ ((Vπ(g^y) + W^) z,z) du)] = b^(x,y) (G*’y’T(s,t)) for s, t ∈ [0,T] and i, j ∈ {l,...,d}. v 7ij (7.19) Remark. To evaluate integrals with more general polynomials in z than in (7.8), e.g., for d = 1 1 := ⅛ (στ γ(3) (⅛y(t))⅜)dt)2 eχp(-⅛∫τvπ(gτy) + w⅛) z2(a) du)) rp2 0 0 one can use Lemma 20.4 in Simon [5] that says: Let X^, ..., X2k be jointly Gaussian random variables. Then -128- <Xχ where ∑ X 2k^ ^~ <W"<∖∙x⅛> ∑ pairings denotes the sum over all (2k)! ∕t^i k! ways of breaking (1, ..., 2k) into ∙ ∙ LJCLAL lllgo ∕ XvT ∙ k pairs. So we get with G = Gr,j, (see also Mizrahi [1]) I = b? (χ,y) iτjτ V<3⅛(.)) v<3∖⅛'(.), 0 0 (9G(t,t) G(t,s) G(s,s) + 6G^(t,s)) dt ds. Since G(t,s) = 0(e integrand is 0(e ') uniformly the — ω1∣t-s∣ τ, x i ) and hence 1 = U(bf (x,y) i) Generally if α∣, ..., αn ∈ Nθ and ∣α∣∣ + ■·· + ∣an∣ = 2k we get (7.20) 2eJ, T = Σ j0 " j0¾ "■ it») dtl '" eXP(~~2 + WT z’z> du bT(χ,y) ∫τ ... JT G( ) ∙∙∙ G( )dt1 -. dtn = O(bT(x,y)Tn) finite 0 0 unformly for (x,y,T,r) ∈ B(xθ , Sq) × B(b,6θ) × [Tθ , ∞) x [0,rθ]. We base a proof of sublemma 3 on the following Proposition (Ellis and Rosen [1], Lemma 4.4). Suppose that A∣ is a symmetric strictly positive trace class operator on a real separable Hilbert space H∣ and Λ is a symmetric operator in B(Hj) such that Aj~^ + A > 0 on D(AJ-1). defined and > 0. Then ∆∣ := det(I + A∣ A) = det(I + JA^ Λ J Aj) is well- -129- Also B j := (A j 1 + Λ) * exists and is given by B∣ — J A^ (I + J A^ Λ B1 is the covariance operator of a mean zero Gaussian measure Pn 151 on Hj and dpB1(y) = (7∙21) eχp(-2 (Ay,y» dpA1(y) Remark. Here B(H∣) is the space of bounded linear operators and <∙, ∙> is the inner product on H∣. A Proof of Sublemma £ (see also Davies and Truman [1]). Let ∕irp be the Brownian bridge measure on Cθθ([0,T], Rd) as in (5.10). Then (7∙22) ∫zi(s) zj(t) dμτ(z) = <5ij s(l - ‡) =: (p(s,t))ij ifO≤s≤t≤T and i,j ∈ {I,..., d}. Define Aχ : L2([0,T],Rd ,dt) → L2([0,T],Rd,dt) by Aj f(s) = ∫J μ(s,t) f(t)dt. ∕ >2 _ -1 Then A, = I —on [0,T] with Dirichlet’s boundary conditions) 1 is symmetric, v dtz strictly positive and trace class. Now we use an idea from Stroock [1] and Chevet [1], for instance, to embed Cθθ into L 2 and work with the induced measure on L 2 . So let Φ : Cθθ —+ L 2 be the natural embedding, which is continuous. Then Aj is the covariance operator for the induced measure P^ := μ-p ο Φ ∖ by (7.22). -130- Define Λ : L2 —⅛ L2 as the multiplication operator f w (V,,(g^y ) + W∑p)f for each (x,y,T,r). Then A is bounded and symmetric. With H1 = L2([0,T]) we get, using (7.4), B1 = B*’y,T’r := on [0,T] with Dirichlet’s boundary conditions) V,,(gψr) + operator of a mean zero Gaussian measure Pg is the covariance 9 on L such that dpB1(α) = ∫⅜ exP(~2 Jθ <(v"⅛τ) + Wt) du) ⅛a (a). Since 1 = Pb (L2) = Γ∆^ ∫ exp( ∙ ) dPA (a) = 1 l2 1 = J∆^ ∫ exp ( )dpτ(z) C00([0,T]) = J^∆χ Ej,(exp ( )) we have (7.23) + = (E?(exp(-^ ∫θ {(V"(g*y) + W⅜) z,z) du)))“1 With a = Φ(z) Ej(zi(s) Zj(t) exp(-^ ∫θ ((V"(gyy) + W⅛) z,z) du)) -131- which along with (7.23) gives (7.17). §7.3 Keeping track of the large T dependence. We will do the usual Taylor expansions (see Schilder [1] and Davies and Truman [1]) paying attention to large T’s. We write (7.1) with r replaced by ψ as F(x,y,T,λ,ψ) = (⅛)2 exp(-λA(x,y,T) (F1(x,y,T,λ,f ) + F2(x,y,T,λ,⅜)) where Fj(x,y,T,λ,⅜) = Ez1 (exp(-λ ∫θ (V(g^,y + λ χyλ) - λι2 χy> z z) - V(g^∙ z xr V∣, ,(g^ y )z)dt)χi(z)) where χi(z) = {z ∈ Cθ0[0,T]= λ _1 2∣∣z∣∣oo ≤ ‡ and (-l)i ∫θ (V(gτy + λ~2 z) - V(g*y) - λ^2 V'(g*y) - z - A∑I <V"(g*y) z,z))dt ≥ 0} Then we expand the integrand in F^ and F2, using exp(x) = n —1 i ∑ ⅜ + R∏(x) i = 0 1∙ where Rn(x) ≤ ∕‰ exp(x) if X ≥ 0 ⅛ιn n! if X < 0, collect the terms and we get (7.24) d F(x,y,T,λ,f ) = λ2 exp(-AA(x,y,T)) (Iθ + Iχ + ⅛ + ⅛ + Jχ + J2) -132- where (7.25) Ik = ⅛ (2τrT) _d 2 eJ ü-λ J0τ(v⅛y + λ~2 z) - V(g*y) - A~2 V'(g×y) z - AZ! (v"(g×y) z,2>) dt]k T -1 exP(~⅜ ∫θ {V"(g×y) z,z) dt) χ(z : λ 2 ∣∣z∣∣0θ ≤ ^)) _d for k ∈ {0,l,2,3}, Jχ = 0((2τrT) 2 eJ (∣-λ∫θ (V(g⅜y + λ~2 z) - V(g×y) - λ~2 v'(g×y) ∙ z - A∑2 <v"(g*y)z,z)) dt∣4 exp[-A ∫ τ (V(g⅜y + A and J2 = 0((2πT) _I _1 2 z) - V(g*y) - A 2 V,(g*y) ∙ z)dt)χ1(z)) _d 2 eJ (∣-λ∫θ (V(g⅜y + λ^2 z) - V(g⅜y) - λ^2 V'(g⅞y) ∙ z - A∑2 <v"(g⅜y) z,z)) dt∣4 eχp(~5 ∫θ (v"(gψy) Z∙,z) dt) χ2(z)) . For the error terms we use λ(v(g*y + λ^2 z) - V(g⅜y) - λ^2 v'(g*y)z - A∑i <v'⅛y) z,z)) = = o(λ~2 ∑ v ∣α∣=3 ∣zα∣ = o(λ~2 ⅛ (l + zP(t))) k i = lV 11 -133- uniformly for λ _1 2 ∣∣z∣∣θo < ψ. Therefore we have a crude estimate —— T d J1=0((2πT) 2eJ[(∫ ∑ (1 + z?(t) dt)4 1 0 i=l 1 exp(-λ ∫θ <(V''(gτy) + w⅝) z,z) dt] ∙ λ,-2 “2 = = 0(b?(x,y) ς∣) = θ(bj(x,y) (1 + 0(r)) ⅛) = θ(bj(x,y) ‰) λ A λ where we used (7.20) and sublemma 2. Similar J2 = 0^bJ(x,y) uniformly. Using (7.5), (7.18), (7.20) Holder’s inequality and (see Simon [2] or Stroock [1]) (7.26) (2πT) _d 2 2 Ej(χ(z : ∣∣z∣∣0θ > ⅛)) = θ(exp(- 6)) one can show there exists cθ > 0 such that (7.27) 5 τ λr2εnk(d) Iθ = (2π)2 bj(x,y) (l + 0(exp(--------^f-)) ’ 11 = -⅛⅛τ) 2 eJ[∫tvi4∖g*y∙,z,z,z,z)dteχp(-J∫τ(V"(g*y) 2,z> dt)] 0 0 + bj(x,y) [0(i) + 0(1 exp(- ^-^-b)] λ2 1 i2 = iΛ (2!)(3!p 2T4 λ γi3⅛y; z’z’z) dt)2 exp(-2 J,j<v''(gτy) z>z) dt)] + b0M (0(X- exp(- and I3 _ θ(bj(χ,y) . Λ -134- v‰ ς (χ,y,.",y y)= ) Σ Here 8xα α, We will give the details for I∣ but first we note by (7.24) and the estimates for Il , ∑2 , I3 , T1 and T2 above we have F(x,y,T,λ,^) = (^)2 bj(χ,y) [1 + - + 0 (⅛) Λ = <-<-⅞¾ where (7.28) bj(x,y) af(x,y) = T 2E∏ ⅛ ^2 Go γi3⅛y; z’z’z) dt)2 ~ ⅜ Jθ γ(4)(ετy; z’z’z’z)dt eχp(-⅛ Jθ (v,,(gτy) z,z) dt) Here bj(x,y) is as in sublemma 2 and a∣"(x,y) = 0(T) follows from Remark (7.19) but we refer to the proof of lemma 7 in Chapter 11. So now the proof of the asymptotics of I∣ in (7.27). By Taylor’s theorem (Hörmander [1]) and (7.25) -135- (7.29) 2 Eæ([A ∫J I1 = (2πT) (g*y; A 2 z, A 2 z, λ 2 z) ÷ ⅛i ⅛y(t)5 λ"2 z,..., λ~2 z) + ⅛ (g*y (t); λ~12 z, _ λ-2 z) + ∫ 1 (6) -I (gψ, + sλ 2 z; λ -I 2 z, ..., λ -1 2 z)(l — s)^ ds) dt] exP(~5 iT (V"(gψy) z,z) dt) χ(z : i⅛2 < ‡)) =: ι1,1 + I1 2 + ∑1 3 +I1 4 λ2 0 which defines I1x,l1 ∙∙∙ Ii√± 1 , (in the same order), By symmetry (7.30) Ij , = 0 = I13 . _1 Since A 2 llz∣lσo ≤ ψ, derivatives of V are uniformly bounded on compacts and ∣z0(t)l < Σ z.θ(t) if ∣α∣ = 6 we get i=l (7.31) I1 4 = 0(λ~2 £ J1 T~2 Ej(zθ(t) exp(-I JT(V"(g£y) z,z) dt) i=l = 0(λ^^2 0 1 i 0 <1 τ ∑ Jθ bθ (x,y) G?j(t,t) dt) = bθ (x,y) 0(^) uniformly. (7.27) now follows from 136- (7.32) λ~1τ 2 eJ, (∫θ √4∖g*y z,z.z.z) dt exp(- U T Ο <v''(g×y) z,z) du) χ(z : A = O(A^1 _1 5 l∣zll0θ > ‡)) ό T? *--_ rτ∕∖r∣∣f×y> £ ∫ T 2 Ej⅛(t)exp(-A1 f(V''(g^ y ) z,z) du) χ(z : λ i= l Ο 2 0 i 2 ∣∣z∣∣ > ‡)] 1 a ... _d ∕Z±εO — -Τ 2 ηΤ/ί 4,.xJ Τ = O(λ-1 ∑ ∫ [Τ 2 Ej((z4(t)) i=l Ο 1 /Τ + £r»\ Τ .. XV exp(-5(~"T-0)∙f0 (v (gτy> z'z> du)] .d Τ 2 eJ(z = A (. -Τ X -1 2 ∣∣z∣∣ >Χ) ε0 T+εf Τ x,y,T.t .4 Jτ+ε(2 = 0(A~1 ∑ JT(bJn(x,y)(l + (G^∩,i(t,t)?;)) dt ;_1 θ Ο ϋ b* (exp (->A(d)) T+.Q = 0 I rpθ Λ z±⅛h ∖ 2Τ4 where we used IV^4∖g^,y-,z,z,z,z)∣ = 0 ( ∑ z^(t)) i= l T-f"£n T-{^5∩ Holder’s inequality with P = —≈τ-y and q = —≈—- where without loss of much 1 t∩ -137- generality T+e∩ Ι, e (______ U) -τ-θ ≤ 2 for T ∈ [Tθ , ∞) so (z^(t)) T ≤ 1 + zf(t), sublemma 2 with Wψθ(t) = ψ V,,(g^y(t)), (7.20) and (7.26). (7.29)-(7.32) imply (7.27) which completes the proof. §7.4 A sketch of a proof of Lemma 8.2. Equation (6.3) of Lemma 8.2 follows from: B(b∕0)) X B(b,6θ) If (x,y,T,S) ∈ (B(xθ , 6θ U X [Tq,∞) × [Tq , oo) then inf{A(x,z,T) + A(z,y,S)} = A(x,y,T+S) is attained at the unique point z = g^y^g(T) ∈ B(b, ^) (uniqueness by Lemma 2 Chapter 2 and the estimate by (2.39)) and d d ∫ (À)2 b0 (x>z) exp(-λA(x,z,T)) (^)2 bθ(z,y) exp(-λA(x,z,S)) dz B(b,6θ) d = (<⅛)2 b(f + S(x>y) exp(-λA(x,y,T + S)) (1 + 0(^)) uniformly for (x,y,T,S,λ) ∈ (B(xθ , <$0) U B(b, 0θ)) × B(b, όθ) × [Tθ , ∞) × [Tθ , ∞) x (0 , ∞). A Proof uses Hörmander’s stationary phase theorem in §4.3 with k = integer part of (^ + 2). The argument of the important uniformity is the same as in (6.24), that we refer to. d The prefactor being follows from (4.10) and Sublemma 2.2. -138- 8. A Proof of Lemma 9. It suffices, by lemma 8.1, to show (8.1) τH(A) G(x,y,T,λ,r) := e Λ (x,y) - F(x,y,T,λ,r) = d = 0(λ2 exp(-AA(x,y,T)) exp(-λkg r)) for some kg > 0, uniformly. By sublemma 1, chapter 5, (8.2) d G(x,y,T,λ,r) = (2pp)2 eχP (-AA(x,y,T)) T -i E^(exp(-A ∫θ (V(g*y + λ 2 z) - V(g×y) _1 -λ _2 2 V'(g*y)Z)dt) χ(z : r < λ 2∣∣z∣∣)). For small rθ > 0 and large Rθ , yet to be chosen, we write χ(z : r < λ _1 _I 2∣∣z∣∣) < χ(z : r ≤ A 2∣lz∏oo < ⅛) + X(z : rθ ≤ A _1 2∣∣z∣∣ ≤ Rθ) + X(z : R0 ≤ A _1 2∣∣z∣∣) where y(z : A) = {J ∣ . Accordingly we write (8.3) G(x,y,T,A,r) ≤ G1(x,y,T,λ,r) + G2(x,y,T,λ) + G3(x,y,T,λ) -139- Estimating G2 uniformly is the main problem now since we know how to handle G^ from chapter 7 and we estimate G g using (7.26) that says (8.4) (2πT)~2 E?(Z : IMILoo[0/r] > 0) = 0(exp(~⅛⅛ (recall T ∈ [Tθ , 2Tθ].). More precisely, we pick R,θ such that Ro > 2 sup∣gψ,y(t) - ((1 - i) X + i y)∣ (x,y,T,t) ∈ B(xθ , <5θ) × B(b,6θ) × [Tθ , 2Tθ] × [0,T] and R∩ k(d) 9 —⅛ψ— > sup (A(x,y,T)) + 100 <5∩ 8T0 (x,y,T) U and we get (8.5) Gg(x,y,T,λ) = 0(exp(-λ(sup(A(x,y,T)) + 100 5^)) uniformly, but we leave out the details. Now fix rθ ∈ (0,0θ] such that (8.6) ∫θ (r∕2 + (1 + rθ) (Vπ(γ)τ∕,η))dt > ⅛ ∣M∣l∞[0,T] for all η ∈ Dθ(0,T), γ with ∣∣γ - g^y „θθ ≤ rθ and (x,y,T) ∈ B(xθ , <5θ) × B(b,6θ) × [Tθ , 2Tθ]. -140- Then using the mean value theorem to estimate the exponent in this integrand, Holder’s inequality with p = (1 + rθ) and (8.4) (similar to (7.32)) one can show G1(x,y,T,λ,r) = 0(λ2 exp(-λA(x,y,T)) exp(- (8.7) Ak(d)r∩ 7 0 .)) uniformly for 2iθμ + rθj (x,y,T,λ,r) ∈ B(xq , όθ x B(b,δθ) × [Tθ , 2Tθ] × [l,oo) × (0,rθ] (after taking slightly smaller όθ). Finally we look at G2 ∙ Put L:= {z∈ C0 θ([0,T]) : rθ < Uzl∣oo ≤ I⅛} (8.8) and set f(z) = fx'y,τ(z) = ∫θ(V(⅛y) + z) - V(g*>y) - V'(g⅜y>) dt (8.9) near L and extend it to all of Cθ θ([0,T]) so that it is bounded and continuous into R. Also set (8.10) H(x,y,T,λ) = Ej(exp(-λf(A _1 2 z)) : z ∈ L) and Λ(γ) := ∫J(¾ 72 + V(γ)) dt if γ ∈ Cx,y([0,T]) and γ ∈ L2([0,T]). Then d G2(×,y∕Γ,λ) = (2pp)2 exp(-λA(x,y,T)) H(x,y,T,λ) and we will show there exist εθ > 0 and λθ such that -141- (8.11) H(x,y,T,λ) ≤ i e Af° if λ > Aθ for all (x,y,T) ∈ B(xθ , 0θ) × B(b,6θ) × [Tθ , 2Tθ]. That will complete the proof of (8.1). m I9 1 J With ε = i and Pf (A) = Pψ(λ^ A) (μ^ is the measure corresponding to τ Ez ( ’ )) we have according to Schilder and Varadhan (see §5.2) jrj- log H(x,y,T,A) _ A-→oc = ∏m ε log ∫ exp(-ε-1(f(z)) dpf(z) ≤ - inf (f(z) + I(z)) ε→0 L z∈L where I(z) - 5 jτ∣z∣2 dt∙ £ 0 By the Euler-Lagrange equations f(z) + l(z) = ∙Λ(gjy + z) - -4(g^y) and therefore by (2.33) inf (f(z) + I(z)) ≥ k1 min(rθ , z € JL = k1 r(j and we have: For each (x,y,T) ∈ B(xθ , 6θ) x B(b,6θ) × [Tθ , 2Tθ] there are constants C(x,y,T) and λθ(x,y,T) such that (8.12) H(x,y,T,λ) ≤ C(x,y,T) e if λ > λθ(x,y,T). -142- To prove uniformity for (x,y,T)(×1 ,y1 ,T1) ∈ B(xq ,6θ) × B(b,6θ) × [Tθ ,2Tθ] (s°i ≤ T_ ≤ 2) we denote T1 g(t) = gyy(t ÿ-), g1(t) = gτiyι(t), II ∙ II = II ∙ II 1 1 ∞ L∞([0,T1]) ∣H(xχ , yχ , Tχ , A) — H(x,y,T,λ)∣ = and we write T τι -I -A = ∣Ez 1(exp(-A ∫θ (V(g1 + A 2 z) - V(gχ) - A 2 V,(g1)z]dt) χ(z : r0 ≤ A 2∣jz∣∣oo ≤ R0)<it - T τi [V(g + (λ≠) il - Ez 1(exp(-λ i ∫ il 0 -A 2 z) - V(g) - (λX) 11 -A 2 V,(g) z] dt) _1 ∣∣z∣'oo~ R0)} ~ h(xi ,yj >τι ’ A)(l + x(z ∙ r0 ≤ sup ro≤λ 2I∣z∣∣oo≤R0 τ1 _i _i _i [exp(-A ∫θ ([i V(g(t) + A 2(ιJ-) 2 z(t)) - V(g1(t) + A 2 z(t))] - ⅛ξ∙ V(g(t)) - V(g1(t))] -(λ _1 1 2) [((i)2 V,(g(t)) - V'(g1(t))) z(t)]) dt)] τ _I _I + E⅛p(-λ ∫θ (V(g*y + A 2 z) - v(g*y) - A 2 v'(g*y) z) dt) -I (y(z : 2 _1 _1 2 rθ < A w ≤ rθ) + χ(z : Rθ ≤ A 2∣∣z∣∣ OO v v 2∣∣z∣∣ ))] oo -143- 9 where we used z(at) = a z(t) (see Simon [5], chapter 2) and {z ; r0 ≤ (^) 2 l∣z∣lθo ≤ R-θ} ≤ {z ■ 2 U {z : rθ ≤ λ ×y, _1 2∣∣z∣∣ τ∖ 1 2 2 rθ ≤ λ ≤ Rθ}} U {z : Rθ ≤ λ 1 2∣ zl∣z∣∣0o ≤ γq) _1 2∣∣z∣∣ } . xιyι "loo[0,T∣] → θ as (x,y,T) ^^+ (xj » >,ι » Tχ) By (2.37) ∣∣grp ( ∙ ψ-) — gj^ Hence for (x,y,T) near (xχ , yχ , Tχ) ∣H(x1 , y1 , T1 , λ) - H(x,y,T,λ)∣ ≤ H(xχ , yχ , Tχ , λ)(l + 0(eλε)) + 0(exp(- 4τθ(1 + rζ))) + 0(exp(-λ 100 ⅛) where we used also (8.7), (8.5) and the definition of G<χ . Hence we can use the same exponential bound in a neighborhood of (xχ , yχ , Tχ) and so (8.11) follows from (8.12) by compactness. -144- τH(Λ) We want an estimate of ∣j^-e 3xj (x,y)∣ s*m^ar f° e (x,y) ≤ (27rT^ bOIQ- 9. Differentiating the Heat Kernel. exp(- ⅜^~y∣2) when ∣x∣ and T are bounded and a replacement for the upperbound in H(A) (see (1.17)) lim ⅛---- 5---- = -A(x,y,T) u.o.c as λ → ∞ . λ→∞ ^ Proposition 2. Let V be a double well with ∣V,(x)∣ = θ(e A-∣x∣ 2 ) a-∏d i ∈ {1 , ..., d}. 1. With k(d) as in (7.26) d j∣H(A) (θ∙υ λ 4⅛e E⅛⅛i2 + ∫θ exp(-λ ∫ 2 =(2⅛)5eχp(-⅛≠) 2T (d - ‡) X +4 y + a V((l — ‡)x + ‡ y + A 0 1 1 2 z(t)) (1 _ dt} 1 22 z(t)) dt)] whenever (x,y,T,λ) ∈ Rd × Rd × (0,∞) x [γΛJ , ∞) and obeys κ(d) (9.2) ∣^ i ⅛e ä = 0(λ τ∏(A) λ (χ,y)' = _ .Ο ∣χ, _ y. I 2 exp(------2τy ') {^ i^T * + τ2 exp(4A(∣x∣2 + ∣y∣2))}) uniformly. -145- 2. ___ (9.3) lθε∣i⅛e τH(A) λ (×>y)l lim -------- î------ ?--------------- ≤ —A(x,y,T) u.o.c. in λ→∞ λ Remark. Let |x| < x x (0,∞). , ∣y∣ > R and T ≤ T∣ for some R and T∣ . Then -T H(A) (9∙4) λId ⅛e (×,y) τH(A) A , ≤ 0(exp(- ⅛- )) X. uniformly for λ > some λθ , since in (9.2) λ2 exp(- ⅛⅛) exp(4A(∣x∣2 + ∣y∣2)) = »(A? exp[- ¾√1 - S2AT))) 1 = 0(λ2 exp[- ⅛- (1 _ 32AT -^⅛) 4T Remark (9.5). ∕ A∣x∣2 When ∣Vz(x)∣ = 0(e ) one can use Proposition 2 to prove the second part of (1.20). p(x,b)} u.o.c in . Namely, for each n lim l°g∣^7⅞(χ,A)∣ < —min{p(x,a), A→∞ λ -146- H(λ) ,En(A) -T Write ∣VΩn(x,λ)∣ < e λ ∫∣Vx e λ (x,y)∣∣Ωn(y,λ)∣ dy for x in a compart which is inside B(0 , (1.21), that says ∣Ωn(y,λ) ∣ = 0(e for some large R. —λco∣y∣ When ∣y∣ ≥ R use (9.4) and ) for ∣y∣ and λ large. For ∣y∣ ≤ R use (9.3) an, (1.19) that says lim A(x,y,T) = min{p(x,a) + p(a,y), p(x,b) + p(b,y)} u.o.c. 1 →∞ and the first part of (1.20), i.e., Finally we recall by (1.2) E f A) Jim λ-→∞ < — min{p(x,a), p(x,b)} u.o.c. = 0(1) as λ —► ∞ , so fixing a large T and taking λ→oo gives the result. Proof of Proposition 2.1. Write the Feynman-Kac formula (5.5) as (9∙5) e _ rp,H(λ) v< ∕ Q 2 λ (x,y) = (^pf)2 exp(-λ'x~y' ) Ej,(k(x5z)) T -1 with k(x;z) = k(x,y,T,A;z) = exp(-λ∫ V((l — ⅛)x+iy + λ 2 z(t)) dt) 0 i i whenever z ∈ Cθ q([0,T]). -147- To differentiate we write k(x + εepz) — k(x;z) ∫θ ds k(x÷sei>z)ds (9∙6) ε τ _1 = -z Jθ{Jθ ⅛((1 - ⅛)(χ + sej) + 4 y + λ 2 z(t)) (1 — τj¾)dt) k(x + sepz) ds T g: ((1 - ⅜) X + 4 y + λ --2 z(t))(i - i)dt) k(x5z) → -λ(i0 = : —Af∣(x,∙z) k(x;z) as ε -→ 0 for each z ∈ Cθ θ([0,T]) . By hypothesis ∣V,(x)∣ = 0(e^'x' ) so (9∙7) ∣fjτ((1 - ⅛)(χ + sej) + τy + λ 4A((∣x∣ + ε)2 + ∣y∣2) _1 2 z(t)∣ = 2 L∞[0,T] By (7.26) with ∣∣ ∙ ∣∣oθ = ∣∣ ∙ ∣∣l∞^ tj) we have (2πT) > 6) = 0(exp(- ⅛¾ 2 Ej,(z ; ∣∣z∣∣ ∞ X -148- and we get (9.8) (2πT) 2 Ej,(exp(2A ∣jz,j2 )} ^ ≤ <x> £ ( sup [eχp(2A∣∣z∣∣2 )])(2πT) 2 eJ(z : n ≤ ∣∣z∣∣ < n + 1) n=0 n<∣∣z∣∣00<n + l A ∞ ∞ ∞ exp(2A(n + l)2 eχp(- ° ^d^ι)) = O(∑ 0 A i = O(exp(2A) + g eχp(- (i _ )) = O(exp(2A) + 53 exp(- ⅛^-) = O(exp(^) + T2) when λ ∈ , ∞)∙ Now (9.6), (9.7) and the dominated convergence theorem give ,„m (9.9) Ej,(k(x + εei , z)) - Ej,(k(x≈z)) τ , -------------------i—g-------------------------- ► —λ E∕ (ij(x;z)k(x;z)) as ε whenever (x,y,T,λ) ∈ × × (0,∞) x » oo)∙ (9.9) and (9.5) imply (9.1) and together with (9.7) and (9.8) they imply (9.2) -149- Proof of Proposition 2.2. Recall ⅛ < (x>y) ~~ ^2πT^ e λl _ P exP( 2T ) J¾f (k(x,y,T,λ∙,z)) and (9.1) can be written as (θ∙iθ) 4⅛e THQ) 2 " λ (x’y) = c2⅛>2 eχP<- Alχ-y∣ ¾r1-) (-iτ^ Ej(k(x,y,TA∙z)) + Ej(£j(x,y,T,A;z) k(x,y,T,Λiz)). By (1.17) (see Simon [2]) -T (9.11) lim λ-→∞ log e H(λ) (χ>y) -A(x,y,T) u.o.c. and we only need to look at the second term. X Rc^ X (0,∞) be a compact set and — 2 I(x,y,T,λ,R) = (57τ)2 eχP(~ ^‰4-) Ej(∣ii(z)∣ k(x), χ(z : IIzI∣l∞ ≤ RÀ >> and II = II(x,y,T,λ,R) = χ-y∣2⅝ E pzτ = <2⅛)2 exp(- A∣ ¾^-) 1(∣L(z)∣k(z)χ(z : IM∣l∞m > R^)). Then the second term in (9.10) is 0(1 + II). M∣H Let L Ç -150By Cauchy-Schwartz (9.6), (9.7), (9.8), and (7.26) 1 II(x,y,T,λR) = 0(λ2(ET(exp(4Δ∣∣z∣∣2oθtθ ^)))χ2 (E?(Z : !∣z∣∣l∞[0,t] ≥ R^)))2 = 0(λ2 exp(- uniformly for (x,y,T) ∈ L and R > 0 where Tmax = max {T} . (x,y,T)∈L R∩k(d) Now fix R∩ such that f∏⅛------ > sup {A(x,y,T)} + 1 and then II(x,y,T,λ,R∩) is u 21 max L u out of the story. So now we consider I(x,y,T,λ,Rθ) since ∣x∣, ∣y∣ and T are all bounded for 2 4A(∣x∣2 + ∣y∣2) 2ARq λ J (x,y,T) ∈ L, I(x,y,T,λ,Rθ) = 0(Tz e"''l^l "∙7' 'e u (⅛)z exp(- ⅛⅛) 1 E?(k(z) χ(z : IIzI∣l∞[0,T] ~ and (9.3) follows from (9.11). Ro) = τH(A) λ (x^)) -151- 10. A Proof of lemma 10 §10.1. Preliminaries (sublemma 4) §10.2. Proof of (6.6) and (6.7) of lemma 10 §10.3. A sketch of a proof of lemma 10.3 -152- 10. Proof of Lemma 10. We will use some basic rough estimates on the minimal action the Agmon distance and their paths that we state in sublemma 4 below. These estimates will also be used to prove Proposition 1. §10.1. Preliminaries (sublemma 4). From the definition of a double well follows there are constants Cθ, C∙p C2 > 0 such that (10.1) C2 < V "(x) ≤ C2 if x ∈ B(a,Cθ) U B(b,Cθ) (10.2) C2∣x-c∣2 ≤ V'(x) (x-c) ≤ C2 ∣x - c∣2 A 2t if X ∈ B(c, Co) for c ∈ {a, b} and (10.3) C C2 V(x) > -Ar-Θ ifx 0 B(a,Cθ) U B(b,Cθ) We will assume B(a,Cθ) and B(b,Cθ) are far apart compared to the “diameter” of B(a,Cθ) and B(b,Cθ) in the Agmon metric, more precisely we picked Cθ such that (10.4) inf{p(x,y)} < max{2sup{p(x,a)},2sup{p(x,b)}} (x,y) ∈ B(a,C0)×B(b,Cθ) x ∈ B(a,Cθ) x ∈ B(b,Cθ) We notice if γ(t) ∈ B(c,Cθ) for t ∈ [0,T] and 7 satisfies the Euler-Largrage equations γ(t) = V V(γ (t)) then α(t) := ∣γ(t)-C∣2 satisfies α(t) = 2∣γ(t)∣2 + 2(ι(t)5(7(t)-C)) ≥ 2(W(γ(t)), (γ(t)-C)) ≥ 2C2∣γ(t)-C∣2. This makes easy to get rough estimates for minimal action paths and Agmon gedesics using differential inequalities (Protter-Weinberger [1]). -153- Sublemma 4. (1) C∩ η —C9t ç If ∣x-C∣ ≤------ L 1 then ∣x-C∣ e 2 ≤ ∣gx,(t)-C∣ (i÷⅜) q 2 c 2 ≤ ∣x-C∣ e-Ct and ⅛ l×~C∣2 ≤ p(x,C) < ∣x-C∣2 (2) If 0 ≤ tχ ≤ t2 ≤ T, 0 ≤ α ≤ 1 and ∣gxy(t) - C∣ ≤ a Cθ for i ∈ {1, 2} and ∣g^,y(tj) —C∣ ≤ a Cθ for t ∈ [tj, t2] «hen ∣gγ(.) -CP ≤ -CP "°hbccι⅞^ ÷ ⅛y⅛>-c∣2i⅛⅛¾ ≤ λo2 (3) Co cosh C9T If ∣x —C∣ ≤ Cθ then A(x, C, T) ≤ -f ∣x - C∣ 8ΪπΚΤ|τ (4) IfO≤α≤β≤l then ∣x-C∣ = aCθ A(x, y, T) ≥ ∣y-C∣=∕7Cθ τ>o The proof of this sublemma is in Chapter 12. (β1 - a2) -154- Now we prove (6.6) and (6.7) of lemma 10 will only use rough estimates valid _TH— (10.5) ~th(A) (x, y) and ∣ —e for both e τH(A) ___ a ∖ lira l°g e___________(×.y) λ-*∙oo λ _ λ→∞l°β∣4⅛' (x, y)∣. Namely ≤ -A(x,y,T) thw λ <×∙rf∣ uniformly on compacts (see(1.17) and (9.3)) and: If ∣x∣ ≤ ⅞jι, ∣y∣ ≥ R and T∈[Tq,2Tq] the for i ∈ {l,...,d} exp(-T¾^) (x,y) = θ(exp(-⅛)) (10.6) mH(λ) .1 ; e λ ôxi (×,y)l uniformly for λ > λθ (λθ maybe getting larger) (see (9.10)). Finally we notice one can take 6q smaller and Tθ larger in lemmas 2-9 without a loss of generality. §10.2. Proof of (6.6 and (6.7) of lemma 10. We will assume p (10.7) '-'O. ⅛ ∈ (θ, C0⅛ + J) __I o> 2) -155- and put ^i=⅛Z2 ½ (10.8) Recall Proposition 1 says A(x,y,T) — min{p(x,a) + p(a,y), p(x,b) + p(b,y)} + 0(e _ ) uniformly an compacts in Rd X Rd as T → oo, so we may assume (taking Tθ larger) C15r ∣A(x,y,T) - (p(x,C) + p(C,y))∣ ≤ -⅛ (10.9) if (x,y) ∈ (B(xθ,6θ) X B(C, 0θ)) U (B(C, Cθ) × B(C,Cθ)) J* i<jγ Ί ∕~ι -- ( _ l m <fc, m e {a,D∫ ana i 1θ and (10.10) sup sup{p(x,c) + p(c,y)} + 100 Cj Cθ C∈{a,b}(x,y)∈(B(xθ,iθ)×B(C,Cθ))U(B(C,Cθ)×B(C,Cθ)) τ c√*2⅛> 10 2 sup sup{p(x,c) + ρ(c,y)} + 100 C1 Cθ C∈{a,b}(x,y)∈(B(C,Cθ))2 By (10.1) and (10.2) (10.11) δ V(x) ≥ C1-^β- if X £ B(a,∕J<5θ U B(b,∕J6θ) and for T1 ∈ [Tθ, 2Tθ] x ∈ B(xθ,0θ) and w B(a,6θ) U B(b,0θ) we have: -156- (10.12) A(x,w,Tj) > To g θ no m*n*mal acti°n path gψv enters B(a,∕J6θ) U B(b,∕36θ) and A(x,w,T∣) ≥ inf {A(x,z∣,S∣) + A(z2,w,S2)] (z1,z2) ∈ (B(a,ββθ))2U(B(b,∕3δθ))2 sl>s2 otherwise. In the first case we only keep the potential term of the action and use (10.10) and (10.11). In the second case we only keep the contribution until the first time that a path enters B(a,βδθ) U B(b,∕3δθ) and from the last time it leaves B(a,β0Q) U B(b,^0θ). If (z1,z2) ∈ B(b,0<5θ) U B(b,^60)then A(x,z1,S1) + A(z2,w,S2) ≥ ρ(x,z1) + p(z2,w) ≥ [1]) and the triangle inequality p(x,y) < p(x,z) + p(z,y). By sublemma 4.1 p(z∣, b) ≤ ∣z∣-b∣'2 ≤ 5 iθ2 a∏d since w g B(a,δθ) U B(b,6θ) implies p(b,w) > C 6% I p(k,z) > - 1 θ by sublemma 4.4. We obtain by above ∣z-b|— oθ 4 A(x,z1,S1) + A(z2,w,S2) > > p(x,z1) + ∕,(z2>w) ≥ p(x,b) + ⅛β - ≥ p(x,b) + ⅛ cι 50 by the choice of β in (10.8). to∣∣-∙ p(x,b) - ρ(z1,b) + p(b,w) - p(b1,z2) since p(x,y) = qi>0 A(x,y,T) (Carmona-Simon β2 √ -157- If y ∈ B(b,/?0Q) then by sublemma 4.1 p(b,y) ≤ 2^ ∣y-b∣x <, 2 fp∙fi < 1 p χ2 - 2C1 0 00 ≤ 256 ci*O o ≥ ∕>(b,y) - ⅛ c1⅛ ι.e., and we get A(x,z1,S1) + A(z2,w,S2) ≥ ρ(x,b) + p(b,y) + C1<Sθ for all y ∈ B(b,j55θ). By (10.8) and (10.10) we get =∈⅛b} {p(χ,O+p(c,y)} + ⅛ c1⅛ (10.13) A(x,w,T1) ≥ A(x,y,T) + cιio for all X ∈ B(xθ , 6θ), T∣ ∈ [Tθ , 2Tθ], w £ B(b,0θ) U B(a,6θ), y ∈ B(b,∕J<50) and T ≥ Tθ . Looking at (z∣, z2) ∈ B(a,∕36θ) × B(a,∕3δθ) and y ∈ B(a,∕3δθ) we get a similiar statement where a and b are interchanged and hence: ⅛wx'ci+',^ + ⅛ cι⅛ A(x,w,T1) ≥ A<x,y,T) + C1<5θ for all X ∈ B(xθ,0θ), T1 ∈ [Tθ,2Tθ], w g B(a,6θ) U B(b,<5θ), y ∈ B (a,∕J<5θ) and T ∈ [Tθ, ∞). -158- Now we consider large ∣w∣ 's. Pick large R so that B(xθ, 6θ), B(a,Cθ) and B(b,Cθ) C B(0,ft). Bearing in mind (10.6) we make j{2 > 16T0 ^ i<p(x.r) + ∕>(c,y)} + 100 Cιcg I A(x,y,T) + 100 C1C⅛ V(x,y,T) ∈ B(xθ , <5θ) × B(C,Cθ) × [Tθ , ∞) and C ∈ {a,b}. So we have the wanted estimates in (6.6) if (x,w,T∣,λ) ∈ B(xθ , 6θ) × (R^-B(0,R)) × [Tθ , 2Tθ] × [λθ , ∞). For this R, we use (10.5). 1 o With ε = jηrQ Cj0q there exist k7 = Cε and λθ = λθ(ε) (still larger λθ!) such that uniformly for (x,y,T) ∈ B(xθ , όθ) × B(0,R) ×[Tθ , 2Tθ] we have -T1Hγ z (x,w) (λ) -T1H ,1 ∂ ,___c vλ,w,, e ≤ Cε exp(-λ(A(x,w,T∣)-ε)) C 5 2 = k7 exp(-λ(A(x,w,T1)------ ⅛⅛-)) which, by above, gives (6.6) of Lemma 10, with kg = C∣0θ. Proof of (6.7) is similiar. We get uniform estimates of A(x,w,T^) for x ∈ B(b,0θ) w £ B(b,<5θ) (not cι^2⅛ just on compacts in w) and T∣ ∈ [Tθ , 2Tθ] by writing A(x,w,Tj) > (Tθ---- ------ ) if -159- no minimal action path enters B(a.50θ) U B(b,∕70θ), A(x,w.T1)> inf{p(z1,z2)} z, ∈B(b,0θ) z2∈B(a,^0) if some g⅜,w enters B(a,6∩) and 11 A(x,w,Tj) ≥ inf{A(x,z∣,Sj) + A(z2,w,s2)) Zj ,z2∈ B(b,∕J6θ) W0B(b,0θ) s J ,s9 >0 if some g ×y enters B(b,∕36θ). τ1 This first lower bound is large since we made it large in (10.10), the second is large since the wells are far apart and 0 is small. Using how 0 was chosen we see the third one is bigger than p(x,b) + p(b,y) A(x,y,T) 53 C16 256 lu0 for all y ∈ B (b,∕36θ) and T ∈ [Tθ, ∞), as in (10.13). Now we can finish the proof as above using (10.6) first and then (10.5). §10.3. Sketch of a proof of lemma 10.3. Recall, sublemma 1 says: (10.14) e τH(Λ) a λ (x,y) = (2pr)2 exp(-AA(x,y,T)) τ _1 _1 ET (exp(-λ ∫θ (V(g^y(t) + λ 2z(t) - V(g⅞y(t)) - λ 2 V'(g⅞y) z(t)) dt)), -160- lemma 2 says gψy is unique for (x,y,T) ∈ B(xθ∕0) x B(b,0θ) x B(b,6θ) × [Tθ,2Tθ) and is differentiable with respect to x. For (x,y,T,λ) ∈ B(×q> × × [Tθ, 2Tθ] x [1, oo] and z ∈ Cθθ([0,T]) set T -1 (10.15) f(x;z) = f(x,y,T,A;z): = exp(-λ ∫θ (V(g^y(t) + A 2 z(t) - V(g⅞y(t)) - A 5 V'(g⅝y(t)) z(t)) dt) Then f(x + hβpz)-f(x,z) ___ jh ds f(x + sepz) dg (10.16) = 4 ί0(0 {rτ j=l ∑ J igτ+sei,y ω +λ ⅛> - β¾ <4+""y<∙>) s'i'y<∙¾dι! ⅛ - A T f(x+sβpz) ds → ~λ ∫0 ( d (g'py(t) + λ 1 2„z(t)) z 5(g⅞y(t))p - &*/<*» - ⅛ 1 ⅛- (⅛y(∙)) ¾W) ωξ⅞γω) dt f<«> J J κ 1 = ■ —A hj(x;z) f(x,∙z) 2A(∣∣g⅞y∣∣⅛+λ-1∣∣z∣∣⅛), Since I W(x)l = 0(e^∣x∣ ) implies ∣VV(g^y + A 2z)∣-θζ∙ -161- the integrand in (10.17) τ (f(x + hei∙,z)-E^(f(χ∙,z))^ ^τz<(f(x-)-hei∙,z)-f(xiz)>1 Ez --------------- e ez(------------ j------------- J is uniformly bounded by an integrable functional of the form 0(e 2A⅛o Bλi∣∣z∣∣∞ λ e z ) for A large. The dominated convergences theorem and (10.16) implies the expression in (10.17) tends to — AEj(hj(x;z) f(x;z)) as h → 0. By (10.14) we have (10.1S) r(λ) r(λ) -⅛e-τκ* (x>y) = (^T))e-T“'i (χ,y) + A 0X∙ 1 λ X 5y d ∕∂Vf ×,y , a^^2 „X _ ∂γ⅛τ ) , ∕ A A2 pτ∕(rτ 1 V + ∖2ττT∕ Εζψθ jSl ,9uj τ +A ' ∂ui (ÔŸ dt) M-<(V(g⅝y(t) ÷ A-^ z(t)) - ~ V(g¾y(t)) -A 2 V'(g⅞y(t))z(t)) dt) exp (—AA(X,Y,T)) =: τH(A) θ λ (×y) + K×, y,T,λ) and we need to expand I(x,y,T,λ). As before we divide the path space Cθθ([0,T]) into 3 pieces for 0 < r < R: Cθθ([0,T]) = {Z : A 1 2∣∣z∣∣oo < r} U {z : r ≤ A 2∣∣z∣∣oo ≤ R} U (z : A z∣∣z∣∣∞ > R} -162- and accordingly (10.19) I(x,y,T,λ) = IIr(x,y,T,A) + IIIrR(x,y,T,A) + IVR(x,y,T,A). Since T ∈ [Tθ, 2Tθ], which is bounded, we estimate 2a1⅛s β B∣∣z∣∣0o χ(ζ:||ζ|| > Rλ5)) IVa(x,y,T,λ) = 0 (e = o( ∑ 2A(n + l)2 -Λi2k<d> e"2T-" ∕i,"+1,ri e 2'1o ) ∖n = [Rj ∕ where [R] is the integer part of R. Here we used (7.26) (2πΤ)5Ε^(ζ:||ζ|| > r) = 0 (exp —ijr^) = 0(exp (-η2ψj^)) , and for the characteristic function we have ∞ χ{z=∣∣z∣∣ > k) ≤ £ χ(z=∣∣z∣∣ > n) n=k and a uniform estimate for the integrand. For R large compared to B we get -163- ΛR2k,d) (10.20) IVj^(x,y,T,λ) = 0(e IVaθ(x,y,T,λ) = θ(e ^T0 ) as λ -→ oo and we fix R,θ such that -λ[ sup A{x,y,T} + l] (x,y’T) ) as λ → oo. 2 Next, using (2.49) we fix rθ such that — ⅛j + V,,(γ) > 0 on Dθθ(O,T) if dt j. ∣∣γ-g^,y∣∣ ≤ for all (x,y,T) ∈ B(xθ, δθ) × B(b,δθ) × [Tθ, 2Tθ]. Then we make a Taylor expansion of the integrand in Π1∙n(x,y,T,λ) similiar as in the proof of lemma 8.1. Bounded T makes it easier and we get: (10.21) Hrθ(x,y,T,λ) = (⅛)≡ [1 1, exp (^5 ∕τ(v',(Sτ,) z. ≈) duj) + θ(⅛)] 0 exp(-λA(x,y,T)). Finally by the bound on ∣V,(x)∣ J λ -164- IΠrθ,rθ(x,y,T,λ) = (10.22) = θ(e 2AR; 0(5⅛τ)2 E^(exp(-∫θ(V(gψ+λ 2z)- V(gx∕)- λ χ(z : rθ < λ 2∣∣z∣∣00 ≤ Ro)) exp(-λA(x,y,T)) = 0( e 2V,(gψ)z)dt) ’^’Τ) + ε0^ uniformly for (x,y,T,λ) ∈ B(xθ, όθ) × B(b,δθ) × [Τθ, 2Tθ] × [λθ, οο) according to (8.11) in the proof of lemma 9. Now (10.19)-(10.22) give (10.23) I(x,y,T,A) = ∏rθ(x,y,T,A) + 0 (e - Qτrτ)2 ⅛ <9χ.) Ez\exp( — λ(A(x,y,T)-∣- ε∩) ) = 2! ∫θ (v''(8τ)z>z) dt) + θ(⅛)] exp(-AA(x,y,T)) λz ∂*>o (χ,y) d 0x∙ = (À)2 (-------- 5------- ■ + θ⅛) e×P(-Wx,y,T)) rρ where we used the definition of bθ(x,y) in sublemma 2. Now (10.18), (10.23) and (6.5) give (6.8). -165- 11. A Proof of Lemma 7. §11.1. Lemma 7.1 §11.2. Lemma 7.2 §11.3. Lemma 7.3 §11.4. A sketch of a proof of lemma 7.4 and 7.5 -166- 11. A proof of Iemma 7. §11.1. Lemma 7.1. We first want to show (4.17): If y = b + 0(e —ωη T 1 ) uniformly on B(xθ, <5θ) ×[Tθ , ∞), as in lemma 6, then for β ∈ Nθ with |/?| ≤ 2. (11.1) J0l t ∂fβ ao<x'y) ∣y= 7 = CO,Z? aθM + θ <e -i1T ) unformily on B(xθ, 6θ) × [Tθ, ∞). Xb Here Cθ^ are constants and aθ(x) — (det X ’ (0)) cy Xb and X ’ as is in lemma 2. For β = 0, (11.1) follow from (7.8) that says (n.2) aj(x,y) = aθ(x) (1 + 0(e 1 + ∣y-b∣)) uniformly on B(xθ, 6θ) × B(b,6θ) x (Tθ, ∞). Now recall (see sublemma 2) (11.3) d d I _T ∑ ωi bj(x,y) = 22 (∏ωi)2e 2 i = 1 aj(x,y) and so it suffices to show <11'4> 373 ⅛<×∙rty=y = (c0jβ + °< "‘"J) t>J<Xy>∣y=y for ∖βI ≤ 2. -167- To do that we differentiate (11.5) bθ^(x,y) = T _d rηp 2 E^(exp(-i ∫θ {Vπ(gxy)z,z) dt)) from sublemma 2 and use (11.6) T __ d rjι 2 Ej[zi(s) Zj(t) exp(-i ∫ (V,'(gxy)z,z) du)] = bθ∖x,y) (Gx,y,T(s,t))jj from sublemma 3. We get (∏.7) bT(x,y) = T B? [(-J J' JJ" ∂jv(gy(t)) Z∏(0 ⅛,C) g(gT - dt) exp(-i ∫T(Vπ(gxy)z,z) du)] 2j0' u⅛τ χyλ)k ,n . τ ∂3v⅛τy(t)) ~χ,y,τ. t, ∂⅛τ = (4 J “2'o k£n ¾≈ dt)b0(x’y) and (11.8) g2 ,T∕ ' ∂y∙∂y∙ <√x^ J .1 ∣∂, ⅛y-( w(∕gτ ×y∕(t)) cχ,y,τ∕t tx g(⅝ ×y∕(t))⅛ ι rτ y g⅛3v dt) 2 ∙Γθ ∑ ∂uk<9un∂um n,m ’ ∂y-ι I 1f rτ V ô3y(gT (t)) ^x’y’Tft ÷) WyW)k + 4∏θ ∑ 0uk0un3um Gn’m (t>t) dt) .χy,(s)∖ rτ v ∂0v(gτy(s)) r<x>y>τp, W κτ(v y) g0 ς 0uk0un0um Gn’m (s’s) d7 ds)J b0 W> 0yi -168- and we need to show _ 11 "^ .χy∕ τ a3v⅛×¾,> x,y,T S(⅛ ,(*))k 0 ∂uk0un0um (G (t>t))n,m l„_„ dt ∂yii y=y 12 :: 0 ∂ui<9uk0un∂um 13 = J 0 9uk9un9um 9yi 9yj 9yi x,y,τ.tl. ⅜⅛⅛, 0 0uk0un0um n,m l ’ ⅜j0yi 9yi ,y=y dt 'y=y τ a3V(⅞y(t)) are all of the form “aconstant + 0(e d, ⅛⅛ l ’ ''-,n'm _ ,τ a3v(⅝y(t)) ⅜⅞τ⅛lι ⅜<⅝χy,z(t⅛ι ,= 4 τ j⅛)) ,c>,y,τ.tt.. d, y=y -6ιT 1 )” uniformly for (x,T) ∈ B(xθ, δθ) × (Tθ, ∞). Take t = T— v, by (2.34) ∂3V⅛Ty(T-v))ι (11.10) 9uk9un9um 94V(gxy(T-v)) 'y=y , '9⅜u>mly=y =constant + θ<e ι(τ-v)) By lemma 5.2 (11.11) Gx,y,τ(T-v,T-v)y=y = (1 - e-2fiv) (2Ω)-1 + 0(e“6l(T“V)) + 0(e_<$lT) and (11-12) 0^^(T-V’T-v)|y=_ = B(ej,v) (2Ω)-1 - (A(ej,0) - A(ej,v) - B(ej,0)) (2Ω)^^1 e~2Ων + 0(e~0l(T~V)) -169- where A(ej,v) and B(ej,v) are diagonal matrices in Cβ([0,∞),Rα ), for each j ∈ <1........ d). By (2.26), (2.22) and 0 < i∣ ≤ -ψ we have (11.13) 0(gi,y(T-v)), Γ -61(T-v) θv -0,T)Ι —--------|y_y = |_(1 + θ(e 1' ⅛ e-fiv (I + 0(e 1 ')Jkj ⅛.e — ω∙v —⅛,(T + v) ■ +0<e > ) Using also (2.56) we get (11.14) ∂2(g*y(T-v)), — <5,(T-v) ---------------------- ~∣y=y = (B(ej,v) + 0(e 1 )) o,, —i1T —0η(T-v) o e-fiv (I + 0(e 1 )) — (I + 0(e 1' ')) e-fiv(B(ej,0) + θ<e — 0ηT ))lk,i = (β(eΓv) - B<ej’°»k,i e —ω∙v — 0,(T+v) 1 + 0(e 1 The estimates in ( 11.7)-( 11.11 ) are all uniform for (x,T,v) ∈ Β(χθ,όθ) × (Tθ,∞) × [0,T] and imply I∣, l%, I3 and uniformly. For example: are of the form constant + 0(e -6,T ) -170- (11.16) _ T ⅜3V(⅞y(T-v)) I4-Jθ ÔUkôUnôUm (G τ ⅛2zgy(T-v)) ×yz ^( t (T v,T v))n,m 0y.0y. ly=ÿdt T -δ 1(T-v) = ∫ (constant + 0(e )) [(1 - e 2ω"v) 6n,m + 0(e 0l(T [(B(epv) - B(ej,θ))kjj e — ω∙v V)) + 0(e 6’T)] — 61(T+v) + 0(e ] dv = JT(constant (1-e 2Wl*V) 6n,m(B(e∙,v) - B(e∙0)), · e -ω∙1 V dv Q JJ*!1 + ∫ T nz -61(T+v) θ(e 1 ) dv 0 = a new constant + 0(e -61T x ). ω-∣ In the last step we used (B(e-,v)), ∙ is bounded and 61 ≤ < ω∙. To prove J K,l J2 I (4.16) ∣-5 ⅛T(x,y) = 0(1) uniformly for (x,y,T) ∈ B(xθ, <5θ) × B(b,6θ) × [Tθ,oo) it suffices (see (11.3)) to show: (11.17) For each a ∈ no ‰ bo(χ,y) = θ(1) bo<x>y) uniformly on B(xθiθ) × B(bθ 5θ) × (Tθ, ∞) -171- For ∣α∣ = 0 (4.16) follows from sublemma 2. For ∣o∣ ≥ 1 we see from (11.7), (11.8) Λ∣θil τ rp and differentiating (11.8) that bθ(x,y) is bθ(x,y) times sum of products of terms that are bounded by Jα1∣) 15==0 f ∣*2∣ v(gxy(t)) 'yωι 0y<*2 α⅛ Gx'y'τ(t,t) U <4yw 0yan ⅞y(t)) dt where n > 3 and ∣ctβ∣, ..., ∣αn∣ are > 1. Using lemma 2 and lemma 4 Λl0l'.., xv, .. √α2Lxv,T.......................... ⅛j∙ vιg-f tυι = »ui, g-^2υ ^ tuυ = lαi∣ -4’(«) = °<^1< τ't,) ‰ and T -ω1(T-t) 15 = °(Jθe 1 ) dt) = 0(1). SO Uniformly for (x,y,T) ∈ B(xθ, 0θ) × B(b,6θ) × [Tθ, ∞) which implies (11.17). §11.2. Lemma 7.2. If (x,y,T) ∈ B(xθ, 5θ) X B(b,⅛0) × [Tθ, ∞) then by (2.26) (11.18) ∂A(x,y,T) ⅜i (g^(T)i, where by (2.34) -172- (11.19) ∣gx∕(t)∣ = 0(e 1 + min(∣y-b∣, 6θ) e ^uniformly, and (11.20) A ",(x,y,T) = (Ÿ(T)-X(T) (X(0)~1Y(0)) (Y(T)-X(T) (X(0))"1Y(0))"1 Here X = Xx’y’T a∏d Y = YX,y,T satisfy (see (2.22)). Xx’y’T(t) = (I + 0 (e~^lt + ∣y-b∣)) e-fit = -XX,y,T(t) Ω~1 and (11.21) YX’y,T(t) = (I + 0(e~5lt + ∣y-b∣)) eβt = YX,y,T(t) Ω-1 uniformly for (x,y,T,t) ∈ B(xθ, όθ) × B(b,5θ) × [Tθ, ∞) × [0,T] Now (11.21) into (11.20) implies (11.22) A "r(x,y,T) = (ω + 0(e _ β rp _6 T 1 + ∣y-b∣)) eOT ((I + 0(e 1 + ∣y-b∣)) efiτ)-1 = Ω + 0(e_0lT + ∣y-b∣) uniformly for (x,y,T),s. If y = ÿ or y = y, that are b + 0(e — ω-, ,T ) uniformly for (x,T) ∈ B(xθ, 6θ), __ β then Ayy(x,y,T) = Ω + 0(e rp 1 ) uniformly for (x, T) ∈ B(xθ, όθ) × (Tθ, ∞). -173- Differentiating Ayy(x,y,T) in (11.20) with respect to y, we get a sum of terms of the form ± L (γ(T)-X(T)(X(0))-1Y(0)} ∂v lα2∣ ⅛ (γ(T)-X(T)X~1(0)Y(0)) (γ(T)-X(T)X-1(0)Y(0))^1 (γ(T)-X(T)X-1(0)Y(0))~1 ∙∙∙ ∂v -⅛I (γ(T)-X(T)X~ 1(0)Y(0)) (Y(T)-X(T)X-1(O)Y(O)) -1 ∣αι∣ ∣αi∣ By(2.24)L(γ(T)-X(T)(X(0))-1Y(0))andL(γ(τ)-X(T)χ-1(0)Y(0)) ∂y i ∂y 1 & rp∖ O(l)enTand by (2.22) (γ(T)-X(T)X-1(0)Y(0))-1 = e_fiT(l + 0(e and, by above, we get Ig = 0(1) and hence 1 ’ + ∣y-b∣)) δ A(x,y,T) = 0(1) uniformly for (x,y,T) ∈ B(xθ, 60) × B(b, iθ) × [Tθ, oo) for each a ∈ Nθ, with ∣α∣ ≥ 3. 171 When y = y= b + O(e -~∙,τ ) ∣α1∣ L- (γ(T)-X(T)(X(O))-1V<O))∣y^ = = (b(q1,0) + O(e~*lT))cfiT Ω, ∣o∣∣ L(Y(T)-X(T)(X(O))-1Y(O))∣ __ = ∂y 1 y—y B(αi,0) + O(e —6 T ” 1 ) by (2.55) and (2.56) and ^Y(T)-X(T)X-1(O)Y(O)^-1 ∣y _y -ωt(I + 0(e^Yr). Hence each _ β I6 y=y + B(α1,0) ΩB(α9,O)∙ ■ ∙B(αn,0) + O(e xp 1 ) and therefore J/?l -⅛1τ) 2—j A(x,y,T)|y_y — constant + 0(e ) ∂yp uniformly for (x,T) ∈ B(xθ, 6θ) × [Tθ, ∞), for each ∈ Nθ with |/?| > 3. -175- To prove (4.20) we write, for ÿ ∈ {y, y}, A(x,y,T) = p(x,b) + A(x,b,T) — p(x,b) + A(x,y,T) — A(x,b,T). By Proposition 1 (2.26 and (2.34) p(x,b)-A(x,b,T) = = s⅛o(A(x,b,s)-A(x,b,T) _ lim f ∣∙S ∂A(x,b,T ) jrpΛ S→∞ Vj QrΓ, ) uniformly for (x,T) ∈ B(x∩, 6θ) × (Tθ, ∞). By Taylor’s theorem, (11.18, (11.19) and (11.22) we get A(x,y,T) - A(x,b,T) = (A^(x,y,T), (y-b))∣y=^ + i J'\Ayy(x’b+s(ÿ-b)’T)(y-b)’(ÿ_b))ds = θ((e Lü T 1 + ∣y-b∣)∣y-b∣) + 1 6 τ + ⅛ ∫ ((Ω + 0(e 1 ))(y-b),(y-b)) ds 2 0 = 0(e — 2ω1 T 1 ) uniformly for (x, T) ∈ B(xθ, 0θ) × (Tθ, ∞). -176- This completes the proof of (4.20) and of lemma 7.2. §11.3. Lemma 7.3. By (7.50), (7,20) and Simon [5] lemma 20.4 a^(x,y) is a finite sum of constants times terms of the following types: Typel: I = Ix,y, γtλ = Γ 'J-' 0 ∂αV(g(t)) G∙ ∙ (t,t) G∙ . (t,t) dt where a ∈ Nc j1j2 j3j4 ∣a∣ = 4, Type 2: II = IIx’y’T = JT JTÖ«V(g(t)) Λ(g(s)) G∙ ∙ (t,s) G∙ ∙ (t,s) G∙ ∙ (t,s) dtds 0 0 JP2 j3j4 -,5J6 where ∣α∣ = 3 = |/?| and Type3: III = IIIX’y’T = JT0αV(g(t)δ^V(g(s))G∙ ∙ (t,t) G∙ ■ (t,s) G∙ ∙ (s,s)dsdt 0 0 j1j2 j3j4 j5j6 where ∣a∣ = 3 = ∣∕J∣. χ,y,τ Here g = g.×y for (x,y,T) ∈ B(xθ, ίθ) × B(b, <5θ) x [Tθ, ∞). T and G = G See also Mizrahi [1], and Davies and Truman [l]. We want to show (11.23) For each γ ∈r⅛ λ∣7∣ t 2— aT(x,y) = 0(T) ∂y' uniformly on B(xθ, όθ) × B(b, 6θ) × [Tθ, ∞) -177- and rp (11.24) If ÿ = y(x,T) = b + 0(e 1 ) uniformly for (x,T) ∈ B(xθ, δθ) × (Tθ, ∞) then a^(x,y)|y_y = (constant) T + f(x) + 0(T e ^ ) uniformly on B(xθ, δθ), x [Tθ, oo) where f ∈ Co° (B(xθ, δθ), R). It suffices to show (11.23) and (11.24) are valid for a term of each of the 3 types. for IX,y,T, IIX,y,T, and I∏x,y,τ we differentiate To see that (11.23) holds under the integral signs and use (2.24), (2.26), and (2.53) to show the integrand is 0(1) uniformly for I xVT — ωι (t—s) ’ and 0(e ) uniformly for the others. To prove (11.24) for each type we use (11.25) Gxy,τ(t,s) = Gxb(t,s) + Gj(s,t) + RX,T(s,t) from lemma 5.2 and the estimates in (2.58) - (2.60) together with (11.26) XV xb —ω1T —u>η(T-1) gxy(t) = gxφ(t) + 0(e 1 e 1' ') by (2.42). We will only do the calculations for type 3, say. The others are similiar. Using (11.26) and δ∣ ≤ ω1 < we get: -178- (11.27) IIIxy’T = ∫τ0αV(gxb(t)(Gxy'τ(t,t)∙ . {(∫τ+ ∫τ)^V(gxb(s)) Ο v j1j2l 0 0 7 cxy'τ,t'⅛rt (gx7 ∙τ<8∙s>)⅛6d-}dt + θ (e^s∙τ) = jVv(g×tω (Gxy'τ<t,t))jιj2{σθ + Joτ) ^v(g×b<s)) (cxb(t,s) + GT(t,β) + Rx,τ(t,s))j3j4 (G×b(s,s) + G∣(s,s) + Rx'τ(s,s))⅛j6 ds} dt + 0(e~0,i ) = = ∫θ 0aV(gxb(t)(GX’T(t,t))jij2{;£l†T(t) + 0(∫θ |RX’T(s,s)|ds)}dt + 0(e_élT) = ∫θ ö“v(gX^)(GX’T(t,t))jij2{ i∑ lfτ(t)} dt+ 0 (Te_6lT) uniformly for (×,T) ∈ B(xθ, 6θ) x [Tθ, ∞). Where ,×-τ(,) = jVv(g×b(,))(G*'b (t,s))j3j4 (GxΓb(∙∙≈))j5j6 "s Ixf(t) = I/',V(e’‘'>(s))(Gx1'b(t,s))jA (GT (≈,s))jsj6 ds Ix3τ(t) = ∕∕'sv(g×b(s))(Gx1∙b(.,s))j3j4 (G? ⅛s>)w6 ds ■ 179- l×∙τ(t) = jSiW><.,>(Gi ,lj,))j3j4 (C∙Γb⅛,s,)j5j6 ds I*jτ(t) = jV V<gxb<.)>(< -T (s,s)) ds ⅛j6 ’■> I1⅛τ(.) = jVv(Λ))(fiJ <' -s))j3j4 (g? M)j5jβ «b and .×∙τ(.) = f δsV(gxb(.,)(<^, (t,s))j3j4 (GT (s,s))j5j6 ds Now we just have to look at each one of them. By (2.58) and (2.14) we have: For a e Nθ^ (11.28) ∂___ ∕p×sb. . Q^a (‰s) e Ω∣t s∣∕∏(~)∖ —1∖ __ ∩∕p (2ii) ) — 0(e 61min{t,s} e ,ι∣t-s∣) and (gX’b(t) - b) = 0 (e (11.29) 1 ) uniformly. Hence ,×∙τ(t) = jVv<Λ))(<^⅛1 (Gιb(≈,≈))jsjβ <b 'J3J4 'j5j6 — w∙ |t—s∣ T λ>0jλz∕k∖ oe j3 = Γ <9MV(b) ------------ δ. . γ j0 4 uj3ωj5 j3j4 <$· . ds + j5j6 -180- ω∙ ∣t-s∣' j3 .x,b + ∫ âpv(b) (G*,b(t,s))i ; (G 1,b(s,s))i - - e.ds V 1 v z Z)QJd V 1 v 7 ∕JκJfi 4 u>. ω· ∕J3J4 J5j6 ο ∙13 j5 T 0 + f √V(gx'b(s)) - 9iV<b))(G×1∙b(.,))j , (<f,⅛ ds 0 , v -U. t — ω∙ (T-t) = ⅛0-^j3÷" J:! )) J3 J5 fi -V +λ 1Vλ5u√ (11.30) -L Z» 1 X , T i+ ∖v∕ ΛlτK ω∙ ∣t-s∣ ,x,b f(x, t) = ∫°° <∕v(b) (Gx1,b(t,s)∖ · (G* ,b(s,s)Y ; - eλ J3— V 1 k 7∕J ZJ3J4 q V 1 v 'ZJ&Jfi 4 u. ω, J5j6 0 J3 J5 + f <∕v⅛×∙b(s)) - S<jV(b)) (Gx'b(.,s))j j (<⅞‰⅛ · ds d, and by (1128) and (11.29): (11.31) u· |t—s| VT 00 ft J3-— ,x,b fG x 1 ,b (t,s)Y . fG x,b (s,s)Y · Ç-. — ex'i(t)=∫ c√iV(b) \ 1 v ’ jJi'iJA 4 W. ω∙ ZJ3J4 \ 1 v ’ 7Jd J5jfi 6 T J3 J5 + ζ>',v<ex∙bω) - ∕v,t,), (Gx1'b(t,s))j3j4 (Gib(s,s))j5j6 d» = ds -181- = Γ oo —δηs —w1 |t—s∣ — 6ηmin{s,t} —ω1 |t —s∣ 0(e 1 e 1 +ei e )ds T + ∫ OO θ(e —(*>1 S 1 )dg = θ(e T — 0, T 1 ) —t√ι uniformly. Differentiating (11.30) under the integral sign and using (11.28) and (11.29) gives: For each γ ∈ N d0 (11.32) 0x7 f(x,t) = 0(e — <511 ) uniformly for (x,t) ∈ B(xθ, 6θ) × [0, ∞). Now we state what one gets for the others I G^(t,s) = - e -Ω(2T-s-t) x,T ,s using (11.28), (11.29) and i (2Ω)-1. A'(⅛) ; If C = 6. . 2*V⅛ ⅛i4 J5j6 then «Tz., _ c 12 W = u→S7 j3 j5 ^2l⅞'τ-t> +, n0(e ,.-"ι(τ-t> e.-<lT,) -182- -2ω∙ (T-t) 5 (T-t)ifω. = 2u>∙ j3 j5 -u,j5(τ-t) -2"⅛(T-t) — e if u∙ ≠ 2ω∙ (2ω∙ — ω∙ ) j3 j5 l J5 √ e ιx7(t) + 0(e =c — w1(T—t) —δ∙∣T P e 1 ) -<51T Ixf(t) + ⅞τ(t) = 2Sr + O(e uii) j3 and I3⅞τ(t) + I3⅝τ(t) = gg- + O(e WlT) Adding up gives: (11.33) y —w∙ t —ω∙ (T—t) ∑ IX.’T(t) = d1 + d2 e 3 + d3 e 3 i=l 1 -ω∙ (T-t) _s τ 3 + f(x,t) + 0 (e ) + (d4 +c⅛ (T-t)) e for some constants d∣... d3 with d∣ = g <∕v(b) 2 Now we put (11.33) into (11.27) and we get: and f satisfying (11.32). -183- IIIX’y’T = ∫τ(5aV(b) + ∂oV(gxb(t)) - 50V(b))(Gxjb(t,t) + G∑(t,t) + Ο 1 z W∙ + Rx,T(t,t ))j j e (dj + d2 e UJ, t ζ rp__ †\ ψ (d4 + d^(T—t) 3 + d3 e ~ωj,(T-t) -61T —ό,Τ 5 + f(x,t) + O(e 1 )) dt + O(e 1 ) where (G†b(t,t))j j = ⅛2 + O(e 1 2 Λ 1 ) + O(e 1 ) -2ω. (T-t) j2 z _ £ fj' = ------ «----------- 0; ; and ∣Rx,τ(t,t) | = O(e~ 1 ) "j1j2 zwj1 j1j2 (GÏ(t,t)); : ' z by (11.28), (11.29) and (2.60). Hence Inx,ÿ,T= /5 Zv1t) √[l,) i Uθ WjιWj2 w⅛ j1j2 , τ t ( j3j4 J5jθ√ + constant + Γ (9aV(gx,b(t)) — 5αV(b)) d1 + 0 i ∂α(gx,b(t)) (f(x,t) + d2 e — ω∙ t 1T ) J33 ) dt + 0(Te -0 Uli which implies (11.24) holds by differentiating under the integral sign and apply (11.29) and (11.32). -1 KI­ SH.4. A sketch of a proof a lemma 7.4 and 7.5. Recall that in the definition of *■ . in (6.48) L1(∙) is a differential operator of ∣ ⅜J -*order 2. Jα∣ + I0l τ Hence (4.23) follows from the results above and by showing - -------- -5- b∩(x,y) ∂ya∂x θ = 0(1) bθ∖x,y) which can be done by differentiating in (11.5) under the integral sign. _ X b —^-∣ T To prove (4.24) one uses W = g (Tθ) + 0(Te ) uniformly by (2.42). J^Lτo ∂' lbnυ(x, W) |/?| χb -ω T Hence, for instance, --------—^------- = ----- -≈ bi(x,g ’ (Tθ)) + 0(e ) ∂∖∖' ∂Wp since all W — derivatives of order |/?| + 1 are uniformly bounded. (4.25) is proven in the same way as (6.23). -185- 12. The Proofs of Sublemma 4 and Proposition JL. §12.1. Sublemma 4.4 §12.2. Sublemma 4.1. §12.3. Sublemmas 4.2 and 4.3. §12.4. The upperbound in Proposition 1. §12.5. The lowerbound in Proposition 2. -186- §12.1. Proof of sublemma 4.4. We need to prove: If 0 < o < ≤ 1 and c ∈ {a,b} then inf A(x,y,T) > ¾.√2 - α2) |x —c∣ = qCq ∣×-c∣ = ∕3cθ τ>o (12.1) ι where and cθ are such that ( 10.1 )-( 10.3) hold. Recall T A(x,y,T) = inf{Jθ (1 7 2 + V(γ))dt : γ(0) = x, γ(τ) = y} and note we only need to consider paths γ in (12.1) with αcθ < ∣γ(t) — c∣ < 0Cq for t ∈ [0,T] since Υ∣[t t j , where tj := max{t ∈ [0,T] : ∣γ(t) - c∣ = αcθ}, and t2 := min{t ∈ [t1 , T] : ∣γ(t) - c∣ = 0cq}, has less action than γ. 2 C1 9 From (10.1) and Taylor’s formula we get V(x) > -^(x — c)z if ∣x — c∣ ≤ cθ. 9 For this harmonic oscillator we solve the Euler-Lagrange equations γ = c∣(γ — c). xy sinhc-∣(T-1) The minimal action paths are given by: γ>p (t) := c + (x — c) sin⅛ c ψ— ÷ sinh c-∣ t (y ^^ c) sinh c1T . Hence the R.H.S. if (12.1) is greater than -187- inf fcl∕[(χ-c)2 + (y —c)2]∞sh c1T - 2((x-c),(y-c))∖⅜ > X — c∣ = αc∩l∙2∖ sinh c^T )} ∣A-V|—UVθ ∣x-cj = ∕7c0 T>0 2„2 (,÷g)∞shc1T-2g - T>0 t 2 k sinh c1T )j which attains its infimum value when cosh c∣T = ^(^ + g) > 1 and leads to (12.1). §12.2. The Proof of Sublemma 4.1. We recall the statement: If ∣x — c∣ < ——c∩uc^ then (5 + q) (12.2) ∣x — c∣ e 2 ≤ ∣gx,c(t) — c∣ ≤ ∣x — c∣ e and (12.3) 2⅛∣x - c∣2 ≤ p(x,c) ≤ ⅛ - c∣2 2c1 First we show that (12.4) |gX’C(t) — c∣ ≤ cθ if t > 0 and x as above and then we use (10.2), c∣∣x — c∣2 ≤ (W(x),(x-c)) < C2∣x — c∣2 for ∣x — c∣ ≤ cθ to prove (12.2) and (12.3). Taking γ(t) = (χ - c) e as a trial path for -c2t + c for t > 0 -188- p(x,c) = inf{∫°°(l γ2 + V(γ))dt∣ γ(0) = x, 7(00) = c}. 0 2 We get, by (10.1), (12.5) J∙°°(⅜ 72 + V(γ))dt ≤ ∫°°(I γ2 + ^(7-b)2)dt 0 0 = y∣x - c∣ c c2 c2c0 < -r^ 2⅛+⅛> _ î___ r_____ r,.-( . .ι,. VV line g<Jlllg HΛJΓI1 λ λ∣ ∣Λ— C∣ ïo 1 r., . ι,, „I _ VQJ^ „ Ί tlllU «„J τ ∕ vv, ^va ∙ ίΛ—vi 1 + ⅛)2 ( 2 + c1j then to c makes o(χ∙c) = r⅛tex'c(t)2 + V(g×''(.)))d. ≥ (ι - τ⅛) + ⅛S = '∙2 + ci', and so by (12.5) we get (12.4). Now (12.6) ⅛gx,c(t))2 - V(g,c(t)) = = ⅛oo⅛(δx,c(t))2 - V(gX’C(t)) - i(g-(T))2 - V(gx'c(T))) = lim JT<(gX’C(t) - W(gx,c(t))), gx'c(t))dt = 0 1 →∞ t xc o and so when we put f(t) = ∣g ’ (t) — c∣ we get c2c0 2⅛+⅜ -189- f(t) = 2(gx,c)2 + 2{gX,C,(gX,C—c)} = 4V(gx,c) + 2(VV(gx't ),(gx,c-c)) By (10.1) and (10.2) (12.7) 4c∣ f(t) ≤ f(t) ≤ 4c2 f(t). For n ≥ 1 put fn(t) = f(t)∣jθ, sinh2c1(n-1) h-<t> = ,<°> sinh-¾. sinh2c1t + f(''> !hΠΓ2⅛ and k-w = f(θ> sinh 2co(n-1) sinh 2cot il,ι⅛ + f(") SSh2⅜ Here hn and kn solve the comparison equations kn(t) = 4c∣ hn(t), hn(0) = fn(0), hn(n) = fn(n) and kn(t) — 4c2l⅛(t), kn(0) — fn(0), kn(n) — f∏(n)∙ By Protter-Weinberger [1] k(t) < ⅛(t) ≤ kn(t) if t ∈ [0,n], hence r⅛ — 2cot — 2cot ∣x-c∣2 e = f(0) e =n⅛00 k∏(t) e∖ —2C-∣ t ≤ 1⅛ fn(t) = f(t) ≤ nlimoo hn(t) = ∣x - c∣2 e which gives (12.2). Using 2 Cj∣x — c∣ ≤ V(x) ≤ c2∣x - c∣ lx - c∣ ≤ co (12.6) and (12.2) -190- we get p(x,c) = 2∫°°V(gx,c(t)) dt≤ ∫°° c∣∣x-c∣2 e ∩ n & dt < ⅛L≤ ^Cι and z X .∞ 9 9 —2c9t c?|x—c∣2 P(×,c) ≥ ∫n c2∣x-c∣2 e 2 dt = -1∙^— which finishes the proof. §12.3. The Proof of Sublemma 4.2 and 4.3. Put Then fτ(t) := ∣gφ,y(t + t1) - b∣2 for 0 ≤ t ≤ t2 - t1 . fτ(t) = 2(g⅞y)2 + 2<g*'y(t),(g*⅜)-c)) ≥ 2(VV⅛y(t)),⅛y(t)-c)) > 2c2∣grp,y(t) — C∣2 > C2 frp(t) where used the assumption ∣g^,,y(t) — c∣ ≤ cθ if t^ ≤ t ≤ t2 , (10.2) and the Euler- Lagrange equations. Therefore (Protter and Weinberger [1]) ⅛(t) ≤ fτ(0) sinh Cjt smh Cx(t9 t-I t) 1 sinh cι(t2.tι) + fT<t2 - tP sinh c1(t2-t1) if t ∈ [0,t2~tj] and so ∣g^,y(t) — c∣z = f∙p(t) '' !cx,yr÷ 1 Γ∣2 sinh cl(t2_t) l ∣cχ,y<t ) ≤ ∣gτ (t1) - c∣ sinh cι(t2-ZIp + ⅛T <t2) ,∣2 sιnh cl(t-tP <r c∣ sinh c1(t2-t1) " -191- < - f ιc V οί /»inh c,(to-t) -+- sinh c1(t-tη)∖ max {∣g⅛',y(t.)-c∣2∣ (--------12l ,1------------------ -) i∈{l,2}ι 6Τ ' J ' sιnh c1(t2-tl) ' ≤ max 2-}{∣gτy^ti^-c∣2} for a,1 t e ftl ’ t2^ and sublemma (4.2) follows. sinh c9(T-1) To prove sublemma (4.3) take 7ψ(t) = (x — c) ——_ l _ rp— + c as a trial sinh C2T C<j ty path for A(x,c,T) and use V(x) ≤ τf∣× — c∣~ if ∣x — c∣ ≤ cθ . §12.4. The Upperbound in Proposition 1. We prove the upperbound in 3 steps. First we consider x and y = b with p(x,b) < p(x,a) + p(a,b) then x = a and y = b and finally the general case. If p(xθ , b) < p(x∩ l0 , a) + p(a,b) then there exist 0(x∩) 0> > 0 such that Step L A(x,b,T) < ρ(x,b) ■+- 0(e —2c1T ) uniformly for x ∈ B(x∩ , i) as T → ∞ . To prove step 1, pick ό > 0 such that (12.8) p(x,b) < p(x,a) + p(a,b) for all x ∈ B(xθ,6) . Claim. There exists 6<-> > 0 s.t. (12.9) ( U {gx,b(t), t ∈[0,∞]}) ∩ B(a,62) = 0 . x∈B(xθ,6) z -192- Otherwise, we have xn ∈ B(xθ,6) , xn → x and yn : = g n,b. ’ (tn) → a as n → ∞ . This implies (proof below) (12.10) p(x,b) ≥ p(x,a) + p(a,b) which contradicts (12.8) and hence proves (12.9) by contradiction. xn b We write gn(t) := g ’ (t) and to prove (12.10) we notice (12.11) lim tn > 0 since n→∞ ∣gn(t∏) - xn∣2 — ∣g∏(tn) — gn(θ)l^ ≤ tn ∫ g∏(u) 0 ≤ (sup{p(xn,b)}) tn , xn → x ≠ a and g∏(t∏) -→ a as n → ∞ By lemma 4, the triangle inequality and (12.11) (12.12) 1 ∫ “(A g£ + V(gn))dt ≥ A(xn , yn , tn) ≥ A(xn , a, 2tn))dt - A(yn , a, tn) 9 o∣yn-ar cosh cotn ≥ ∕>(xn , a) - c-⅛ λ SHΓ⅛⅞ as n → ∞ , since xn → x and yn → a as n → ∞ . Now pick sn → ∞ as n that sn — tn → ∞ and with zn := g∏(s∏) → b as n → ∞ . Then ∞ such -193- (12.13) ∫ OO 1 1 ο (i g2 + V(gn))dt > ∫ t∏ Ο (I g∏ + V(gn))dt ⅛ ≥ A(yn , zn ,sn — tn) > A(a,b,3(sn —tn)) — A(a , yn , sn — tn) — A(zn , b , sn — tn) 9 9 ∕ L∖ c2∣a-y∏l cosh C9(sn —tn) CQ∣zsn-b∣ cosh C9(sn-tn) ≥ P(a,b)----------2------------------------------------------------------------------------sinh C2(sn — tn) 2 sinh C2(sn-tn) p(a,b) as n → ∞ . Now (12.12) and (12.13) imply (12.10) and by above we have (12.9). Pick L∩ <$2 > 0 with ⅛2 ≤ ------—î sθ that (12.9) holds and 63 > 0 such that V(x) ≤ 63 implies (1+52)2 l'2+cι7 X ∈ B(a,δ2) U B(b^2) ’ x,b. If x ∈ B(xθ,0θ) t^ιen *3y above V(g ’ (tθ)) ≤ <5g implies ∣gx'b(tθ) — b∣ ≤ ⅛2 ≤------5iθ—ϊ and by sublemma 4.1 fI+52)2 t2+c1l ‰ e~Cl(t tθhft> - tθ ∣gx,b(t) - b∣ ≤ (1+52)2 l'2+cι' Since 63∣{t : V(gx'b(t)) ≥ 63}∣ ≤ ∫°°(i(gx,b(t))2 + V(gx'b)) dt ≤ ma^_____ p(x,b) j ύ 0 z x∈B(xθ,<5) we get (12.2) ∣g×-b(t)-b∣ ≤ -J½ e Cl<‘ τ°, (j+q)2 ∏√¾(*'k>l it l ≥ τθ := ⅛⅛--------- for all X ∈ B(xθ,6) -194- Now set, for T > To + 1, (t): = gx,b(t) ift ∈ [O,T-1] (b-gx,b(T-l))(t-T + l) + gx,b(T-1) ift ∈ (T-1,T]_ A(x,b,T) ≤ |T(J γ2 + V(γτ)) dt = Then 0 τ τ = ∕>(x,b) -T (i(gx'b(t))2 + V(gx,b(t))) dt + J (iγ2 + V(γτ))dt τ-l l τ-l T < p(x,b) + ∫ (l∣κx⅛-l)-b∣2 + ÿ|gx,b(T_T)2)dt - - ' ' z' ≤ p(x,b) + ( e -2c1(T-l-Tθ) for all T ≥ Tθ + 1 and x ∈ B(xq,Îj) which completes the proof of step 1. Step 2. If K∣ is compact then (and similar when b is replaced by a). 2c1T (12.13) A(x,b,T) < ρ(x,b) + 0(e ) uniformly for x ∈ K1 as T → ∞ . Proof. Given x ∈ Kj. If p(x,b) < p(x,a) + p(a,b) then (12.13) holds near c by step 1. If p(x,b) = p(x,a) + p(a,b) then for some xθ with p(xθ , a) = i p(a,b) = p(xθ , b), p(x,b) = p(x,a) + p(a,xθ) + p(xθ , b) and (12.13) holds near x by applying step 1 to -195- A(x,b,T) ≤ A(x,a,j) + A(a,xθ,∑) + A(xθ,b,∑) . Uniformity on compacts follows also. Step 3. Given a compact K∣ × K2 C R3 × R3 . By the triangle inequality and step 2 we have: Uniformly for x ∈ K∣ and y ∈ A(x,y,T) ≤ min {A(x,c,ι) + A(c,y ,∑)} c∈{a,b} 2 2 = = c1T c1T 3 ) + p(c,y) + 0(e 3 )} min Jp{x,c) + 0(e c∈{a,b} min c∈{a,b} _2iZ {p{x,c) + p(c,y)} + 0(e 3 )} as T → 00 . §12.5. Proof of the Lower Bound in Proposition 1. x I<2 be a compact subset of Let (12.14) E = × . Put sup {A(x,y,T)} (x,y,T)∈Kj×K2x[l,∞) and pick δ⅛ > 0 such that (12.15) V(x) ≤ <54 implies x ∈ B(a,y) U B(b,y), where cθ is as in (10.1). We reduce to proving -196- (12.16) There is a Tθ such that for each T ∈ [Tθ , oo) there exists a time interval [tp , t-p + Sψ] Ç [0,T] rp E i -τ~ with Sy > —ιgj∕ c 3ciCq + 1 for (x,y,T,t) ∈ Kj X K9 X ∣Tθ , 00] X [tτ , trp + Sy] . Assuming (12.16) we now prove the lowerbound. By sublemma 4.2 fT>171 (12.17) lcx,yftl ∣gτ (t) cl2 " lcx,yft 1 c∣ < ∣gτ (tτ) ∣2 sinh cl(tT+δT-t) c∣ sinh cjSτ 2 sinh Cj(t-tp>-1) + ⅛τy(tT + sτ)"c∣z -----sinh c1^S 9 2 cθ /sinh c∣(t<p + Sφ—t)4~siπh c^(t—trp)∖ ~ 4 ∖ sinh c∣Srp ) ~ 4 if t ∈ [tτ , t«p + Srp] where c ∈ {a,b}. Put Rrp = tτ + (12.18) Sψ then by (12.16) Rrp 1 T — Rrp ≥ 2(c3T — C4 ) and by (12.15) c =: c^l - c4 and so that g^,y(t) ∈ B(a,-^) U B(b,-^) -197- cιsτ = 0(e 2 ) = o(e ∣⅛y(Rτ>-≈l≈ (12.19) clc3τ ∣g*,y(τ-Rτ)-< uniformly for those (xty,T),s. χ9y Now using the triangle inequality A(x,y,T) = A(x,gy^(t)ft) ÷ -^(δτ W,y,T-1) if t ∈ (0,T), p(x,y) = rmf {A[x,y,T]}, sublemma 4.3, (12.18) and (12.19) gives A(x,y,T) = A(x,g⅜,y(Rτ),Rτ) + A(g£y’(RT),y’T-RT) ≥ A(x,c,T) — A(g⅜y(Rτ),c,T-Rτ) + + A(c,y,T) — A(grp (Rrp),c,Rrp) ≥ co X V ∣2 cosh(c2RT) ≥ p(x,c) + p(c,y) - ^∣gτ,y(⅛)-c∣ sinh(c2T) c2∣ x>y,τ r ∖ .∣2 c0sh(c2(τ rt)) - ^2^∣gτ (l-Rτ)-c∣ sinh(c2(τ-Rτ)) = p(x,c) + p(c,y) + 0(e uniformly for (x,y,T) ∈ 4 ) x K2 X [Tθ ,∞). This gives the lowerbound in Proposition 1. XV ∙ > c∩ z c∩ x We prove (12.16) by estimating the total time gy (t) is inside B(a,-^) U B(a,-^) -198- from below and the number of times gψ,y(t) is there from above. With E and (12.20) as in (12.14) and (12.15) ∣{t ∈ [0,T] : g^,y ∈ β(a' u β(a' y))∣ ≥ ∣{t ∈ [0,T] : V(g×'y(t)) ≤ M = T - ∣{t ∈ [0,T] : V(gτy(t)) > M∣ ≥ T - ξ if (x,y,T) ∈ Kq X K2 × P>∞]∙ The last inequality is valid since 64∣{t ∈ [0,T] : V(gx∕(t)) ≤ ⅛4}∣ ≤ ∫θ V⅛y(t) dt ≤ sup {A(χ,y,T)} = E. (x,y,T)∈K1×K2×[T,∞] XV C∩ If c ∈ {a,b} then by sublemma 4.2 gj never leaves {x : |x — c∣ ≤ -ÿ} and comes back to that set without leaving {x : |x—c∣ ≤ cθ}. Doing so contributes at least c2 c1 -Q (1 — A) = ‡g Cj cθ to A(x,y,τ) by sublemma (4.4). Hence, the number of times g^,y is inside B(a,^) UB(b, y) ≤ (number of times g‡Y can go from {x:|x-c| = y} to {x : |x—c∣ = c∩} for c ∈ {a,b}) + 1 ≤ + 1. u 3c1cf Now write -199- {t ∈ [0,T] gτ,y(t) N B(a,-^) U B(a,-^)} = . Ujtj , tj + sJ s∙ > O and N — N(x,y,T) ≤ where + 1 3clc0 then by (12.20) T - E 6, which implies (12.16). < N ∑ s. ≤ N(x,y,T) max{sj(x,y,T)} i-1 1 -200- 13. Some Basic Properties of Agmon Geodesics and Minimal Action Paths. §13.1. Infimum is attained and so on (sublemma 5). §13.2. Proof of lemma 1. X Ö §13.3. On the uniqueness and nondegeneracy of g ’ (sublemma 6). §13.4. There is a unique and nondegenerate minimum of the action for large T’s (sublemma 7). -201- 13. Some Basic Properties af Aeιιιoιι Geodesies and Minimal Action Paths. Infimum is attained and so on (sublemma 5). §13.1. If T > 0 let J = [0,T], (-T.T]. [0.∞) or (-∞,∞) and for γ ∈ C(J) define Λ(γ)== ∫ (Iγ2 + V(7))dl J z (13.1) which we will only write when Λ(">) < oc . We start by showing infimum in ∞ 1 o (13.2) p(xθ , b) = inf{∫θ (⅛ 7~ + V(7)dt∣γ(O) = xθ , γ(∞) = b} is attained when (13.3) ρ(xθ , b) < p(x∩ 0 , a) + p(a,b) Sublemma 5. 1. Assume (13.3). If γ∏ ∈ C([0,Tn]) for Tn ∈ (0,∞] where Tn → ∞, γ^(0) → ×q, 7n(Tn) → b, or 7n(t) → b as t → ∞ if Tn = ∞, and J,(γ^) -*, p(×θ > b) as n —► oo then there exists g that ∣∣γ g njc — g χ∩,b xθ,b ∈ Coo([0,oo)) minimizing (13.2) and a subsequence γ ∣∣τ oo. , -→ 0 as k -→ oo . l lu,1njfJ such Hence by continuity of p(x,b): xQ,b . . Xb is unique and for χ near xθ , g ’ is some minimizing path for /?(x,b) then l∣ex’*■ - sx°‰0,oo)) - » “ X - ×o ■ If -202- 2. Assume (13.4) p(xθ , b) + p(b,yθ) < p(xθ . a) + p(a,yθ). lf 7n e c([θ'τn]) where T» < x τ∏ -* ∞ ’ 7Q(°) → x0 ’ T,n(τn) → y0 x∩,b and J,(γ^) → P(xθ ’ *5) + p(b,yθ) as n → ∞ then there are g and g θ in Coo([0,∞)) minimizing p(×q , b) and p(yθ , b) in (13.2) and a subsequence γ such that ∣∣g xθ,b nk y∩,b and ∣∣g ü — γ n, k (Tn, k Hence by Proposition 1: __ j∙ ∏i L∞[0,^⅛ If g x∩,b — γ ∣∣ oo nk"L∞[O,Tn ] 0 as k → ∞ . x0’b and g are unique and for (x,y,T) near XV. (χ0 , yθ ,∞) gτ, is a minimal action path then ⅛y - 8it°∙‰10,il - 0 and ιι^∙x,yz'rτ ∖ y0,b∣ι ι∣gτ (T- ∙ ) - g ∣∣l∞[0 ι1 → 0 (x,y,T) → (xθ , yθ , ∞). -203- 3. X b Let g ’ be a minimizing path for (13.2), where x satisfies (13.3), and Xb . ^∩,b . x∩,b Xq = g ’ (tθ) for some tθ > 0. Then there is a unique g (t) given by g u (t) = εx,b(t + t0)∣ [0,oo) Remark. Moreover its second variation is positive definite. The last part of sublemma 5.3 implies (see (2.9)): There is no nonzero solution of the Jacobi’s equations. (13.5) f(t) = V,,(gx,b(t)) f(t) on [0,∞) vanishing twice on [0,∞). Since every solution of (13.5) that vanishes at ∞, decays exponentially, this can by (13.8) below, be strengthened to: [0,∞]. No nonzero solution of (13.5) vanishes twice on So we remind on Jacobi’s condition (see Hestenes [1], p. 124) that says: An endpoint and a point between endpoints of a geodesic, without corners, are not conjugate. Proof of Sublemma 5.1. We can assume γ e is defined on [0,∞), by extending it to be (γ (Tn) — b) — ft—T 1 n + b on (Tn , ∞), if Tn < oo . Going to a subsequence we may assume γ → γ weakly in L (0,oo), since a bounded set in a reflexive Banach space is weakly sequentially compact (Berger [1], p. 31) and then ∫ OO ο OO o γz dt ≤ lim ∫ γ∏ dt (also Berger). -204- Now Ascoli’s Theorem (Royden [1]) implies there is a subsequence (call it also) γn such that 7 η → 7 uniformly on compacts if we show {7 } n, is an equicontinuous n n fc ” family and pointwise bounded. Equicontinuity follows from sup ∣f(u) - f(s)∣2 ≤ (t - s) ∫t∣f(v)∣2 dv, 7 (0) → xθ u∈[s,t] s n U and sup A(y ) < ∞ . Pointwise bound follows from ∞>sup Λ(7n) ≥ -A(γn∣[θ t]) ≥ P(7n(t),7n(0)) ≥ p(7n(t),b) - p(b,7n(O)) > λj^∣ ∣7n(t)∣ — constant where we used: There exist 6 > 0 and R > 0 such that p(x,b) ≥ ∣x∣ if ∣x∣ ≥ R (see beginning of §13.2). Then 7(O) = lim 7jl(0) = ×q and for any L > 0 by the Fatou lemma and above Jθ (⅜ + v(γ) dt ≤ ⅛q Jθ(⅜ T,∏ + v(7n))dt OO -1 9 ≤ 1⅛ Jθ ⅛ K + v⅛)dt = P(×0 ’ b)’ for any L > 0. Therefore ∫θ (A ∙γ2 + V(7)) dt ≤ p(xθ , b) so 7(t) → a or 7(t) -→ b -205- .OO η for any L > 0. Therefore ∫ as t → oo . (⅛ γ 0 z ο + V(γ)) dt ≤ p(x∩ , b) so γ(t) → a or γ(t) → b Recall 7^(Tn) -→ b as n —* ∞ and so if γ(t) →a as t → ∞ there is a sequence 0 < Sn < Tn such that γ (Sn) → aas n → oo and we get S∏ oo p(xθ , b) = lim Λ(γn) = lim J ( ) + Uffi ∫ ( ) ≥ p(xβ , a) + p(a,b), u u no o u which contradicts assumption (13.3). Hence γ(t) → b, γ is a minimizing path of (13.2) , .. x0'b and we write γ = g Now we want to show ∣I7 — g n χ0,b II oo -→ 0 as n —+ 00 for this L (0,∞) subsequence, which we do by help from sublemma 4. χ∩,b Given a small ε > 0 pick T > 0 such that ∣g υ (t)-b∣ < ε if t > T and pick X b M such that Tn > T, l∣7n-g θ’ ∣∣loo[0 T] < ε> ∙4(7n) - p(×0 ’ b) + ∣p(7n(θ)>b) - P(×o ’ b)∣ ≤ ε2 if n ≥ M. Then for n ≥ M ∫θ (J ‡2 + V(7n))dt ≥ p(7n(O),7n(T)) > l invf∣^ (p(7 (θ),b) - p(y,b)) ∣y-b∣≤ε n n>M ≥ Xxo > b) - (1 + ~⅜) ε2 and -206- by sublemma 4.1. By above ο T oo p(xθ , b) + ε2 ≥ Λ(γ ) = (∫ + ∫ ) n 0 T ≥ p(×o , b) - (1 + ^) ε2 or C ⅛ 7n + v(7n}) dt ≤ (2 + ⅛) ζ2 ,p z η n 2c∣ and ∣7n(T) - b∣ ≤ ε . By sublemma 4.4 this implies sup sup ∣7n(t) n≥m t∈[T,oo) 1 + q(2 + _2_ 2c1 ) ε ∙ Assumption (13.3) makes the minimizing paths g χ0,b for (13.2) stay outside a neighborhood of a and by sublemma 4.1 stay away from b on any bounded subinterval χ∩,b of [0,∞) and by Hilbert’s differentiability theorem (see Hestenes [1]), g ∈Coo([0,∞)) as function of t. Proof of Sublemma 5.2. As in the beginning of the proof of sublemma 5.1 we get 7∣ , 72 ∈ C(0,oo) and a subsequence (call it also) γ with 7nl[0,ιn] "* 7l ’ 7"<T" ^ ' >'∣θ∙ ^≠1 "* 72 -207- uniformly on compacts in (0,∞), ∫ y^, ο — ∙,2-√ γ ∣ dt ≤ lim ∫ γ∏ dt Tn Γ and OO o Z 7o dt < lim Γ 0 2 n 0 o 7 (T — t) dt as n n ∞ Using Fatou’s lemma we get o° J 12 ∙>0 ⅛ -h + V(71)) dt ≤ p(x0 , b) + p(b,yθ) and ro° 1 ∙2 ∙∣n ⅛ ^2 + V(7o)) dt ≤ p(x∩ , b) + p(b,yn) ‘ V ' 4,' ■ — - ' V ' w Therefore γ1(t) → c1 ∈ {a,b} and 72(t) → c2 e {a>b} as t → 00, implying there is a subsequence u^ → ∞ as £ —+ 00 and sequences un and Tn ≤ vn^ such that 7n^(u∏P → cι and vn with un^ ≤ ~4 <⅛ as £ → 00 . Hence p(xθ , b) + p(b,yθ) = = Tn, *£ 1 9 n£ Um ∫ (i 7in + V(γ )) dt ≥ lim ∫ ( ) + lirn ∫ £—>00 0 2 n£ £ 0 £ 0 Un£ ≥l⅛∫ 0 Tn, vng * £ ()+lim∫ () + lim∫ t jo ()≥ £ un, ≥ Xxo ’ cP + p(cl ’ ⅛) + z,(c2 ’ yθ) and we conclude from (13.4) that c 1 - c2 -- b. £ ( ) ≥ -208- Now γ∣(0) = Xq , ?2^θ) = anc' T,l(∞) = b = 72(∞) implies ∕>(x0 , b) ≤ ∫∞(i 7∣ + V(γ1)) dt and p(y0 » b) ≤ ∙f∞Φ2 + v(τ,2)) dt∙ We also have for any L > 0 i2 n + V(7n)) dt + Jl(A + V(72)) dt . ≤ Um ∫⅛ rn + V(7J) dt + lim ∫ T, (±(7*(T-t)∕ + V(γ (T-t)) dt n 0 n n n q η n Tn ≤ l⅛2 ∫θ*^(i y2 + V(7n)) = p(xθ , b) + p(b,yθ). Hence ∫∞(⅜ 7ι + V(71)) dt = p(xθ , b) and ζ°⅛ 72 + v<72)) dt = P(yθ , b) 71 = g Since χ0,b and 72 = g y0,b p(x0 , b) + p(b,yθ) < p(xθ , a) + p(a,yθ) implies P(x0 , b) < P(x∏0 , a) + p(a,b) and P(yθ , b) < P(yg > a) + p(a,b) from the subsequence 7 we get, by sublemma 5.1, a new subsequence γ nk above such -209that χ∩,b l∣g θ - nk L°°[0,-?J and x∩,b ∣∣g θ -γn (T- ■ )∣∣ X b ∙ x∩ ,b g ’ (t) := g 0 as k -→ oo . L∞[0⅛] (t + tθ)∣ [0 ∞) min'm*zes (13.2) with xθ = χ and to prove the uniqueness one can use an argument similar to one given in Kobayashi [1], p. 99. χb Now assume the second variation of g ’ is not positive definite (and we will χfl,b get contradiction to the second variation of g υ is non-negative). Then by (2.9) ,, χ∩,b there are tθ ≤ tj < tg < ∞ and a solution f ≠ 0 of f(t) = V''(g υ (t)) f(t) on (tj,tg) with f(t∣) = 0 = f(t2). Then f(t1) ≠ 0 and so there is some h ∈ Cgo((0,∞),Rd) with (f (tχ),h(t1)) < 0 but h(t2) = 0. Now for small 6 > 0 put ^(t) _ + 0h(t) where f ig eχtended to be zero outside (t∣ , t2). Then 0 < Jo0⅛∕ + (V,,(g )ψb , ⅞p6)) dt = ∕2(If l2 + <v"(gxθ,b) f,f)) dt tl + 2< ∕2((i,h) + <v"(gx°∙bwι)) dt tl + i>2 ζo(∣h∣2 + <v"⅛1⅛'b)hjh,) dt = 2« ιf(t1)Λ(t1)) + s2 iθ (IN2 + <v"fex0'b) h h)) dt -210- which is strictly negative for 6 small enough. §13.2. Proof of Lemma L· ~} is compact. By continuity First we show K = {x : p(x,a) = p(x,b) = of X ι→ p(x,a) and x ∣→ p(x,b) K is closed and it is bounded since p(x,a), ρ(x5b) “* ∞ as ∣x∣ → oo . To prove the last statement pick δ > 0 and R > 0 such that V(y) ≥ δ if ∣y∣ > R. When ∣x∣ ≥ 2R, p(x,a), ρ(x,b) ≥ inf p(x,y) = inf ∣y∣=R {inf [∫1 ^2V(γ) ∣γ∣ds : γ(0) = ×, 7 ∣y∣=R 0 7(1) = y} ≥. ⅛fτ, {i∏f {>∣20 ∫1∣γ(s)∣ds : 7(O) = x, ∣y∣=R 7 0 7(1) = y} ≥ 425(∣x∣-∣y∣) ≥ λ[j ∣χ∣∙ If Xq ∈ K then using sublemma 5.1 we see the problem p(a,b) = inf{∫°° (Aγ2 + V(7))dt∣7(-∞) = a, 7(00) = b, 7(O) = xθ} —∞ i (13.6) has a minimizing path gxθ . Pick tθ ∈ (0,∞) and put x = gxθ(-tθ). minimizes p(x,b) in (1.3.2). x,b. Then g ’ (t) := gxθ(t - t0)l[0jθc,] xb Since xθ = g ’ (tθ) and p(x,b) < p(a,b) < p(x,a) + p(a,b) we get from sublemma 5.3 g x∩,b xb (t) = g ’ (t + tθ)∣is the unique minimizing for p(xθ , b) in (13.2) and has a positive definite second variation, x∩,a Similarly gχθlζ-oo qj (— t) = g (t) is the unique minimizing path for p(×q , a) and has positive definite second variation. Therefore this gxθ is the unique minimizing path for (13.6). -211- Xä Xb Now we apply lemma 2 and we get unique g ’ and g ’ for x near xθ and their second variations are strictly positive. Moreover p(x,a) and p(x,b) are Cσo near xθ and p'(x,c) = -gX,C(0) . By above g θ' = g ∣ and g °’ = gχ∩∣∕ oo ∩1 U x0l[0,∞] θ ( ,υj so p,(x,a) - √(x,b)∣χ = χ0 = 2gxθ(0) which is not zero since 1 g2θ(0) = V(xθ) ≠ 0. The continuity of p,(x,a) — ρ∖x,b), implies it’s nonzero near xθ and lemma 1 follows, since K is compact. xb §13.3. On Uniqueness and Nondegeneracy of g ’ fSublemma 6). We will show that if (13.3) holds and x b there is a unique g ’ minimizing (13.2) (13.7) and the second variation of xb g ’ is positive definite holds at x = xθ then (13.7) holds for x near xθ . Sublemma 6. Assume (13.3) and (13.7) for x = xθ . Then there are 6θ > 0 and k > 0 such that if x ∈ B(xθ , όθ) then there is a unique gX,b and if ∣∣γ - gX’b||Loo^0 qq) ≤ <5o then (13.8) ∫°°(i)2 + (V"(γ)v,r∕))dt ≥ k∣∏∣2 0 L∞[0,∞) for all η ∈ Dθ(0,∞) := {η ∈ AC([0,∞),R^) j}(0) = 0 and η, ή ∈ L2([0,∞),Rc^)}. -212- Proof. By sublemma 5.1 ∣∣g x,b — g xob. . L . → 0 as X → x∩ so it suffices to [0,∞) υ prove (13.9) x∩,b There exist δ, > 0 and k, > 0 such that ∣∣γ — g υ ∣∣ 0o ≤ 61 l[o,∞) implies ∫°°(⅛2 + (V"(ι)η, η)) dt > k,....... for all η ∈ Dθ(0,∞). 0 L∞[0,∞) X L· To see (13.9) implies there is a unique g ’ for x near xθ , we assume γ∣ and 72 are two minimizing path for p(x,b) in (13.2). By sublemma 5.1 they are both close to g for x close to xθ . Hence, 7∣ = 72 + η for η ∈ Dθ(0,oo) and we get by (13.9), using 7^ and 72 satisfy the Euler-Lagrange equation 7 = W(7): 0 = -Λ(71) - -4,(72) = -Λ(72 + τ)) ~ ∙^(72) = J∞[¾^ - ⅛ + V(,2 + ,) - V(72))dt -∞.r)2 = ∫q ⅛ - ^2 ' η + v^72 + rti ~ v(72)) dt -∞z⅛2 ∫ (-y + V(72 + η) - V(72) - VV(72) ∙ η) dt •2 5 ∫θ (¾- + (V,,(72 + z),7'Ι)) dt where ΙΙ^Ι^οο^ ≤ II'7∣Il∞[0,oo) ÷ l∣71 - 72∣∣L∞[0,∞) ' Hence ^2 + Z is close -213- x∩,b to g and by (13.9) 0 = ,Λ(γ1) - ∙X(γ2)≥ ∣kι∣l∏l∣2θoiθ ooj = ⅜k1l∣Υ1 - 72∣∣l∞[0,∞) and we have γ∣ = γ2 ∙ To prove (13.9) we split [0,∞) into two pieces [O,tθ + 1] and [tθ + 1, oo), for tθ > 0 to be chosen. On [0, tθ + 1] we use the positivity of the second variation of x∩,b g υ and on [tθ + 1, ∞) we use V '(b) > 0. Recall (10.1), there are cθ , c^ and c2 > 0 such that if c ∈ {a,b} and |x—c∣ < x∩,b c∩ /. _ fbr⅛p r? <' ∖f∣∣(γ∖ r½ Pirl· > ∩“ such that 1er u ft,l — bl < -^ if t > tr. . U ~χ . v~, ~∙1 . -------- u _ " ----------'° V ' , — 2 — U Then (13.10) there exist ⅛2 > 0 and k2 > 0 such that ∣∣γ - gxθ,bH1<xwθ + 1] ≤ *2 tθ-∣-l o implies Jθ (⅛2 + (V,,(γ)z∕1z7)) dt ≥ k2∣H∣L∞[0,tθ + l] Vz∕ ∈ Dθ(0,oo) and (13.11) If∣∣7 -SX°’b|lLoo[to+l50o) ≤ ⅛θ then ∙ftθ + l^ dt ≥ ^hllL∞[tθ + l,∞] + Vz? ∈ Dθ(0,∞) Now (13.10) cr and 0j = min{02 , v0 and (13.11) imply (13.9) with k∣ = min{k2 , cι -214- To prove (13.10) set kg := cninf,n , q∈Dθ(0,∞) tn+1 Jθ t0 + 1 . o . x∩,b. {∫" (√2 + <V"(g θ'‰>)dt} 0 ∣j)∣2dt = l Then kβ > 0; proof by contradiction. Assume kθ = 0. Then there exist ηa with = 1 and IVnllτ2 Lz[0,tθ÷l] Γ t∩ + l x∩,b (h2 + (V,,(e θ 'lHn.Hnb dt → 0 as n By Alaoglu’s Theorem, a bounded set in a reflexive Banach space is weakly sequentially compact (see Berger [1]) and we may assume ηn wβa∣kly → ή ∈ o L^[0,tθ + 1]. Since ∣Ιn(t) - τ7n(s)l^ ≤ (t — s) ∫ ⅛2(u) du < (t — s) if t — s > 0 and for all n and i∕n(0) s = 0 we may assume by the Ascoli’s theorem (R,oyden [1]) that ηn → η in C[0,tθ + 1]. If we show η = 0 we get contradiction, by above, since (∫ t0 + 1. ,,. x∩,b. <v"(g )Ιn,ΙnMtl = θ(∣∣⅛∣∣2∞r∩ t 0 b , 1l) → 0 as n → ∞ . Lυ,tθ÷lj We have a small problem in using the positivity of the second variation of g X0’b . j 9 since we only know 77nl[θ tθ + i] ∈ D(0,tθ + 1) := {η ∈ AC([0,tθ + 1],R ), ⅛ ∈ l∕ and q(0) _ Qj a∏d j‡ js nθt necessarily in Dθ(0,tθ + 1) (we don’t have qn(tθ + 1) = θ)∙ -215- Define D(O,tθ + l)f 3 ι-, f ∈ Dθ(O,tQ + 1) by f(t) if 0 ≤ t ≤ tθ f(t)(tθ + l-1) if tθ < t ≤ tθ + 1 f(t) = If we show ή = 0 then η = 0 and we are done. On [tθ , tθ + 1] we have xn,b „„j „2 τ∣∣∕r, 0υ f2 ≤ 2f2(tθ + 1 - t)22 +I or2 2fz and c^≤- ∖ V"(g hence ∫ tθ-(-l (t)) ≤ c% , X b (f2 + (V"(g θ’ )f,f)) dt ≤ (2 + ^^2)) ∫ θ+ (f2 + <V"(gxθ,b)f,f)) dt . cl t0 η Therefore, by uniform convergence of ηn and weak L -convergence of ⅛n , we have ∫ t0 + 1 .9 „ x∩,b (⅛2 + <V"(g θ ) η,η)) dt ≤ 0 (∣√n∣2 + <V,,(g θ’ ) ηn , ijn)) dt ≤ Uffi ∫ ° n 0 ≤ (2 + (⅛)) lim ∫ θ+ (⅛2 + <V"(gxθ,b) ηn , r7n>) dt = 0 c∣ n 0 By the positivity of the second variation of g x∩,b ,ij = 0 and we conclude k^ > 0. -216- Therefore tQ + l a jθ (⅛2 + {VZ/(gx∩,b. 0’ } η, ηyi dt > k3 IMI 2 L∞[0,t0 + l] (13.12) for all η ∈ Dθ(0,∞) where k3 > 0. Having in the mind ∣τ∕(t)∣2 ≤ t ∫ ∣⅛2(u)∣ du if 77(O) = 0, hence 0 llr>wllL∞[0,t0 + l] ≤ (t° + υll^llL∞[0,tθ + l] lw‰o+l]si02 ⅛[0,t0 + 1]∙ we pick implies for xθ,b 61 > 0 such that ∣∣γ - g ∣∣l∞[o θo) ≤ δl k3 -^ V"(γ(t)) ≥ V"(gxθ,b(t)) - (to+1)2 t ∈ [0,∞) (V is uniformly continuous on compacts). For such γ,s tθ +1 ∫ (⅛2 + (v"(γ) η, η}) dt ≥ 0 ≥ k3IIOII 2 _ k3 IMl2L2[0,t0 + l] L2[0,t0 + l] (⅛ + l)2 kq 2 > k3 ≥ ⅛h∣ι bll L∞[0,t0 + l], L2[θ,t0 + 1] ~ (t0 + 1)2 which completes the proof of (13.10). -217- Now we prove (13.11). From the choice of tθ and since ×∩,h [∣7-g cn ∣∣L∞[tθ + l,0o]≤ T we have, (ή2 + {V,,(γ) η,η)) dt ≥ ∫°° (⅛2 + c2 η2) dt. tθ+ 1 to+1 ∫°° (13.13) ∣η(t)∣ is continuous and tends to 0 at ∞. Pick tj and t2with tθ ≤ t^ < 00 h(t1)l = M1oo[tθ+ι,oo) b(t2)l - ⅜lli,llL∞[tθ+ι,∞) and ∣Ι(t1)∣ ≥ b(t)∣ > ∣τ7(t2)∣ for t ∈ [t1 , t2]. Then by (13.13) Θ2 + <V',(γ) η,η)) dt ≥ ∫ 2(⅛2 + c1 η2) dt ≥ ∫°° t0 + 1 t1 lh(t2)l-H(t2)∣∣2 2 . r ι z i x ≥ —(⅛ - τ1τ + cι t11⅛t2w°lit2-tι) I ll,llL∞[tθ+l,oo) ⅛2-t1) + ⅛t2~tl)) ≥ y ∣W∣L∞[tθ+l,oo) ∙ -218- §13.4. There is a unique and nondegenerate minimum of the action for large T’s. The results stated below imply lemmas 3.1, 3.3 and a part of lemma 2.2. Sublemma 7. Assume (13.4) ∕>(×θ , b) + p(b,yθ) < p(xθ , a) + p(a,yθ) t . x0’b , y0'b there are unique g and g , each one having positive definite second variation. Then there are positive constants 6θ , Tθ and k such that if (x,y,T) ∈ B(xθ , 6θ) × B(yθ , 6θ) × [Tθ , ∞) then « « XV 1. There is a unique minimal action path gψ and if ∣∣γ - ∣∣l∞1 ,oo) ≤ δ0 τ then ∫θ (⅛2 + (Vz'(γ) η, η)) dt ≥ k∣∣η∣∣L∞[0,T] ∀r∕ ∈ Dθ(0,T) = {η E AC([0,T],Rd) : ⅛ ∈ L2[0,T] η(0) = 0 = ι7(T)} 2. Λ(gψ,y + η) - -A(gψ,y) ≥ k min{0θ , (|»7||^οο[0,Τ]^ fθr a11 η 6 D(/0’T)· 3. If (xn , y∏ , Tn) → (x,y,T) as n → ∞ then ∣∣⅛Tyn( ∙ ⅛) - SXryHLoo[0,T] θas ∞ . -219- Proof of Sublemma 7.1. (t) — b∣ < Pick tθ > 0 such that ∣g θ and ∣g θ (t) — b∣ ≤ -Q if t > tθ where cθ is as in (10.1)-(10.3). By (13.10) there are (13.14) ∣∣7 - g θ > 0 and ∣∣1∞[t0+l] > 0 such that If ≤ 04 θr ∣∣7 - S θ ∣∣L∞[tθ+l] ≤ δ4 (tθ-(-l) then ∫θ (⅛2 + (V,,(γ) η,η)) dt > k4l∣Ι∣∣L∞[tθ+1] for all η ∈ D(0, tθ + 1) = {η £ AC([0, tθ + 1], Rd) : ⅛ ∈ L2[0, tθ + 1] and >∕(0) = 0}. By sublemma 5.2 there exists 65 > 0 and Tθ > 4(tθ + 1) such that (13.15) If ∣x - Xq∣ ≤ ⅛5 , ∣y - yQ| ≤ 65 and T ≥ Tθ X b then ∣∣g⅜y - g θ’ ∣∣l∞ Τι ≤ ⅛ min(04 > c(p lO,2∙∣ and ∣∣g⅞,y(T - ■ ) - g °’ l∣1oθ[θ Tj ≤ ⅜ min(64 > 5) · Now we prove that if (13.16) ∣x - xθ∣ ≤ 65 , ∣y - yθ∣ ≤ δ5 and T ≥ Tθ then ∣∣γ - gτy∣∣L∞[0-T] - 3 min(04 ’ cθ)> XV. for some minimal action path grp , implies -220- ∫θ (02 + (V,,(γ) η,η}) dt ≥ min(k4 , , y )∣∣Ι∣∣2 οο[0;Τ] for all η ∈ Dθ(0,T). (13.14) and (13.15) imply (to+1) ∫θ (√ + (V,'(7) η,η)) dt > k4 IH⅛∞[0,tθ+1] (13.17) and ∫ T-(tθ + l) (ή2 + (V,,(γ) η, 77)) dt > k4∣∣η∣L oθrm∕t , ∙∩ ψi b l for all η € Dθ(0,T) and γ⅛ like in (13.16) as can be checked using the triangle inequality. On [tθ + l,T-(tθ + l)] we have V,'(γ)>c^ since ll7-bllLoo[£+1)T_(£+1)] ≤co There we split into two cases depending on whether l^(t)∣ ≥ >∣∣L∞[t0+l,T-(tθ + l)] for all t ∈ [tθ + 1, T — (tθ + l)] or not. In the first case we get T-(to÷1) (13.18) (∫ t0 + 1 9 „ 9 τ-(tθ + l) (⅛2 + (V"(γ) η, η)) dt ≥ cf ∫ √(t) dt t0 + 1 > TllηllL∞[tθ+l,T-(tθ + l)](τ0 - 2<t0 + 1)) 9 2(tη + l) 99 cl 2 > —_— c1ihiiLOO[to+1)T_(to+1)] > 2-h∣∣L∞[t0+l,T-(tθ+l)] ∙ -221- In the second case do as in the proof of (13.11) and get T-(tθ + l) (13.19) J t0 + 1 (⅛2 + (V,,(->) η. rj)) dt ≥ y∣∣t∕∣∣ 2 Loo[tθ+l,T-(tθ + l)] ∙ Tθ+1 rp Now write = Γ ∫ 0 T — (tθ + l) + ∫ tθ + l 0 rp + ∫ τ-(tθ + l) and (13.16) follows from (13.17)-(13.19). (13.15) implies that if (x,y,T) ∈ B(xθ , 6g) × B(yθ ,δg) x [Tθ , ∞] then the difference of any two minimal action paths between x and y in time T, has uniform norm less than (13.20) 1 XV min{6^ , cθ}. Hence the uniqueness of gy follows from If (x,y,T) ∈ B(xθ , ig) × B(yθ , 65) × [Tθ , ∞] and grp,j is some minimal action path, then t? € Dθ(0,T), ∣∣i∕∣∣^oojθ 'p] ≤ 3 min(62j , cθ) implies ^∙(gj'y + η) — ->t(gjy) ≥ kg∣∣t7∣∣2∞[q,T] c where kg = τ> min{k^ , c2 , -¾} ∙ To prove (13.20) we integrate by parts, use gψ,y = W(gψ,y) and we get J∙(g∕p,y + τf) — J∙(gy,y) = = jTφ2 + v(⅛,y + Ι)- V(gτ,y) - VV(g*'y)η) dt = ⅛ iτ(⅛2 + {V"(gτ,y + 2S) η, η) dt z 0 where II^I∣l∞[0,t] ≤ ∣∣Ι∣∣l∞[0,T] ' •222- Now (13.20) follows from (13.16) and sublemma 7.1 from (13.16) and the X V uniqueness of gy . Proof of Sublemma 7.3∙ We prove a slight ∣y stronger statement that we use in proving sublemma 7.2. (13.21) If γn ∈ C([0,Tn]), Tn - T, ->n(0) → x and γn(T) → y, where (x,y,T) ∈ B(xθ , 05) x B(yθ , i5) x [Tθ , ∞), τ and ∙Λ,(γn) = Jθ (^7n + ^(7n)) dt → A(x,y,T) then sup ∣7n(-⅞β) - gτ^(t)l → 0 as n → ∞ . 0<t≤T 1 If (13.21) is false then we have for a subsequence, as in the proof of (13.10), 0<tP<Tl7n(^a) - gT>(t)l ≥ ε0 > 7n⅛) → 7(t) 1 ♦ rF in C[0,T] (by Ascoli)and g⅛7n(-ψβ) —- 7(t) weakly in L2[0,T] (by the Alaoglu theorem). Then 7(O) = x, 7(T) = y, ∣∣γ - ≥ ε0 but Jθ ⅛ i2 + V(7)) dt ≤ ⅛ Jθ «ft 7⅛)2 + v<7n⅛)) dt = ⅛ T“ J (⅛(⅛)2 7n(u) + V(γn(u)) dt 1n θ z 1 - A(x,y,T) xy which contradicts the uniqueness of gψ for (x,y,T),s as in (13.20) and completes the proof of (13.21). -223- Proof of Sublemma 7.2. By (13.20) it suffices to show that if (13.22) 6 = g min(6ij , cθ) then {⅜x∕ + Ιψ) - ∙^(gψy)} > θ ^inf (x,y,T)∈B(xθ,-^)×B(yθ,-^)×[Tθ,∞] »?T € Dθ(0,T) and ∣∣r∕-p∣∣ ≥ δ . Proof by contradiction: If not we have (xπ , y∏ , Tn) → (x,y,T) ∈ B(xθ , 65) × B(yθ , 65) × [Tθ ,∞) U∙(gy1^,yn + with -*, θ as n -*, 00 ll^Tn∣∣L∞[0,Tn] ≥ δ > θ ∙ If T < ∞ we get a contradiction using (13.21). ,. ∙ . x0'b , y0'b contradiction, since g and g are unique. If T = ∞ sublemma 5.2 gives 224 14. Some Estimates (Sublemma £_). Let Co , C∣ and C2 be as in (10.1) and put (14.1) '0 C3 = (i+-)2 t2 + c√ Then we have Sublemma 8. 1. Uniformly for x ∈ B(b,c3) and i ∈ {1. ..., d} /1 A O∖ x b ∕~,u∕+λ k∖ Z,, — u∖ λ vr v ~u>∙t 1 I n∕rι.. u∣2o c_ — min{w∙ l 1 ,2ω-ll jft ∖ T u(∣λ — uj — 1,. x,b, „ = ~wi 1(g (t))i and hence (14.3) ρ(x,b) = i(Ω(x-b),(x-b)) + 0(∣x — b∣0) uniformly. 2. For any compact Kj Ç {x p(x,b) < p(x,a) + p(a,b)} there is a constant E I such that (14.4) ∣gxb(t)∣, ∣gxb(t) — b∣ ≤ D∣ e — u∙ t 1 uniformly for (x,t) ∈ K1 x [0,∞) and (14.5) xqb x9b x1b xob ∣g 1 (t) - g 2 (t)∣ ≤ D1]Jg 1 - g 2 I∣l∞(0,oo) e ^ωl2 uniformly for (x∣ , ×2 , t) ∈ K2 × [0,∞). -225- 3. For any compact K∣ C {(x,y) ∈ Rt^ × R*^ p(x,b) + p(b,y) < p(x,a) + p(a,y)} there are Tθ and D2 such that uniformly for (x,y,T,t) ∈ K2 × [Tq , ∞) × [0,T] (14∙6) ∣gτy(t)l, ∣gτy(t) ~ b∣ ≤ D2^e 1 + min{∣y-b∣,cθ}e ∖ Hence (14.7) Eτ(x,y) = ⅛⅛y(t))2 - V(g*'y(t)) = 0(e_WlT) and (14.8) A(x,y,T) = p(x,b) + p(b,y) + 0(e — ω1 T ) uniformly for those (x,y,T),s. 4. Let K∣ be as in part 2 then there are 6θ > 0 and Tθ such that (14.9) XV —2ix¼ T —u>ι T λ ⅛y(T) = Ω(y - b)(l + 0(e 1 )) + 0(e 1 + ∣y-b∣2) uniformly for (x,y,T) ∈ K∣ × B(b,5θ) × [Tθ , ∞). 5. Assume p(xθ , b) < p(xθ , a) -(- p(a,b), there is only one g definite second variation. χn,b and it has positive Then there are 6θ > 0 and Tθ < ∞ such that (recall xb XV sublemma 6 and 7 say there are unique g and gψ ) (14.10) x,b . ⅛Ty(t) - 8τ x,b. ∣gτy(t) - g^ ,"(t)l — 0(e — ω1T — c-∣t -it —C1(T—t) 1 + ∣y-b∣2 e r ,} 1 e C∖ if t ∈ [0,T] -226- ⅛y<t) - gx,b(.)!2 (14.11) + e ⅛τy(τ-t) - ey'b(t)∣1 'cιi⅛^t∖ ) e if t ∈ [Ο,ΐ] and (14.12) ∣⅛i'1(>) _ g⅛y¾)∣2 = 0(∣∣gxlyl - 42i2||2Loo[0,T] e^'1* + ∣y1 - y2l2 e everything uniform on B(xθ , όθ) x B(b,5θ) × [Tθ , ∞). Proof of Sublemma 8.1. Write V'(x) = «Ï (χ-b)ι u¾(x-b)d where 0 < ω∣ ≤ u∣2 <■ ■■< Wi and (14.13) R(x) ≤ Dθ ∣x — bp for ∣x — b∣ ≤ cθ . X t) With g = g ’ (14.14) we set g = f + r where (14.15) i.e., fi(t) = (x - b)i e 1 + b; + R(χ) 1(T-t∖ 1 ') if t ∈ [o,τ] -227- (14.15) fj(t) = (x — b)j e i + b∣ i.e., fj(t) = ωp(f — bj), f(0) = x and f(∞) = b so f(t) = ω? r∣(t) + Rj(g(t)), ri(O) = 0 and Γj(∞) = 0. Therefore (14.16) .t „ ~ω,it .. sinh ω∙t oo —w:s Γj(t) = --ι,z- Jθ sinh(ωis)Ri(g(s))ds------- σp-l- Jζ e Rj(g(s))ds and <14.17) -ti,j∙,√ f∙(t'' iι∖v∕ = *"e -ω∙t t 1 ΓJ βιnbf∕∣).s^ ds —*v-∙-7 —j∖O∖'∕∕ — 0 — cosh ω∣t ∫ oo e —ω∙s R∣(g(s)) ds Recall sublemma 4.1 says (14.18) ∣gx,b(t) - b∣ < ∣x - b∣ e c ''11t'' if ∣x - b∣ ≤ c Without loss of generality we assume nc∣ £ {ω∣ , ..., ωd} for n = 1, 2, ... (take smaller c∣ if needed), If Cj < 2c2 <∙∙∙< 2 n∩ + l cj and we put <l(n) = Σ i⅛l (√-(21> + 1 c1)2) for n = 0, 1, ...,nθ and for r > 0, βθ(r) = 1 and /?j + ^(r) = 1 + 2Dθ /??(r) q(j) γ, we get -228- (14.19) If 0 ≤ j ≤ nθ and ∣x — b∣ ≤ c^ then xb —25c1t ∣g ’ (t) — b∣ < ∣x — b∣ /?j(|x — b∣) e implies Xb —min(ω1, 2i + lc1)t ∣g ’ (t) - b∣ ≤ ∣x - b∣ /?j(|x —b∣) e For j = 0 this follows from (14.14). For the induction step, we assume ∣gx,b(t) ~ N ≤ β∖× — b∣ e αt for t > 0 and 0 < a < ω∣ and 2α ≠ ω∣ , for i = 1, ..., d. We obtain by (14.9) and (14.12) t t1 ,td∙S , c, t oo — _ 4√∙S ltd 1 lri(t)l ≤ e ⅛,. J e 1 lRi(g(s))| ds + 2_ ∫ e 1 ∣R1(g(s))∣ ds 1 10 it 1 ~~ (a)* 1 D,∕⅛-b∣ι fe-2αt-e-"it ~ ωi I e-2αt , (ωj+2α)J ∣Wj-2α∣ 2Dθ∕72 —min{ωp2α}t ≤ ∣ω2-(2α)2∣ 6 Hence ∣gi(t) - b∣ ≤ ∣fi(t) - bi∣ + ∣ri(t)∣ ≤ ∣xi - bi∣e — ω∙t 1 + 2D∩∕12 9 —min(ω∙,2α)t 9 ° 9 ∣x - b∣2 e ∣ω2-(2α)2∣ which proves the induction step and therefore implies (14.19) with j = nθ (14.20) ∣gx,b(t) - b∣ ≤ ∣x - b∣ β whenever ∣x — b∣ ≤ c^ and t > 0. +1(∣x - b∣) e -229- Now (14.20) and (14.13) put into (14.16) and (14.17) imply (14.2). (14.3) follows by putting (14.2) into p(x,b) = ∫°°(i(gx,b(t))2 + V(gx,b(t)) dt = ∫°°(gx,b(t))2 dt. 0 2 0 Proof of Sublemma 8.2. Let xθ ∈ {x : p(x,b) < p(x,a) + p(a,b)} and pick δ > 0 such that B(xθ,6) C {x : p(x,b) < p(x,a) + p(a,b)}. By compactness it suffices to prove (14.4) for x ∈ B(xθ , 6). By (12.9) there exist ε > 0 such that (14.21) U {gX’b(t) : t ∈ [0,∞]} ∩ B(a,ε) = φ. x∈B(xθ,6) xb Once g , is inside B(b,cθ) it doesn’t leave B(b,Cg), by (14.14) and we bound from xb above the total time the g ’ ,s spend outside B(b,Cg) by p(x,b) ≥ ∫ V(gX’b(t)) dt ≥ ¾⅛t : |gX’b - b∣ ≥ ε}∣ . {ti∣g×,b(t)-b∣≥ε} X ∣} - l —.... Therefore there is atθ < ∞so that ∣g ’ (tθ)-b∣ ≤ cθ for all x ∈ B(xθ,<5) and (14.2) implies (14.4) for x ∈ B(xθ,6) . To prove (14.5) we pick ε > 0 such that ∣x — b∣ ≤ εθ implies V (x) ≥ such that by (14.4) ∣gx,b(t) — b∣ ≤ D∣e Now we get ≤ -<p for x ∈ K∣ and t > T∣. and T^ -230- (14.22) x1b x1b , x1b , xηb g 1 (t) - g 1 (t) = V'(g 1 (t)) - V,(g 1 (t)) ∫ 1 ,, x9b x1b x9b x1b x9b V"(g 2 (t) + s(g 1 (t) - g 2 (t))) (g 1 (t) - g 2 (t)) ds 0 ,, x9b x1b x9b V"(g 2 + Z(t)) (g 1 (t) - g 2 (t)) where x9b x9b x9b x1b |g 2 (t) + Z(t) - b| ≤ |g 2 (t) - b| + |g 2 (t) - g 1 (t)| εn x9b xηb ≤ -f + ∣g ~ (t) - b| + ∣g i (t) - b∣ ≤ εθ if Hence t≥ T∣ and Xj 1×2 ∈ K∣ j2 x1b x9b 9 |g 1 (t) - g 2 (t)∣2 dt2 ,, x9b xηb x9b xηb x9b = 2(V"(g 2 (t) + Z(t))(g 1 (t) - g 2 (t)), (g 1 - g 2 )) 9 xηb x9b 9 ≥ ω2∣g 1 (t) - g 2 (t)∣2 ift ≥ T1 As in the proof of sublemma 4.1 we get xηb x9b 9 x1b x9b 9 —ω1(t-T1) ∣g 1 (t) - g 2 (t)∣2 ≤ ∣g 1 (T1) - g 2 (T1)∣2 e p 1 ≤ e — ω1Tη x1b x9b 9 —u>ηt 1 1∣∣g 1 -g 2 I∣l∞[0,oo) e * if t > T-£ which gives (14.5). -231- Proof of Sublemma 8.3 and 8.4. It suffices, by compactness, to show (14.6) holds near any (xθ , yθ) with p(xq , b) + p(b>yθ) < p(χθ 5 a) ÷ p(a,yθ)∙ Given such (xθ , yθ) we pick (14.23) > 0 such that p(x,b) + p(b,y) < p(x,a) + p(a,y) for all x ∈ B(xθ,61) . Now we claim there exist 62 ∙> θ and ^*1 > θ such that (14.24) u T≥Tj ( u {gτy(t) '∙t e Lθ>'rJ}) x∈B(xθ,<51) n tJ(a,∂2) = Φ- y∈B(yθ∕1) We prove the claim by contradiction. Otherwise thereexist Tn —► ∞, 0 ≤ tn < T∩ xn → X ∈ B(x0,δ1) , yn → ÿ ∈ ¾P7) and gτ∏,yn(tn) → a as n → ∞ . Then by Proposition 1 p(x,b) + p(b,y) — min {∕,(χ,c) + p(c,y)} c∈{a,b) = nl⅛0 A(xn , y„ . T„) > 1™ .∣0"⅛⅛l"'y")2 + V(g⅛'yn)) <U + ⅛ J " ⅛(⅛yι,)2 + v⅛τ", ">) K ≥ p(x,a) + p(a,y) which contradicts (14.23). -232- Put (14.25) 6g = min^2 ∙>------ - —i} an^ pick ^4 > θ such that d+⅛)5 V(x) ≤ Ô4 implies x ∈ B(a,δ3) U B(b,63) since 63 < <$2 and (14.24) we have ∣{t ∈ [0,T] : ∣gfp^(t) — b∣ ≤ 63}∣ > ∣{t ∈ [O,T] : V(g*'y(t)) ≤ 04}∣ ≥ = T- ∣{t ∈ [0,T] : V⅞y(t)) > ⅛}∖ ≥ τ - 04 max {p(×,b) + P(b,y)} =: T - tθ x∈B(xθ,61) y∈B(yθ,<51) (defining tθ) for all x ∈ B(xθ , 64), y ∈ B(yθ , 5∣) and T ≥ some T2 ≥ T∣ Here we used ⅛∣t ∈ [0,T] : V(g⅜y(t)) ≥ 64}∣ ≥ A(x,y,T) and Proposition 1 that says A(x,y,T) → min {p(x,c) + p(c,y)} c∈(a,b) = p(x,b) + p(b,y) ≤ max{p(x,b) + p(b,y)} uniformly on B(xθ,5^) x B(xθ,6∣) as T ∞ . -233- For T ≥ T2 put t1 := jnin^O : ∣gj,y(t)-b∣ < 63} T 2 if |y —b∣ ≤ <S3 max{t∈[O,T] : (g*∣'∙' (t)-b∣ = 63} if |y—b∣ > 63 Then t2 — tj ≥ T — tθ and we now show ∣gψ'y(t) — b∣ ≤ Cq , Vt ∈ [t j . t9] and then we use sublemma 4.2. (14.26) The path γrp(t) — grp,j(t) if 0 ≤ t ≤ t1 or t2 ≤ t ≤ T ∖ ι,∖ sinh c2(t2-t) ,~×,y,i λ vλ sinh c2(t-t1) b + (gT <tl)~b) sinh c2(t^-Γj + (gT (t2)-b) sinh c2(t2-t1)if t1 < t < t2 shows (⅜(gx,y(t))2 + V(g*'y(t)) dt ≤ ∫ tl c2f[∣Sτy(tl)-b∣2 + ⅛y(t2)-b∣2i COS h(c2(t2-tf))-2((grp,y(tι)-b),(gy,y(t2)-b))∖ 2l sin h(c2(t2-tl)) c2 63 cos h(c2(t2-1∣)) + 1 )≤ 2c2 sin(c2(t2 —tj)) if T > some T3 > T2 , ∣x — xθ∣ ≤ ∣y - yo∣ ≤ a∏d (since t2 - tι ≥ τ - t(P- •234 On the other hand leaving jx : ∣x-b∣ < cθ} in [t∣ , t£] would make ∫ t ∙> ^2 + V(gψ,y(t)) dt > 2(-~∣-^(l----- 1)) by sublemma 4.4. Then by above tl c0 2c2 δβ > -⅛, θ(l — -^)) or (1 + 4f2) ∣kθ > cθ which contradicts (and explains!) (14.25). co 1 Therefore we have (14.26) and sublemma 4.2 implies sin h(c.(t9-t)) . 2 2 sin h(c1(t-t,)) ⅛y(t) - b∣2 ≤ 02 ⅛h(⅛rψ) + m,n<⅛∙∣y-b∣ )s⅛h(c1(t2-t1))∙ X V We have uniform bounds on ∣gy, (t) — b∣ on [0,t^] and [t2 , T] and then since t2 — tj ≥ T — tθ where tθ is independent of T we obtain (14.27) ⅛y(t)-b∣2 ≤ 0(e~c!t + min(02 , ∣y-b∣2)e~Cl(T_t)) uniformly for ∣x — xθ∣ ≤ ⅛j , ∣y — yθ∣ ≤ 5∣ and T > T^ . Now we want to improve this and get (14.6). As before we write V,(x) w? (×-b)1 ⅛j(χ-b)d + R(χ) where R(x) ≤ Dθ∣x — b∣2 if ∣x — b∣ ≤ cθ and 0 < wj < ω2 <∙∙∙< Wj . -235- On [tj , t2] put gy^(t) = f(t) + r(t) where -v 1)-b)i smh uj∙ Γ trt fi(t) := ¼ + (gy x∕(t t) + ⅛ γv sιπh (aj∙>(t2 t ,¾ t -∣ ) y(t2)-b)i sinh satisfies fj = ω∣ (f∣ — b∣) .×,y, f(tl) — 6τ,y(t1) and f(t2) — g'p j(t2) Therefore r∣ satisfies ri(t) - ufri(t) = Ri(g⅞y(t)) ri(t1) = 0 = ri(t2) (14.28) ri(t) = - sinh ω∙(to-1) t xv s,nh ^(t22,tp ∫tι sinh u,j(s-t1) Rj(gT’ (s)) ds sinhω∙(t-11) t2 xv ωi sinh ⅛t2-t1) Jt ™l> u,i(t2-s) Riteτ' (s)) ds and (14.29) fi(t) = cosh ω∙(to-1) t xv s-nh w,(t2-tp Jt^ sinh ωj(s-t1) Ri(gτ,y(s)) ds cosh w∙(t-11) t2 xv sinh √(⅛-t1) it si"h "i⅛-s> Ri<ST «> ds since t∣ ≤ tθ and T — ⅛2 ≤ tθ where tθ is independent of T and γV Cηt q — C-∣ (T~1) Ri(g⅜’yW) = 0(e 1 +γ2e 1 ) «1 uniformly if 7 = min(6β , |y —b∣z), we obtain -236- ∣ii(t)∣, ∣ri(t)∣ = 0(e 1 1 + e -w1t t W∙S —c1s 9 —C1(T-s) ) ds 1 ∫ e 1 (e + √ e 0 -w∙(T-t) T ω.(T-s) -c1s 2 -c1(T-s) )ds) 1 (e + 7 f, ir t “e 1 = O(e — c1t 9 — C1(T-t) 1 + γ2 e p ) . Since ∣f(t)∣, ∣f(t) - b∣ ≤ O(e XV — Wit 1 + ye 1' XV ) we get -cι(τ-t∖ -clt ⅛Ty(t)∣, ∣gτy(t) - I ≤ O(e ) ∙ + 7e —min(ωι ,2c∣)(T-1) Using this estimate in (14,24) and (14.28) yields jrj(t)l, lr∣(t)j = O(e Assuming nc∣ £ {w^ , ..., w^} for n — 1, ∙∙., 2, ... we get by induction Ki(t)t, ki(t)∣ = 0(e — Wit ο — w∣(T-1) 1 + y2e ) and hence ■V V y V — W1t lt∕-∣(T t) ∣gτ,y(t)∣, ∣gτ (t) - b∣ = 0(e 1 + ye 1 ) uniformly. This completes the proof of (14.6). If K2 = K1 X {y : |y — b∣ ≤ b⅛} as in part 4, then t2 — T and we get ⅛T,y(T))i = fi(T) + ri(T) XV f = -u.iteιy(t1)-b)i (sinb + 0 (e 1 ∖ + u,i(y - b)i cosh w∙(T-11) -- W1T Q 1 + ∣y - b∣2) = wi(y - b)i (1 + 0(e — 2w∙T — w∙T 1 )) + 0(e 1 + ∣y-b∣2) uniformly in I<2 × [Tg , ∞) which proves (14.9). -237- By the Euler-Lagrange equations 1 XV Ο XV (gψ (t)) — V(g-p, (t)) is independent of t ∈ [0,T] and (14.7) follows from (14.6). To prove (14.8) A(x,y,T) = p(x,b) + P(b,y) + 0(e 1 ) uniformly we use A(x,y,T) = A(x,g^,y(I),I) + A(g^,y(∑),y,^) for the lowerbound (14.30) and A(x,y,T) ≤ A(x,b,^) + A(b,y,^) for the upperbound. (14.31) If p(x,b) + p(b,y) < p(x,a) + p(a,y) then p(x,b) < p(×5a∙) + p(a,b) and so X b estimate (14.4) holds for g ’ (t). Taking 7τ(,)=Γ 1f b<t> f∞t e [°∙τ-'] |_(b-gx,b(T-t))(t-T + l) + gX’b(T-l)_ for t ∈ (T-1,T) as a trial path for A(x,b,T) gives (14.32) A(x,b,T) ≤ p(x,b) × 0(e — 2ω1 T 1 ) uniformly. For the same reason A(b,y,T) ≤ p(b,y) + 0(e —2ω-∣ T ), and by (14.31) we get the upperbound in (14.8) A(x,y,T) ≤ p(x,b) + p(b,y) + 0(e uniformly. 1 ) -238- The triangle inequality, sublemma 4 and (14.6) give the lowerbound by (14.26) A(x,y,T) = A(g^,y(∑),J) + A(gj’y(ï),y,ï) ≥ A(x,b,T) - A(g*'y(I),b Ï) + A(b,y,T) - A(b,g⅞y(J) Ï) > ≥ p(x,b) + p(b,y) - 2A(g^,’y(ï),b,ï) rF 2 cosh(c2‡) ≥ p(x,b) + p(b,y) - c2∣g^,y(ι)-b∣ rp sinh(c2∙±∙) ∕n×,Dj -t- piD,n f∖∕ . υ(e -ω11 T > uniformly. Proof of Sublemma 8.5. We reduce to proving that if tθ is such that (14.43) ∣g θ (t)-b∣ ≤ y for t ≥ tθ then (14.44) x1 y1 γh If g = g<jΓ 1 or g = g ’ then ⅛⅛y(t)-g(t)∣2 ≥ ⅛y(t)-g(t)∣2 if t ∈ [tθ , T] and if g = g^b or g = gX,b and s ≥ 2(tθ + 1) then (14.45) ∣⅛y - g∣∣2<×w = 0(∣⅛y(s)-g(s)∣ ⅛y(s)-g(s)∣) uniformly near (xθ , b , ∞). -239- X V By the differential inequality in (14.44) for g = gy and the endpoint conditions l⅛y(t0) - g^y>(.0)∣2 ≤ ll8xry - ⅛j'1I∣lco[0,t,. ∣8τy<τ) - KX‘y 1<T)I2= ∣y - y1l2τ we get (see Protter and Weinberger [1]). (14.46) ⅛Ty(t) - g(t)∣2 ≤ ∣g⅜,y(tθ) - g(tθ)∣2 + ∣gτy(τ) g(T)∣ sin h c∣(T-1) sin h Cj(T-tθ) _ ein h z» f f . .+ . λ 2 w*w ** ''Pv u0' sin h c∣(T-tθ) - θ(l∣gτ -8τ ∣∣L∞[0,T] e -c1t 9 -c1(T-t) 1 + ∣y -y1∣2e p ') uniformly, which proves (14.12). If we take S = rr Xb XV and g = g ’ or g = gy in (14.44) we get — ω1 T 1 ) ∣g√y(to) - δ(to)∣z ≤ ⅛χ,y tj - 6∣∣l∞[ 0 Ï =0(e uniformly by (14.4) and (14.6). Also in both cases -240- ∣gτy(T) - g(T)∣2 = 0(∣y-b∣2 + e 2WlT) and now by (14.44) we have the first part of (14.46) which implies (recall c∣ < ω^) γv o —ω1T sinhc-∣(T-1) l⅛y(t) - e(∙)l2 ≤ 0«. 1 ) ,inh CJ(T_T|)) o + °<∣y-b'2 + e = 0(e —2ω1T sinh c1(t-1∩) 1 > riπh C11(T-Z⅜) — ω∙lT —c1t o —c1(T-1) 1 e 1 + |y— b∣2 e 1' ') and we have proven (14.10). The first part of (14.11) follows from the differential inequality in (14.44) rp X V ^X ∙ x,y χ,b∣. —ω1^∣'1 T, applied to [tθ , ∙^] and the bound ]∣grp — g ’ ∣∣lτ∞r∩ oo 5j. ) mentioned above. ~j = 0(e b [θ!0r] The second part can be proven by an analog of (14.45) on [T — S, T]. Next we will prove (14.44). Using sublemma 5 in chapter 13 we choose 6θ and XιVι Tθ such that gy γh C∕∖ i(t), g ’ (t) ∈ B(b,-^) for all (×1 1 y∣ 1 T , t) ∈ B(xθ , 6θ) × B(b,6θ) × [Tθ , ∞) × [tθ , T] where tθ is from (14.43). χ1yl If g = gτ1 1 or g = g x,b then -241- (14‘47) ⅛lgτyW " g(t)|2 = 2lgτy - g∣2 + 2<(⅛y - g),(g⅜y - g)) = = 2∣g⅜,y - g∣2 + 2<(V⅛y)-V'(g)),(g⅜y-g)) = 2∣g⅞,y - g∣2 + 2(∫θ V"(g+u(g*’y-g))(g×'y-g),(g⅞y-g)) du = 2∣g⅜,y - g∣2 + 2{V"(g+Z)(g^y-g),(g×>y-g)) . When t ∈ [tθ , T], g(t) + Z(t) ∈ B(b,cθ) and we get by (10.1) and (14.47) ⅛⅛y - g∣=≈ ≥ ⅛⅛2-i∣≈ > ⅛⅛y - sl2 which is (14.44). (14.47) also implies for s ≥ 2(tθ + 1) ⅛gτy(t) - g(t)∣2 dt = ∫8 tθ + l dt2 ≤ 2 ∫8 (⅛y - g∣2 + c2∣ g« - g∣2) dt t0 + 1 ≤ n,in(c1 , ⅛)lle⅞y-⅛oθ[to + l s, as in the proof of (13.16). -242- Kg = g™ O, g = gx'b then j‘° + ' 4∣g×∙y - g∣2 d. 1 0 dt2 1 tθ + l = ∫q (2∣⅛y - g∣2 + 2<V"(g + Z)(g*>y - g),(g⅞y - g)} dt > k1∣⅛y - g∣∣L∞[0,t +l] after taking smaller όθ > 0 and larger Τθ < ∞ if needed. By the above we have: If s ≥ 2(tθ + 1) then 2<⅛τy(s) - g(s)),(gx∕(s) - g(s))> = js 4 ∣⅛y - β∣2 d. n At* a = 2∫θ(∣⅛y - g∣2 + <V"(g+Z)(gx'y-g),(gx'y-g)> dt ■ ∕ι 2m∣ χ,y ∣∣2 t χ,y χ>b ≥ mm(k1 , c1 , q)∣∣gτ∙, - g∣∣^θo[0 s] for g = gτ or g which completes the proof. -243- 15. On the Solutions of the Jacobi Equations. §15.1. Asymptotics. §15.2. Estimates and derivatives of minimal action paths and Agmon geodesics. -244- §15.1. Asymptotics. We assume (1.33) V,,(b) = Ω^ where Ω == diag(ω∣ ,..., Wj) with 0 < ≤ u>2 ≤ Wj and we put (15.1) 51 = min { c -j1 , ω∙ - ω∙1} > 0 . {j=ωj-l≠ωj} j j Sublemma 9. Assume p(xn , b) < p(xn . a) + p(a,b) there is a unique g x0,b and it has positive definite second variation. Then there exist <5θ and Tθ positive such that 1. For all x ∈ B(x∩ , 6∩) (15.2) Z(t) = V,,(gx,b(t)) Z(t) on [0,oo) has matrix solutions Xx,b and Yx,b(t) such that uniformly for x ∈ B(xθ , <5θ) (15.3) Xx,b(t) =(I + 0(e_<5lt))e-m = -Xx,b(t) Ω~1 and (15.4) Yx,b(t) = (I + 0(e~*lt)) efit = Yx,b(t) Ω^1 as t → ∞ xb where X ’ (t) is nonsingular by (13.8). Moreover (15.5) ∣(Xx,b(t)-χxθb(t))e^t∣ = 0(e^ ||gX’b -g*^ II ∞ ) = = |(XX,b(t)-XX°b(t))em| and (15.6) |(Yx’b(t)-YX°b(t))eßt| = 0(e^6lt∣∣gx'b-gxθb∣∣oo) = = |(ŸX’b(t)-ŸX°b(t))em|). -245- If xθ then Xb,b(t) = e_tfi and Yb,b(t) = eβt and (15.7) ∣Xy,b efit - I∣ = 0(e ∣y-b∣) = ∣Xy,b(t) ei2t + Ω∣ and (15.8) ∣Yx,b(t) e-fit - I∣ = O(e ∣y-b∣) = ∣Vx,b(t) efit - Ω∣ uniformly for y ∈ B(b,6θ) as t —♦ ∞ . 2. If W(t) for t∈(0,oo) are (d ×d)-matrices that are integrable with Γ oo (W(t)∣dt<∞ 0 then (15.9) Z(t) = (V,,(gx,b(t)) + W(t)) Z(t) on [0,∞) has solutions X^y* and Yγy, satisfying (15.10) X‰b(t) - Xx,b(t) = 0(jV6l(t_S)|W(s)|ds + ∫t°°∣W(s)∣ds)e-βt = x‰b(t) - xx,b(t) and (15.11) Y‰b(t) - Yx,b(t) = 0(|V0l(t_S)|W(s)|ds + Jt°°|W(s)|ds)eßt = (t) — Yx,b(t) uniformly for x ∈ B(xθ , <5θ). 3. The system (15.12) such that Z(t) = V'⅛y(t)) Z(t) on [0,T] has matrix solutions X^^’^and γx,^,r^ -246- (15.13) (Xx,y,τ(t)-Xx,b(t)) cω, = 0(e "12 + ∣y-b∣) = (XX,y’T(t)-Xx,b(t)) efi, and (YX’y’T(t) - Yx,b(t)) e-fh = O(e (15.14) -ω ï Wl2 + ∣y-b∣) = (Yx,y,τ(t)-Yx,b(t)) e-ih . 4. For (r,T) ∈ [0,r∣] × [Tθ , ∞) let Wγ(t) be continuous symmetric (d × d)-matrices on Γ∏ rΓ 1 «ri ♦ l·» T ∫ ∣Wψ(t)∣ dt = 0(r) as r 1 0 uniformly for T ∈ [Tθ ,∞). (15.15) Then there exists a rθ such that (— 2 + Vπ(gy,y(t)) + Wιp(t)) > 0 on Dθ(0,T) (see (2.30) and (2.31)) for all (x,y,T,r) ∈ B(xθ ,6θ) × B(b,6θ) × [Tθ ,∞) x [0,rθ]. Moreover (15.16) Z = (V"(gx'y(t) × Wγ(t))Z(t) on [0,T] has matrix solutions Xx,y,^ and Yr,y,T satisfying (15.17) XX’y’T(t) - XX,y’T(t) = = 0(∫θ e^*l(t_‰ir(s)∣ds + J^|W^(s)|ds)e-m = = xx,y,τ(t) - xx,y,τ(t) and -247- (15.18) Yr’y’T(t) - YX’y’T(t) = t — <5η(t-s) „ T γη 0(∫θ e 1' '∣W⅝(s)∣ + ∫t |W^(s)|ds)em = = γx,y,τ(t) - γx'y'τ(t). Proof. To prove (15.3) we write (15.2) as Z(t) = v,,(gx,b(t)) z = (Ω2 + v"(gx,b(t))-Ω2) Z(t) as a first order system (15.19) ip(t) = (A + Rx,b(t) ip(t) where Λ = diag(λ∣ , » ■■·■> ^2d^ -^2i-1 = wi Xb and λgj = — u∣j for i ∈ {l,...,d} and R ’ (t) satisfies ∣Rx,b(t)∣ = O(∣V"(gx,b(t))-V"(b)∣) = 0(∣gx,b(t) - b∣) = 0(e^"lt), by (14.4). Then we use successive approximations as described in exercise 29 of chapter 3 in Coddington-Levinson [1]. Xb Notice φ, Λ and R ’ in (15.19) are (2d x 2d)-matrices. (15.2) by defining a (2d × 2d)-matrix through We get (15.19) from -248- 252i-l,2j-l - Zij ’ 2s2i,2j-1 — θ (15.21) z2i-l,2j - θ and z2i,2j = Zij Then set (15.22) φ = CSS where ωd -ωd C = diag and P2j be projection matrices such that For j ∈ {l,...,d} be (15.23) P∣ j + P2j = I 1 1 and λkt V1 j(t) := e^t Pjj contains e κ with λk < λj and Akt V2j(t) := ej^t P2j contains e κ with λk > λj . Then there are constants k∣j and ^c2ij suc^1 (15.24) (λ∙-0i)t ∣V1 j(t)∣ ≤ klj e j λ∙t ∣V2j(t)∣ ≤ k9 ; e j k2j ∣ιl d "*" ⅛ d = if t ≥ 0 if t < 0 (recall (15.1)) where ∣A∣ = ∑ la1J ; d fθr an (n × n) - matrix A. l≤ij≤n Pick tθ such that and -249- (15.25) (2d) ∫ OO 1 V h ∣R ’ (t)∣ dt < i for all x ∈ B(xθ , 6θ) t0 where όθ is as in sublemma 8.5 in chapter 14. With ⅛(t) = 0, ^,0b(t) = e j fj where f = (⅛ij)⅛*1 and (15.26) √,b+1(t) = eλjt fj + / V1 .(t-s) RX’b(s) φj £(s) ds ’ t0 -∫∞V2j(t-s) Rx,b(s) ⅞Pj g(s) ds we get by induction, using (15.24) and (15.25), (15.27) λjt ∣*,j⅛b+l - ⅛W∣ ≤ ^+I and therefore tp x,b ∕j∖ ∕ x,b z.∖ x,b ∕⅛∖∖ j’£( )=kSo ^ j’k( )_¥) M-I(t)) → φj (t) uniformly on compacts as f → ∞ . By (15.27) ∣^,b (t)∣ ≤ 2e Λjt (15.28) (t) = e λjt and then by (15.26) it satisfies fj + ? Vlj(t-s) Rx,b(s) y>pb(s) - t0 - J^∞v2j(t-s) RX’b(s) ^x,b(s) ds and hence it is a solution of (15.19). -250- Using (15.20), (15.24) and ≤ t√ι one can show λ∙t ∣<b(t) - e j' fj∣ = = 0(∫ t e (∖-^ι)(t-s) e —ω1s ds + ∫ oo —-∖(t-s) —ωηs e e ds) (λj-61)t = 0(e ) uniformly. Xb Extend the φ⅛ ’ (t) as a solution of (15.19) to all of (0,∞) and form a matrix X b solution T ’ (t) whose j (15.29) 4∙ u, X b -column is y>j ’ Tx,b(t) = (I + 0(e . Then )) e^t uniformly as t → ∞ and T,x,b(t) is a fundamental solution of (15.19) on (0,∞), since by Liouville-formula (Hartman [1]) (15.30) det Tx,b(s) = det Tx,b(t) exp(∫s trace(Λ + Rx,b(s))ds) t Xb and det T ’ (t) ≠ 0 for t large. To make it easier to control the x-dependence of the solutions we don’t use these φ,s but put (15.31) Vx÷(t,s) = Tx⅜), P1 ∙(Tx¼))-1 and Vx÷(t,s) = Tx⅛), P2j(Tx'b(s))-1 where P^j and P2j are from (15.23). Then by (15.29) -251- x,b. ∣V^^(t,s)∣ = 0(e (15.32) (λj-i,)(t-s) if s,t ≥ O, t — s > Ο ) λ∙(t -s) ∣V×÷(t,s)∣ = O(e j ) if s,t > O, t — s < Ο again uniformly for x ∈ B(xθ . όθ). x A fl.b ×rvb. Now we define ≠j υ (t) = a (t) and x,b,. b λ γK x b xfl’b x∩,b ,x∩,b 0' ,.x . f.t <.×.b ≠x,°(t) = ^jθ (t) + ∫θ Vj j (t,s)(Rx,0(s) - R θ (s)) ≠. 0 (s) ds (15.33) -∫ OO x b x b X∩.b Vxf(t,s)(R ’ (s) - R ° X∏)b (s))√ (s) ds which solves (15-19) and satisfies λ.t (λj-61)t V>*,b(t) = e j fj + 0(e (15.34) ) ∙ xb xb Going back to the second order system (15.2) √l2j M give a solution Xj ’ (t) (recall λ9 . = — ω∙) satisfying X^,b(t) = e — τj J J — W;t - xb 11 (e∙ + 0(e — 6■*■)) = — X — x,°(t) (t) J j where e∙ = (0∙ -)^-1 and Xn∙∖(t) gives solutions Y^,b(t) of (15.2) satisfying Y-f,b(t) J IJ 1 — 1 2J 1 J J = eωjt(ej + 0(e -*lx t.. )) _ γx,b(t) ω∙ xb 4" l·» xb Hence the matrix X ’ (t) whose jtn column is Xj , (t) is a solution of (15.2) satisfying (15.3). xb xb Similarly we get a matrix function Y ’ (t) from the Yj ’ is that satisfies (15.2) and (15.4). -252- Using (14.5) that says ∣gX ’ b (t) - g×∏ib θ (t)∣ = 0(∣∣gX x'bb - gxrυ θ b ∣∣oo e ^1o l2) in (15.33) gives IV-*,b(t) χ∩,b ∙ (λ; — *1)t X b xκ ≠jθ (t)∣ = 0(e j ∣∣g θ -gx,b∣∣oo) which implies (15.5) and (15.6). If Xq = b then R θ = 0 so ^b,b(t) = x∩,b. . λit θ (t) = e and hence Xbb(t) = e-Ωΐ and Yb,b(t) = efit . Vb In that case ∣∣g ’ (t) — b∣∣0θ = 0(∣y-b∣) and (15.7) follows from (15.5) and (15.8) from (15.6). Proof of Sublemma 9.2. We follow the proof of part 1 above and write (15.9) as a first order system (15.35) ⅛>(t) = (A + Rx,b(t) -(- W(t)) y>(t) with Λ and RX,b from (15.19) and (15.36) W(t) = 0(∣W(t)∣). Xb Successive approximations give solutions φ. ’ of (15.35) such that (15.37) e -λ.t , λ∙t j ∣⅛(t) - e j fj∣ = 0(e^0lt ∫t e6is(∣Rx'b(s)∣ + ∣W(s)∣)ds + J°°(|RX’b(s)| + ∣W(s)∣)ds) uniformly for (x,t) ∈ B(xθ , ^θ) × [tθ ,∞), fθr some tθ ≥ 0. -253- Now extend ⅜c∣^,b as a solution of (15.35) on all of (0,∞) and let it be the Jιw Xb column of a matrix Tw, (t) on (0,∞). According to (15.37) (15.38) f,w^(t) = (I + θ(l)) uniformly, thus by an analog of (15.30) f'w'b(t) is a fundamental solution of (15.35) on (0,∞). If F∣j and ,. , ,×,b z, x are as *n (i5.'Z0) ana we put wit>si := Pi j(,S'wb(s))~1 and V*÷w (t,s) := T^b(t) P2 .(Tw,b(s))^1 then, by (15.38) (15.39) xb (A--<51)(t-s) ∣vγ,■ w(t,s)∣ = 0(e j ) -*∙j9w if t,s ≥ 0 and t - s ≥ 0 xb λj(t^s) ∣v2j,w (t's)∣ = θ(e ) if t,s > 0 and t — s < 0 uniformly for x ∈ B(xθ » <5θ)∙ With (15.40) x,b ’ (t) from (15.32) we set ^(t) = ≠pb(t) + / V*÷w(t,s) W(s) ≠pb(s) -j0° vι i√t's) w) <'bωds which satisfies (15.35) and using (15.39), (15.34) and (15.36) we get 5w il -254- xb xb i,t ∣<∕g(t) - ≠pb(t)∣ = 0(∫θ e j (15.41) + ∫ ∞ t = 0(∫ s) ^is I W(s)∣ e j ds λ∙(t-s) _ λ∙s e j I W(s)∣ e j ds) = t e 0 —61 (t — s) OO 1 I W(s)∣ ds + Γ ∣W(s)∣ ds)e j . t Now ≠9,∙∖li(t) in (15.39) gives a solution X^,∖t) of (15.9) that by (15.40) J9W obeys (recall λgj = ~ ωj) x,b, . - Xf vx,b,.. , rt e -,5li Xfw(t) ,u(t)= n 0(∫^ i' t-s)z∣ W(s)∣ ds + ∫t I W(s)∣ ds) e J,W J θ "j θ = xj⅛(t) - Xx’b(t) where X^’^(t) are as in the proof of part 1 and (15.10) follows if we let X^’J^(t) be the J J5W jth-column of Xw∖t). X b Xb Similarly the ^∣9,.η ’s give a matrix solution Yw, (t) of (15.9) satisfying -mw (15.11). Proof of Sublemma 9.3. We will follow the proof of sublemma 9.1 closely. Write (15.12) as a first order system (15.42) ⅞px,y,τ(t) = (Λ + Rx,y,τ(t)) /,y,T(t) on [0,T] where Λ is as in (15.19) and -255- (15.43) |RX’y’T(t)| = 0(∣V'⅛y(t)) - V"(b)∣) = O(⅛y(t) - b∣) -Wjt = (e + ∣y-b∣ e -w1(T-t) uniformly for (x,y,T,t) ∈ b(xθ , 6θ) X B(b,0θ) × [Tθ ,∞) x [0,T], by (14.6). As in the proof of part 1 we get, using successive approximations, T of (15.42) such that if P∣and P2j are as in (15.23) then VXJ’T(t,s) := TX’y’T(t) P1J(3'X’y’T(s))-1 and v2jy,τ(t,s) := Ψx,y,τ(t) P2j(Ψx,y,τ(s))^1 satisfy (15.44) xVT (∖-0l)(t-s) ∣VxJ, (t,s)∣ = 0(e j ) if T ≥ t, s ≥ 0 and t — s > 0 and xvT -∖(t~s) ∣v2j ’ (t,s)∣ = 0(e j ) if T > t, s ≥ 0 and t - s < 0 uniformly for (x,y,T) ∈ B(xθ , 6θ) × B(b,<5θ) × [Tθ , ∞). For (x,y,T,t) ∈ B(xθ , 6θ)x B(b,<5θ) x [Tθ , ∞) × [0,T] put χ,y,τ (t,s) 256- (15.45) ≠*'y,τ(t) = ^jx'b(t) + ∫θ Y*J,τ(t,s)(Rx,y,τ(s)-Rx,b(s)) ^’b(s) ds -fr Vx^τ(t,s)(Rx'∙v∙τ(il) - Rx,b(s)) ≠pb(s) ds XH with V,j ’ (t) as in (15.33), which solves (15.42). rp On [0,∙^∙] we estimate |RX’y’T(s) - Rx,b(s)∣ = O(∣gx,y(s) - gx'b(s)∣) Il δ rp Γp —⅜^ ~cι(⅛~s∖ -ωι ⅜ = 0(e - + e i 2 ) e i 2 uniformly by (14.11) and on rΓ , T] we get |RX’y’T(s) - Rx,b(s)∣ = 0(∣gx'y(s) - gx,b(s)∣) = 0(∣gx'y(s) - b∣ + ∣gx,b(s) - b∣) = 0(e — wηs — ω,(T-s) 1 + ∣y-b∣ e 1 ) uniformly by (14.4) and (14.6). Those estimates and (15.44) put into (15.45) give (15.46) .x,bb(t) + 0(e ~12 + ∣y-b∣) e λ,t ttX’y’T(t) = Vf j which implies (15.13) and (15.14). -257- The Outlines Of A Proof Of Sublemma 9.5. The positivity of (----- ^ + V"(⅛τ ) + Wj) on Dθ(0,T) follows from the dtz jz positivity of ( —-a-κ ^*^ dtz »j x⅞y ⅛T (see sublemma 7.1 in chapter 13) and the estimate (15.15). We write (15.16) as a first order system (15.47) χ,y9T - «» φ = (Λ + R, ’ ’ + W⅝)¾p and we get, using successive approximations, fundamental solution T(t) = ΤΧ,^’Τ’Γ(ί) so that v*J’T’r(t,s) := T(t), P1 j(T(s))~1 and 7x,y,T,r V^J, ’ (s) := T(t), P2j(T(s)) -1 satisfy the estimate in (15.44) With ≠jt,y,T(t) as in (15.45) we set ≠f,y,τ(t) =vf,y,τ,r(t) + ∫θ Vjj,,τ,r(t,s) W⅜(s) ^,y,τ(s) ds - ∙fτ vijy,τ,r(t>s) W⅜(s) ήΧ,Υ,Τ(8) ds which solves (15.47) and we get ∣W⅜(s)∣ds + ∫τ∣Wtr(s)∣ds) eλjt ∣vfy,τ,r(t) - ≠×∙y'τ(.)∣ = o(ιf, .^iι<t^s>r j j o t which implies (15.17) and (15.18). -258- §15.2. Estimates And Derivatives Of Minimal Action Paths And Agmon Goedesics. Sublemma 10. Under the assumptions of sublemma 9 and with όθ and Tθ as there, we have 1. and gX’b are differentiable w.r.t. x and y for (x,y,T) ∈ B(x0 ,6q) × B(b,6θ) × [Tθ ,∞). With X = Xx’y’T and Y = Yx,y’T as in (15.13) <15∙48) ⅛)j ⅜ sτ,yw = ⅛⅛>i ⅛yω = (⅛)j(X(t)-Y(t)(Y(T))-1X(T))(X(0)-Y(0)(Y(T))-1X(T))-1 ei and a*·*» ⅛)j ⅛ ⅛y w = ⅛⅛)j ⅛yω = (A)j(Y(t)_X(t)(X(0)) —lγ(0))(Y(T)-X(T)(X(0))-1Y(0))-1 and with Xx,^ as in (15.3). (15.50) <A>i ⅛ ⅛'y(∙) = ⅛(⅛)i ⅛i(∙) = <A>i Xx'bW(Xx∙b(0))-1 ei for i ∈ {l,...,d} and j ∈ {0,l}. 2. Moreover (15.51) ∣gψ,yi(t) - gψ2,y2(t)∣, ∣gτ1,yi(t) - gτ2,y2(t)l = 0(∣x1-x2∣e — Wit —ω1(T-1) 1 +∣y1-y2∣e ) -259- (15.52) xη,b xo,b xη,b xo,b — ω,t ∣g 1 (t) - g 2 (t)∣, ∣g 1 (t) - g 2 (t)∣ = 0(∣x1 - ×2∣ e 1 ) and (15.53) xb xb xb xb ∣gτ, (t) - g ’ (t)|, |έτ’ (t) - g ’ (t)| = O(e w⅛T 1 e t*iη(T--1) P ') uniformly for x,x∣ ,x2 ∈ B(xθ ,δθ), y, y∣, y2 ∈ B(b,6θ) and T ∈ [Τθ , oo). A Proof Of Sublemma 10.2. We observe that (15.52) follows from (15.51) and (15.53) by writing Λl «U g 1 Art ∙O (t) - g 2 A 1 »U (t) = (g 1 A^∣ U (t) - gτ1 (t)) xη ,b xo,b xo,b x<>,b + (gτ1 (t) - sτ2 (t)) ÷ (gτ (t) - s (t)) (similar for t-derivatives) and taking T -→ oo . Also (15.53) follows from (15.51) since gx,b(t)∣[0,T] = ^(t) where ÿ - gX’b(T) = b+ 0(e 1 ) uniformly by (14.4). To prove x1,y1 x∏,yo —ωηt — ω,(T-1) ∣gτ1 1(t) - gτ2 2(t∣ = 0(∣x1 - x2∣ e + ∣y1 - y2∣ e ) in (15.51) it suffices to show (15.54) and x1,y x<i,y —ω1t ∣gτ1 (t) - gτ2 (t)∣ = 0(∣x1 - x2∣ e 1 ) -260- x,y1 x,yo ∣gτ 1(t) - gτ 2(t)∣ - θ(∣yι - y2∣e (15.55) ι(τ-t)) uniformly, as we see from writing xυyi x2,y2 _ xpy1 δτ ~ Srp — l&T ml grf ) teT1 J⅞∙s⅛ g,r )∙ Now we show (15.54), (15.55) is proven in a similar way. Write h(t) = grpl’^t) — gj? y then h(0) = x^ — ×2 , h(T) = 0 and h = V'(g*1'y) - V⅛y(t)) r1 ,,∣∣, x2<y + s(gj , xpy — grp x2’y" z x2,y — grp xυy×) ds j — — ∫θ V (grp )) (grp (V,,(g-pθ ) + W)h where T T 1 χι,y ∫ ∣W(t)∣ dt = 0(∫ ∫ ∣sgτ1 + (l-s) gτ2 0 00 (14Λ2) - χo,yχo,y - gτ2 I ds dt) —c11 τ 1 x1,y χ0,y 1 + θ(∙∣θ ∙Γθ s∣lgτ - gτ ∣∣L∞[o,τ] e + (1-s)l∣g,j'2 — g>pθ '∣loo[0,T] dt) — θ(f(^))> if we define TO = s»p (∣∣g×∙y - β*fl'y L∞[0,T]) ∙ x∈B(xθ,<5) By sublemma5.2, f(<5) —> 0as6 ∣0, and by sublemma 9.4, Z = (Vz, on [0,T] has matrix solutions X and Y such that *P∙y> + W)Z -2G1- χ∩,y,τ ot ■ ∙ ×∩,y,τ X(t) - X θ (t) = O(f(i)) e~nt = X(t) - X 0 (t) (15.56) and χ∩,y,T o . x∩,y,T Y(t) — Y θ (t) = 0(f(Λ)) pli, = Y(t) - Y u (t) (15.57) and by taking 6θ smaller X(0) is invertible for all x∣ and ×2 in B(xθ , όθ). Now (15.54) follows from ετ1,y(t) - 8τ'y(t) = h<t> = (χ<t) - γ(t)(γ(τ))~1 χ(τ)) (X(0) - Y(0)(Y(T))^^1 X(T))-1 (x1 - x2) (15.56), (15.57) and the estimates in lemma 2. The second part of (15.51), ∣g-p^ ∖t) — gp? ^(t) = 0(∣x∣ — ×2∣ e x1 ,y x9,y 1 follows from grp (t) — gpΓ (t) = ∫ I -ω11 ^ ) (1—s)xη+sxo,y gp, (t) ds and (15.48) that we prove next. Proof of Sublemma 10.1. For small ε we are interested in x + εe∙,y xv fε-(t) := (grp (t) — grp’ (t)) ε We observe that fε(t) = ∫1 V''(g™+S£ei’y(t)) ds fε(t) = (V''(gI,y(t)) + wj(t)) 0 fε(t), fε(0) = 0 and fε(T) = e∣ where ∣wj∖t)∣ = 0(εe ) by (15.54). -262- By sublemma (9.4) (15.58) fε(t) = (Yε(t) - Xε(t)(Xe(O))^1 Yε(O)) (Yε(T) - Xε(T)(Xε(0)"1 Yε(O))-1 ei where Xε(t) - XX,y,T(t) = O(ε) e-fit = Xε(t) - XX’y’T(t). and Yε(t) - Yx,y,τ(t) = O(ε) em = Ye(t) - Yx,y,τ(t) X y Now we take ε to zero in (15.58) and we get (15.48) for grp . (15.48) for ⅛y we differentiate (15.58) w.r.t. t and then take ε → 0. To obtain -263- 16. Proofs and Sketches of Proofs of Lemmas 2 to 6. §16.1. A sketch of a proof of lemma 2. §16.2. A proof of lemma 3. §16.3. A sketch of a proof of lemma 4. §16.4. A sketch of a proof of lemma 5. §16.5. A proof of lemma 6. -264- 16. Proofs and Sketches of Proofs of Lemmas 2 to 6. §16.1. A Sketch of a Proof of Lemma 2 . Xb Proof of Lemma 2.1. Sublemma 6 in section 13.3 says there is a unique g ’ and it has positive definite second variation for x ∈ B(xθ , 6θ) for some 6θ > 0 (which will be getting smaller as the proof goes on). (2.14) with ∣α∣ = 0 is (14.4) in sublemma 8.2 (2.15) is (14.2) and (2.16) is (15.3) and (15.4) is sublemma 9. d -t Τ' ω∙ ∙* 1 V (2.18) that says det XX’ (t) ≥ constant e 1-> 0 follows from XX’ (t) = ι (I + 0(e f )) e Of x b (see (2.16)) and X ’ (t) is nonsingular by (13.8) of sublemma 6. More precisely, we can find tθ such that d -t∑ ωi -i,t -'-ς∙"' Λtλ _ i = l * >le i = l det X ×,’ b(t) = l(1, +, nz.~ 0(e u1Γu )) e„ if (x,t) ∈ B(×q,6q) x [tθ , oo) (slightly smallr 6θ). Then by compactness of B(×q,0q) « . xb x [0,tθ] and continuity of det(X ’ (t)) (its nonzero and positive for large t) min det(Xx,b(t)) > 0 B (xq >6q ) x [0,t q] so (2.18) follows. (2.19) is (15.50) of sublemma 10 and (2.20) follows from (2.19), (2.14) with ∣α∣ = 0 and p(x,b) = ∫°°(⅛gx,b(t))2 + V(gX’b(t))) dt. 0 2 -265- (2.14) for ∣α∣ > 0 follows from (2.19) and (2.17) and so to complete the proof it suffices to prove (2.17). Xb We use the pr∞f of lemma 9.1 and we let φ. ∖ (t) be as in (15.26), φpb(t) =k∑θ(⅛b(t) - ⅛1(t)) as in (15.28), (16.1) V*jj(t,s) and j5(t,s) as in (15.31) and as in (15.38) γ κ> X∕∖*b t γ ¼ γ¼ Xr⅛,b x∩,b ≠j (t) = V>jθ W÷∫ V*∖ (t,s)(Rx,b(s) - R θ (s))≠j0 (s)ds (16.2) X∕Λ »b V ¼⅛ ∙vr Î-» ίΎ'ΐ J” V^^(t,s)(R^^(s) — R υ X/t »b (s))V-jυ' (s) ds for X ∈ B(xq , <5θ) and t ∈ (0,∞). f⅞∣θi∣ y b (2.17) follows if we show that for each a ∈ Nq with |.| > ⅛ ≠×∙b(t) = (λ1-51)t 0(1 ) uniformly on B(xθ , -^) × (0,∞), which can be done by induction. Let m = 2, 3, 4, ... and α, /?, γ ∈ Nθ and let Im , ∏m and IIIm denote the following statements. If 0 ≤ ∣α∣ ≤ m — 2, 0 ≤ |/?| ≤ m — 1 and 1 ≤ ∣7∣ ≤ m — 1 then ×+εe∣,b ⅝=>j,ι+ι ∂× w ∂*a a ∣α,∣ a∣ a1’1 √⅛b ∣α∣ . p √∂ lk -( uniformly x+εe∣,b ∂ for φ j,« 0xc (x,t) Em-1 -ei(< + 1) √t) ∈ xj, y>j,g+ι X (t)J . <λj-iι>t ≤ ‰-M+ιe B(xq , (1 _ (1 +...+ 4n))⅛) where Dm_l f+1 < m —1 for i = 0, 1, 2, ... , Ei is independent of É and tθ as in (15.25). -266- IIm: J/?l xb (λ.-01)(t-s) —-ä V√,∙ (t,s) = 0(e j ) if t, s ≥ O, t — s ≥ Ο, ΟχΡ d λj(t-s) 0∣^∣ x,b and -—V2j (t,s) = O(e )ift, s>O, t — s<O ôx uniformly x ∈ B(xθ , (1 - (I +...+ -⅛)0θ)). Finally Jïl xb (λ1-<S1)t IIIm: 2-_ (t) = 0(e j ) c*x ' j uniformly on B(xq , (1 - (A + ■·· + 4π)<5q)) × (0,∞). We leave out the proof but we point out that Im implies we can interchange xb (A; —<5j)t differentiation and summation in (16.1) and get -^—5 φ∙ ’ (t) = (e ) which is dxP J used in the proof of IIm . Notice also -^-3 Rx,b(t) = 0(∣ -^—3 (gX,b(t) — b)∣) ∂xp dxP O∣gx,b(t)-b∣ 0( ∑ if |/?| = 1 I ⅛ (Xx,b(t)(Xx,b(0))~1)∣) if |/?| ≥ 1 ∣α∣ = ∣∕J∣-1 ∂xp and IIm gives by induction we can differentiate under the integral sign in (6.2). -267- iA sketch of a proof of lemma 2.2. X V By sublemma 7.1 in section 13.3 there is a unique grp, for (x,y,T) ∈ B(xθ , 6θ) X B(b,6θ) X [Tθ , oo) and it has positive definite second variation. (2.25) follows from (15.13) and (15.14), (2.22) from (2.25) and (2.16) since we assume 6∣ ≤ (2.23) follows from (2.25) and (2.18). (2.26) except . = l(g^,y(T))2 + V(y) follows from (15.48) and (15.49) in sublemma 10. (2.26) follows by showing gy is differentiable w.r.t. T and differentiate A(x,y,T) = ∫θ ⅛⅛τy)2 + V(g*,y)) dt and use g*'y(T) = y for all T, g*,y(0) = x for all T and the Euler-Lagrange equations. xVT To prove (2.26) one can show that ≠j (t) in (15.45) in the proof of Jα∣ + ∣β∣ xyτ sublemma 9.3 obeys - -------- ∙ψ-y (t) = 0(1) e δxα5y^ J one sketched for ^it by an induction similar to that x,b ’ (t) above. Now finally (2.27) follows from (2.24) and (2.26). §16.2. A Proof of Lemma 3. Lemma 3.1 and 3.3 is contained in sublemma 7, section 13.4. contained in sublemma 8.1 and 8.3, section 14.1. from sublemma 10.2 in chapter 15. Lemma 3.2 is Lemma 3.4 except (2.41) follows x1 >5r1 (2.41 follows by setting α(t) = ∣gj 1(t) — t2,^2∕ gψ" z(t)∣z then ö = 2ά2 + 2(α,α) > c∣ a, by use of Euler-Lagrange equations and -268- the mean value theorem. Now (Protter-Weinberger [1]) this implies α(t) ≤ ∣x^ — ×2∣^ sinhc1(T-1) ο sinh c1t 9 9 siπh c1 T ~ + lyl^y2∣ ⅛Γh c1T so α(t) ≤ max(∣xl ~ x2∣ ’ ^1^21 )’ §16.3. A Sketch of a Proof of Lemma 4. Lemma 4.1 and the positivity in (2.49) of lemma 4.2 follows from sublemma rp 2 9.4. That Gr,y, (t5s) defined in lemma 4 is a Green’s matrix for (—4. V,zfe⅜,,y ) dt2 '~1 ' + oπ [0,T] with Dirichlet’s boundary condition can be easily checked but see also Heimes [1]. Notice Z(t) := Y(t) — X(t) X-∖t) Y(t) satisfies Z(t) = —X(t) X~^^(t) Z(t) and is nonsingular at t = T, by the asymptotics in (2.46) and (2.22) and so it is always nonsingular by the Liouville formula (Hartman [1]). One can obtain the estimates in (2.51) and (2.52) by inserting (2.46) and (2.22) into (2.50). The estimates in (2.53) follows from (2.50) and (2.24). §16.4. A Sketch of a Proof of Lemma 5. (2.55) and (2.56) for ∣α∣ = 1 are obtained by going though the successive approximations to obtain (recall (15.44) and (2.42)) -269- xvT — <51(T- v) — <51(T-u) V^p1(T-V,T-u) = (I + 0(e 1 ))Vlj(u-v)(I + O(e 1 ')) XÿT and similar for V2j ’ (T—v,T —u). Then by differentiating (15.45) and by showing xvT λ∙(T-u) -δ1(T-u) V-Jy,1(T-u)=e j (fj+0(e 1' ')) χ,y,τ y,b — ⅛1(T-u) e~ωlu)) ÔR (T-u)∣y=y = ¾-(u)ly=b + θ(e 0yi and XvT xb Rx'y'1(T _ u) = RX’°(T - u) + 0(e T üj-t u 1 (e 1 +e ω1 (T 1 we get t∙o° ∂ιbx,y,T aτty,⅛ ¾- (T-v) = [∕v Vlj(u-v) ¾- (u)|y=b e >y>b -√θ v2j(n-v) (u)|y = b e ^iu j d« -λ∙u λ∙T j du]ej fj (A.-51)(T-v) +0(e j ) -i1(T-v) = : (q(v)fj∙ + 0(e -λ=(T-v) )) e whereby (15.24) and ¾- (u) = θ(∣⅛ gχ√u)D <2=4) θ(e q(v) = 0(e ~λiv (∫ 1 ) ’ we have ∞ (λi~<5ι)(u~v) — ω1u -λ∙u e j e e du v ∙¾(u-v) -ω1u ~λiu 4- ∫ e j e 1 e du) = 0(1) 0 u) )) -270- which gives (2.55) and (2.56) with ∣a∣ = 1. (2.55) and (2.56), for ∣α∣ > 1, are proven similarly by induction. The first part of lemma (5.2) follows from the formula for Gx,y,^(t,s) in (2.51) and the estimates in lemma 2. The second part is then obtained using (2.55) and (2.56). §16.5. Proof of Lemma 6. By (2.26) and (14.9) . XV — 2ωη T — C√∙,T ey Ay(x,y,T) = ⅛y(T) = Ω(y - b)(l + 0(e 1 )) + 0(e 1 + ∣y-b∣2) uniformly for (x,y,T) ∈ B(xθ , 6θ) x B(b,<5θ) × [Tθ , ∞). implies y — b = (e Hence Ay(x,y,T) = 0 -u-∣ T 1 ) uniformly. Now (4.18) that says β τ Ayy(x,y,T) = Ω + 0(e -(- |y — b∣) uniformly implies B(b,6) 3 y ∙→ A(x,y,T) attains its absolute minimum at a unique _ ___ — WiT point ÿ satisfying y = b + 0(e ) uniformly for (x,T) ∈ B(xθ , δ) × [Tθ , ∞) some 6 ∈ (0,6θ] (taking Tθ larger, if needed). The proof of the statement for B(b,<5) 3 y μ A(x,y,T) + ^(Ω(y-b), (y-b)) is similar. -271- References S. Agmon [1], Lectures on exponential decay of solutions of second order elliptic equations: Bounds on eigenfunctions of N-Body Schrôdinger operators, MathematicalNotes 29. Princeton Univ. Press, Princeton, N.J. 1982. R. Azencott [1], Grandes dérviations et applications. Lecture Notes in Math., no. 774, Springer, Berlin-New York, 1980. A. 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