Effects of a Magnetic Field on the Trace of the Heat Kernel for a Schrödinger Operator with a Potential Well Citation Odencrantz, Kristiana (1987) Effects of a Magnetic Field on the Trace of the Heat Kernel for a Schrödinger Operator with a Potential Well. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/vd33-hs32. https://resolver.caltech.edu/CaltechETD:etd-09072005-130533 Abstract This paper looks at the effect of a uniform magnetic field on the trace of the heat kernel for a Schrodinger operator with a well type potential. Using weighted Sobolev space techniques and noticing the gauge invariance of the perturbation, I show that the magnetic field first appears at a higher term in the small time asymptotic expansion of the trace of the heat kernel than might be naively expected. 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) Taylor, Michael E. Anderson, Michael Thomas Luxemburg, W. A. J. Defense Date: 16 March 1987 Record Number: CaltechETD:etd-09072005-130533 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09072005-130533 DOI: 10.7907/vd33-hs32 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3371 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 12 Sep 2005 Last Modified: 02 Jul 2025 23:21 Thesis Files Preview PDF (Odencrantz_k_1987.pdf) - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

EFFECTS OF A MAGNETIC FIELD ON THE TRACE OF THE HEAT KERNEL FOR A SCHRODINGER OPERATOR WITH A POTENTIAL WELL Thesis by Kristiana Odencrantz In Partial Fulfillment of the Requirements . for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1987 (Submitted March 16, 1987) ii Acknowledgements I wish to thank my advisor, Dr. Barry Simon, for his many helpful remarks and suggestions. Further, I wish to thank Dr. Michael Taylor for some suggestions relating to the use of pseudodifferential operators for heat equations. I also want to thank Dr. W. Luxemburg for his enthusiastic discussions with me and for the motivation he gave me. I also wish to thank in general the committee members for their time. Among my fellow graduate students, I particularly wish to thank Bart Huxtable, Askell Hardarson, Clemens Glaffig and John Lindner for various comments and suggestions as | was preparing this thesis. Finally, I wish to thank my parents, Kirk and Annaliese, and my fiancé, Robert, for their continued support and encouragement. lii Abstract This paper looks at the effect of a uniform magnetic field on the trace of the heat kernel for a Schrédinger operator with a well type potential. Using weighted Sobolev space techniques and noticing the gauge invariance of the perturbation, I show that the magnetic field first appears at a higher term in the small time asymptotic expansion of the trace of the heat kernel than might be naively expected. iv Table of Contents AcknowledgementS .occcc secre enw e renee ante eee er eee eeeeeeneees Abstract ... oom remo enero nee e sree e sree eseereeseeeee ese eereereneesezresese fone: ba ko) | aaa Introduction occ c ccc eee reece cece ee eect cree twee eee eesceeseees Weighted Sobolev Spaces and Pseudodifferential Operators .......... A Weighted Canonical Order Calculus and some Global Results cee. The Pseudodifferential Operator for the Heat Kernel .............. . Asymptotic Estimates Some Further Remarks References ee eee sree eee eee oaere meso veeeesneeeereoerreerereee eer eene renee eo rane eeoerereeraeoeeresenesreneaneerene Li iii il 23 29 36 37 Notation x or y will usually refer to position space coordinates. £ will usually refer to momentum space coordinates. a i) nnwon 3 oom —j . — | : OL n .Q; a os a ) = % 8,1 for a multi index a, and similarly for D™ . 1= (curl A); j = 8,4; ~ 8jAj is the exterior derivative. . —1 —] f ~g if both f g and g f are bounded. i <->=(1 4+]. |%? O(x,f) = <x>k + <€> f: the Schwartz space, co functions of rapid decrease. )’: the space of distributions, the dual space to COUR"). A, th ; H™: the uz Sobolev space with norm || : Ihas . 3; the class of symbols of order u . ops” : the operators corresponding to symbols in s¥ . m log ® ~ it = . ; in H Il - Wn | la log ® the norm in Introduction. Recently there has been much interest in asymptotic expansions for eigenvalue distribution functions, the heat kernel and the wave equation. The index theorem relating spectral quantities with geometric quantities from the underlying manifold has aroused this interest. This paper examines the effects a uniform magnetic field will have on the trace of the heat kernel. I start with operators of the form —A + V acting on L*(R"™), where the potential V has polynomial growth at infinity and is bounded below so that the heat kernel of the operator is indeed trace class. Instead of the magnetic field directly perturbing the equation, the magnetic vector potential, A, enters and one has H(A) = (-iV + A’? +V, where the curl of A is the magnetic field. Although there has been a tremendous amount of work on various properties of Schrédinger equations, the magnetic field case has often been excluded. Leinfelder (9) determined the essential spectrum of a broad class of magnetic potentials and has a comprehensive reference list of some previous work in this area. The additional complications introduced by the magnetic field lead to separate considerations of many cases, see for example (4), (8), and (10). For the more general case of a Yang-Mills potential, Schrader and Taylor (12) proved that there is a complete asymptotic expansion for the trace of the heat kernel in the parameter A. The magnetic field in dimensions other than three is typically defined as a 2-form or a skew-symmetric matrix. The curl or the exterior derivative is thus Oy Ag = OpAy for the (k,@) entry in the matrix form. I will use the matrix form of notation for the magnetic field B, denoted by B*. For the uniform magnetic field, the standard choice for A(x) is ~} B*sx; however, A is not uniquely determined. Leinfelder (9) proved several results that clarify this situation. He showed that if A € L: »>V-. AE L and V € L. Loc foc loc then H(A) is essentially self-adjoint on CHER”. He also proved for two functions A, A’ € L,* with V-A, V-A’ € L,* , related by a distribution Loc foc h € ® with A’ = A + Vd then H(A) is unitarily equivalent to H(A’) by -2- eih H(A}je —i4 = H(A’). This is called a gauge change and one easily sees curl A = curl A’. Leinfelder also verified that the Coulomb gauge condition, VY -A = 0, can always be satisfied by some gauge change. For the potentials considered in this paper, these results follow from the diamagnetic inequalities. Simon in (13), (14) showed that, for a large class of potentials and vector potentials, that for t > 0 —tHtaA _ fe Mg < le Lg for all pa, where these are the operator norms from LP to Lt . For convenience, H will refer to H(A) and Ho to H(O) in the remainder of this thesis. The motivation for this work originates with the simple case of a two- dimensional harmonic oscillator. In this case the eigenvalues are explicitly known in both the perturbed and unperturbed cases. For Ho = —A + {x/?, the eigenvalues. are at 2(n + 1) with multiplicities of n + 1. For the perturbed case, H = —A + iB*x . V+ ,iB*x? + jx|? , and the eigenvalues i 1 are at 21 + i B*.B*)? (n+1) + eG(B™ -B*)*) for £2€{—n, —n +2,...,n —2,n}, ne zr . Computing the trace of the heat kernel for these yields as t | 0 tr e tH 2 —tH — tH = O(t*), and tr e — tre = Q(1). In this simple example, however, the t~’ term is not present in either asymptotic expansion. The question arose as to whether or not the perturbation first arising in the second 0 t term, ,» was a general phenominon. I show that it is such and that it is related to the gauge invariance of the magnetic vector potential. For simplicity, | consider only those potentials with uniform polynomial growth in all directions. They must also be bounded below. In Section 6 there are some remarks about further extensions. Let k > 1 and K& C R"™ a compact set be such that V satisfies the following: (1.1a) there isa C > 0 5.t. C— x |? < vx) < Cix|?* tor x © R7 VK. (1.1b) v e¢ C™R") for some m>4 + ty (m =oo if k =1) and m >n(l +p . n (1.1¢) For each multi-index a, with ja| = > la;| < m, there is a i= Ca > Oat [D% VG] < Colxi?* . Theorem 1. For V satisfying (1.1), and all « > 0 then as t!0 —tH 3A 1 trie tH —e 0) = of t? 207) ‘) . If one also requires that V has a homogeneous decomposition, theorem 1 can be extended. Namely, suppose there is a finite set of V jo € CCR") with the property that for + (1.2) x and px € R®"\ % (2 © R' ,x € R®) V (ox) = pv jO0 and O<j S2k Note: j need not be an integer in (1.2); however, in the usual examples j is taken to be an integer. Theorem 2. If V satisfies also (1.2) then as t10 —tH * _ L @.—2¢, +4) tre tH —e 0) — BB Cg t G+) + oft ° +) 4 where ly = 3 — xe max((j: V; 40, j < 2k) U 2k—2) n-i and c, = VOUS 0) pag +3) . de __ . 0 4k 2 k J. 1 (V,,(0)) 72k -4- —tH Remark. It is known that (1.2) is sufficient to show that trfe 0) Ob oft 2 *), (see Helffer and Robert (7)), and I also show in Section 3 that (1.1) —tH 24,4). implies tre 0) = oft att) *). Section 2 has a brief outline of weighted pseudodifferential operators and weighted Sobolev spaces as developed by Beals (2) to set up notation. To actually compute the integral kernels and traces, | combine two techniques for estimating the asymptotics with gauge changes. In Section 3, I use these weighted Sobolev spaces, semigroup and operator norm estimates in a canonical order calculus similar to that developed by B. Simon in (3). Combining these operator norm estimates with repeated applications of Duhamel’s principle and the appropriate gauge transformations leads to the proof of theorem 1. The more powerful technique, which requires more explicit knowledge of the potential, is that of weighted pseudodifferential operators. Having the canonical order calculus, however, greatly reduces the terms necessary to evaluate for my result. Sections 4 and 5 deal with the remaining symbol computations. For the symbol computations, I use transport equations similar to the standard method such as that in (15). An alternative choice is to use the explicit symbol construction already done by Robert (11) for HZ , and to use 0 —tH the inverse Mellin transform as in (5) or (7) to get the expansion of e 0 : The proof of theorem 2 is completed in Section 5. -5- §2. Weighted Sobolev Spaces and Pseudodifferential Operators. As I am interested in operators with growing potentials, such as —-A + ix 2k , this section provides background to the theory of weighted pseudodifferential operators and weighted Sobolev spaces. The type of weights that will be used here originated from many sources; however, the notation I use mostly follows that of Beals’ (2), who consolidated the notation. More details can be found in Beals’ paper. In the usual pseudodifferential operator case, a symbol a satisfies for some m an estimate of the form ioe D8 atx)! < Cagil + ep™ IO Indeed, for the constant coefficient differential operator, this type of estimate clearly holds where € is the symbol of D. However, the potentials with polynomial growth are excluded from the k symbol class. Changing the weight (1 + |€|) to <x>" + <€> allows a larger class of symbols to be considered. In particular, condition (1.1) implies that ¢? + V satisfies an estimate of the form ee Doe +vy) < Ce (<xok cgay? 7! for |[B| < m. Throughout the remainder of this thesis, the weight ® will be <x>* +. <€> with k determined by the growth of the potential V. Beals used two weight functions in his more general situation. For W and & in CUR?"), there are positive constants C, c, 6, so that the following hold: 1) ~<C; (2) Ww > ¢; (3) ¢ < W(x) Wy.7) > < Candee < Wx) wly,7)~ 1 < C -~6- if Ix —y| <cP(x,6) and lf — 71 < cW(x,€); (4) R(x,0) < C<x>°, where R = Wo '; . 1 (5) c < R(x) Rly < Cif IE — nl < cR(xe))*? and —ti 2 . Ix — yl < eR(x,6)° Ryn) These weight functions then generalize the symbol classes to functions a € C*(R2"), satisfying the following for all a, 6: (ot DE al < Cog dh wit pl (2.1) for some order function . Although the order functions can be generalized, the common ones are } = k, log W + k, log w, for k, and k, € R. The \ ° symbol class corresponding to the order \ is denoted Sy v , From the above estimates, it is clear that equivalent weight functions, i.e, V ~ W’ and » ~ y’ , generate the same symbol classes as WW , and the similar terms are bounded, affecting only the constants in the above estimates. As I do not need the full generality of Beals’ weighted pDO (pseudodifferential operator) calculus, I take ~ = 1 and WV = ® and use the order functions r log @®. This clearly satisfies the hypotheses. Thus, a € Se, » which will be denoted as sh in the remainder of this paper, implies \go% pS al < Cc ep — lal . The theorems in this section, unless otherwise € “x ag explicitly stated, apply to the general case S$ e . From the symbols, the operators are defined in the same way as in the usual wDO case. For the symbol a(x,€), one associates the operator a(x,D) defined on f by (a(x,D)u(x) = Qn)? fe * atxe) a) de . (2.2) -7- Asa eé sh and u € f, this integral converges absolutely, and it can be shown that the integral is in f. Further proposition 3.11 of Beals states that the mapping defined by a(x,D) is a continuous mapping from f to f. This mapping can be extended to a mapping from #* to #*. I will denote the space of operators with symbols in sh by ops* : Beals showed ((2) proposition 3.12) that there is a bijection between such operators and their symbols, and so there is a seminorm that can be placed on the operator space ops* . This seminorm is defined by the constants Cag arising from the symbol estimates (2.1). And so ops’ is a Frechet space. The main computational use of pseudodifferential operators arises from the use of symbol composition and adjoint theorems. The rules of composition are the same as for the ordinary pseudodifferential operators, as the following theorem shows (Beals (2) theorems 4.1 and 4.6). Theorem 2.3. (a) If A = a(x,D) € ops“ , and B = b(x,D) € ops* , then AB € ops#** and the symbol a o b of AB has the asymptotic expansion ~— 1 aa ar. aob ZX ai 8 2 Px bs that is, acob— YS LatantyeshthM lee laicm &! & (b) A™ restricted to f£ is in ops” and has symbol a® with asymptotic expansion # 1 na na = a x ai o¢ Ps @ - Another useful fact is (Beals’ (2) theorem 5.1) that if A € ops* where \ = 0, then A: L? — L? continuously. -8- The weighted Sobolev spaces are defined with the help of the weighted pseudodifferential operators. The weighted Sobolev space HA is defined by H* = span{Au: u € L? and A € OPS—, (2.4) Endowing H* with the finest topology so that the mapping A: L*? — H is continuous for any A € ops—* ; the following can be proved (see Beals (2) theorem 6.1). Theorem 2.5. For 4, u € O(®), (a) H° _ L? topologically, (b) £ C H* c s* densely and continuously, (c) (HY* =H’, (d) if AE ops” then A: HY Th H? continuously, and (e) there is an A € ops” so that A: H ATH H is a topological isomorphism and in particular H has the topology of L? . With (e) of the above theorem, one can extend the norm of L? to give a natural norm to H for any A. The norm is defined by aN = ||Aul| for u L? $ € H, with the operator A given by part (e) of the above theorem. The difficulty, of course, is in finding an appropriate operator A. The next proposition gives a practical way to determine a norm. Proposition 2.6 (proposition 6.17 of Beals (2)). Suppose a, , a2,...5 ap € s” and Dia, ~ eh . Suppose A € O(®) and ce* < eM < co™ eS for some m > 0 and c,C > 0. Let A; =a;(x,D). Then u eH” iff u eH and each A; u EL? . There is an admissible norm in H” defined by whe, = lull + DIA; ull? . In particular, this proposition allows easy verification of several ‘equivalent norms. For the Sobolev spaces with weights r log ®, I first choose symbols k(r —lal) wx et log ® 8q(x,€) = <x> e~ . Then having > lag (7 ~ ; and taking lal<r \ = 0, one gets 2 k(r-lal) 2 2 ilu || = SS ilKx> uli’ + pu ir. r log ® lace As jlull < il<x>kr ull, another equivalent norm is ul, jog @ =D exe) po uy lal<r which I will denote |llulip . This is similar to the usual Sobolev norm. Use interpolation for non-integer values of r. Among other norms for these weighted spaces, one final set of norms will be useful in this paper. Proposition 2.7 If V satisfies (1.1) for m, then for 0 < s < m and oie —u < spec(—A + V), for each |r| < m there is ac > 0 so that om full ys < I-A + V + w)S/2ulllp < clllulllr ys (2.8) for all u € yt tsilog ® Also, since k > 1, one can prove a similar proposition for the operator with a constant magnetic field. In particular, since B® x . D € ops? log ® ; one has that (iV — SB*x)? + V + wi? ~ (—a + KM + DS? for 2 0O<s<cm. -10- Proof. First note that (1.1) implies that for 0 < s < m there is a c’ > 0 so that (6) (A + [x (7% 41)82 ull < I —aA+Vtu)ull < o/(—A + |x 2* 418” ul. B D®™(—A + Ix [2X +1)8/2 for ja} < m, and all 2k And similar relations hold for <x> 8 . Thus, one need only verify (2.8) for V = |x as the Sobolev norms involve taking at most m derivatives. To establish this, one considers the symbols 2k k(m lad gor bg(x,€) = aglx,é) o of(—A+ix + 1)°/ 2), where ag(x,¢) =<x> Expanding out the terms by shows that _ et +s)log ® From the spectral theorem, one has |lull < I —O + [x 2 41872 ull for s > Q. Combining this with proposition 2.6, one obtains the norm 2 _ 2 k(r—lal) nay 2k , 4)8/2 ,, 12 Whe slog ® = jul? + ore De(—A+ixl +I" ull” , which is equivalent to the norm = A+ [x7 4:1)8/2 ue SE icxok “Ope _a ix pPk yas”? ull? , 2 ull 4s)log ® lore which in turn is equivalent to Mle a sog @ = MHA +I +DS? ullip « And so (2.8) follows. a] ~ll- §3. A weighted canonical order calculus and some global results. In this section I use operator techniques and weighted Sobolev space norms to get leading order behavior of various integral kernels. Combining these results with Duhamel’s expansion, I am able to get some estimates on the significance of terms in this expansion. Gauge changes will also allow identification of many vanishing terms. As these weights have growth in x, I am able to get global results with these techniques. The following is a generalization of the canonical order calculus developed in (3). The parameter Xq will be used for gauge changes and t > 0 corresponds to the semigroup parameter for the heat kernel. Definition 3.1. A family of operators {Ay xq) on L2(R") has weighted canonical order (m, , m,) € R? with respect to ® if (a) A :f + g* t,X9 (ob) for all k > @, there exists c(k,2) > 0 s.t. for0 <t <1 At x, ull, << c(k,€)t <Xg> “Illulllg - If V does not satisfy (1.1) for m =o, I will need a more restricted type of calculus and so will add the requirements k—@ < m, |k| < m, |/€) < m. For m sufficiently large, namely, m > 4 + mo and m > n{1 + it) , this will give sufficient control to prove theorem 1. Note that for operators depending on the single parameter t, one can set Xg = Q and obtain similar estimates. In particular, if there is no parameter Xo specified, | mean the family of operators that is constant for varying Xg and has m> = 0 as the second coefficient in the weighted canonical order. The motivation for this definition arises from analogs to the following proposition and lemma 3.3. -12- —t Proposition 3.2. Let V satisfy (1.1) and Hy = —4 + V. Then e 9 has w.c.o. (weighted canonical order) (0,0) with respect to ® = <x>k + <€>, Proof. This follows from proposition 2.7 and the spectral theorem. Using the fact that INH +8’? ull, is equivalent to ||| ull 4s when —yu < inf spec Ho : one has tHy —tH _ lle ullstr < CeriKHy +n’? e ° ulllp —tH < Cyrl(Hy tw’? e — Nihulle , where the operator norm is from Ht log ® to itself. The isometry established r log ® to itself is the by theorem 2.5 (e) implies that the operator norm from H same as the operator norm from L* to itself. Thus, the spectral theorem can be used to estimate the operator norm. —tH Kp tw’? e ON < sup x +8’? e | x>inf spec Ho < Ct —s/2 3 where C depends upon |—y — inf spec Hg| and s. Thus, lle ull pg SC + Cop tS? halle « . Proposition 2.7 can be extended to apply for H = (—iV + Ay’ + V, as remarked previously since k > 1, thus a similar argument shows e tH has w.c.o. (0,0). To estimate integral kernels, it is useful to know the norm of 6, in various weighted Sobolev spaces. In particular, a simple extension of the ~13+ standard proof (see for example Adams (1)) for the usual Sobolev space norm yields: Lemma 3.3. For u € f, = > 0, kn 1 BGT) lu(x)| = |<dy , u>| < CC <x> e)/|{Iu fl gu +¢ +8) Proof. Let Sy be the unit ball centered at x, and let r € (0,1). Let (0,0) denote x in Sy , and similarly for (s,6). Then <5 u> = u(x) = 00,0) = ulre) — f 1 (4,9) at . ia} Thus, 1 ju(x)] < jure)! + J |Vuct,e)) dt . is} Multiplying by the volume element and integrating over Sy , one has: Ska +i +6) vol Sx - juGa| < (fo &«y luty))? dy}? Sx (fF <yo HG Hite) apie Sx 1 4 i) ) iVu(t,e)| dt r™—! w(0) drdéd , S$ 8) x where r"—! 60) drdO is the volume element. In the first term I have used Holders inequality. Integrating over r in the second term and other simplifications gives: k i Lo +--+ 8} —fu +i+e) vol Sx » {u(x)| < <x>? * ull - <x> - Ci(vol Sy)” 2 +if \Vuty3—Y— Sx Ix—yl ~14- kn 1 Ga +; +e) kn i where C, = (2) Sv=) 20 Tete . 2 sup wt? c, lee Using Hélder’s inequality with the dual indices p and p’ , one has: nl ni kG &) a,b fo wo 8, < exe Dull g <xo TO S ix—yl PS x 1/p’ S (n-L)p x Ix—yl —«iGd + 2) where Cc, = (2) If I choose p’ > 2, the above integral converges for n > 2 and then p < 2. Using the fact that Sy is bounded, I have || - llp,Sy > | - lL Sx >i-l.- —} kn, yt Kn, 4 Thus, lu(x)] < Cy(vol Sq) 2 <x> 20 TET Megs et TET yyy ~t kat kn + Calvol Sx) _ <x> 247 Megs Zk) Dull, , where 1/p” _ dy Cs ~ Ce I, ad __kn, pl kad ) And noting <x> 207, r?) < <x> ar? : —"u/k +2) k&(i +$+) mot + 2) lu(x)| < C<x> 7 vere [\I<x> errs ull + |l<x>? Kr Dull} kod < C’<x> 2% Ha i lj . sa +748) -~15- This last step can be seen by looking at proposition 2.6 and noting that if one n 1 kn 1 k(50 +--+ 2)—j st +--+ 8) 2 k 2 k and adds further operators of the form <x> DY to <x> kn 1 Siete) . i. <x>?k D, one also gets an admissible norm. And as all admissible norms must be equivalent, the conclusion follows. QO With the norm of the delta function in the weighted Sobolev space norms and the operator estimates from the weighted canonical order, the following holds: Lemma 3.4. If fALxo) has w.c.o. (m, , ms), then the integral kernels of At x, ; Ky (x,y), satisfy for all «© > 0 tk 9 +4452 mite lim t? © K (Xp, Xp) dx, = 0. ti0 Je At xo 0° "0 0 Proof. Using lemma 3.3, one has txg kn¢i 2 1 —_— — 2 +(2 —1 —2)) < cg t sm <xXg>"2(<x><y>) 24 TS w considering Ay Xo : H® log ® HS log ® : Taking x = y = XQ and integrating over R" , the integral converges whenever s > aC + b) + x . OO If {Ay xq is independent of XQ > note that lemma 3.4 implies ? (ee +) ~Z +m,— ‘) trad xa) =o for any =« >O,astio. -16- —tH —tHy In order to estimate the w.c.o. of e —e , | need the following expansion, which is derived by repeated application of Duhamel’s principle. r log ® Proposition 3.5 (Proposition 12.49 (3)). Let X, Y be operators in OPS for some r > 0, with K and X + Y self-adjoint and bounded below. Let bn = } é€ ( S1 Sn)X Ye SX oe Yo SnX ds, vee dSn 3,20 2s; St and ry be defined similarly except that in e , replace X with — n j UX+Y) _ —tX _ S (—1) pb, +(—1 74 r,4, for alln € N. j=l X+Y. Thene Another similarly useful relation is Proposition 3.6 (proposition 12.51 (3)). Let X, Y € ops’ log ® with X bounded below. Then t [ye—*X} = —[ et —8)Xry,x7 08% as. 0 Proof of 3.5. It is enough to prove that the expansion is valid when applied to ] functions » € f, as f is densely and continuously mapped into H" og ® . Let U, = eit(X+Y) @ itX . Clearly, Uy w € f and Buy = bth FY) ive —itX y ey, By the fundamental theorem of calculus, . . t et(X+Y¥) .-itX yy — J. 2 Us ¥ ds -17- eT it(X +Y) eg HX +Y) { sei S(X +Y) Ye 7isX y ds 0 t : . _f eit —sMX+Y) Ye ~isX » ids. 0 Analytically extending, one has t 0 As X and X + Y are bounded below, this extension exists. Repeating this procedure proves the proposition. oO The proof for proposition 3.6 is similar, using instead U; _ eitX Ye witx ; Needing to compute the w.c.o. of terms from these expansions, we use the following: Proposition 3.7 (propositions 12.47 and 12.50 (3)). (a) If Ay Xo has w.c.o. (m, ,m,) and T a fixed operator satisfies for some ’ b>o0, WTull, < COcd hail, 4p Nico* for all k, then TAL x, and At xg T have w.c.o. (m; — =, mg). (i) (ob) If A for j = 0,1, ..., 2 are operators with w.c.o.’s (m‘? m)}, and Xq me) > —1 for all j, then — (0) (1) (2) Bex, = At—s,--s, Xp S; Xo ed As, Xp ds, tae dsp ~18- converges and has w.c.o. g : £ ‘ [e +> mo? ; > ms) . j=0 j=0 Proof. Part (a) is a direct result of expanding with operator norms. For part (b) more care is needed to choose the correct operator norms. Let Sy = t{—SsS —~ = Sp - First decompose the integral into integrals over regions R; ; where in R; s% <8; for D <i < @. In the region Rj , one has (0) i (2 Gy) 1 £ m?, my—3(k—p) =m As, Xo AD? tee \ ) } 1 2 by bounding the ag? for i + jas maps from either uk log @ to itself. al 4 Xg LXg is bounded by its mapping of H? log ® | pk log @ Now, using the fact that t > s; > aT on R; ,; each integral can be J bounded above by t aw O_ite —p) 045m? Xk —p) i m — — i= 1 2 f ‘TW sg 'ds,---dsy-t ! 2 P' = ct : ’ gq iJ provided m') > — 1 for each i # j. Combining the results for each R; completes the proof. QO . log ® i log ® Remark. The weights have been defined so that € S and x € S$ . Thus, the estimates De ill; < ellie nl; 41 and |llx@|l] < ojlleit 43 are easy to verify. The machinery to prove theorem 1 has now been set up. Before —tH computing any terms with Duhamel’s expansion for e tH —e 9 , first, let’s ~19- look more closely at H. For the magnetic field B* , I replace —A in Hy with (—iV + A)* where A, the magnetic vector potential, also satisfies V- A = 0 ‘and curl A = B* (the exterior derivative is the curl). This does not uniquely determine A, and so [I am left with some free parameters to determine by convenience. In particular, taking A = A —= —l B*(x —x,) is a valid choice Xo 2 0 for any Xq ER", Thus, I am led to Hx. = —4 — B*(x—xq) - D + ,IB¥(x —xQ)? +V for any fixed Xq .« So I will use Duhamel’s expansion with X = Hg and Yxo = —B*(x —x,) - D+ 1|BM(x —xQ)? . Further, as the Hxys are related by a i ifpQ* i(B*x9-x) te —i(B* xp -x) gauge transformation, one has that e = Hx for H the operator with xy = 0; and as these are unitarily equivalent by a multiplicative —tH — tH: operator, the diagonal of the kernel of e and e “O are the same. And so one has —tH —tH, tre = tre 0 =f Ks — tH, (XQ XQ) dXq - Noticing that (Y ulx,) = 0 for any u € § leads me to use the following: —tH _ —tH e 9) trfe = | Ke tx —tHy (Xo 1Xq) dXq - Using the collected facts about the weighted Sobolev norms and the weighted canonical order calculus, one finds that only the first 4 + rain terms are needed from proposition 3.5 to verify theorems 1 or 2. ~20- In particular, one has to evaluate the kernel of —({t—s,—..-—s. —sH b; =| e (t—s, Sj)Ho Yxp @ , ° ds, --- ds 3,>0 j Ys<t isl fs j . These terms are not computable directly as the Yxq8 are interlaced with the —tH e 0 terms. With the help of proposition 3.6, we commute the Yxq’s to the left. This leaves formulas such as —tH —(t—s,—s,)H (—s,—s.)H bixq = —Yxg ¢ Oo, —f e eee” so? ° as, 3,>0 Dest Again, bringing the interlaced terms to the left, one finds from proposition 3.7 that at most, second-order commutators are needed. More precisely the terms that lead to critical behavior are: Yio ak : for j<4+ rot ; [Yx5 » Hol Yy, eo emi for j< 2+ m4 ; [Yxq » Hol YX e 0 oo for j < ch,» and UYx, » HobHgl YX, e 0 cheater i<4- (3.8) Remark. Some commutator terms need to be factored with one factor —sHg considered acting on the e term before it and the other after it to satisfy the hypotheses for proposition 3.7 b. However, as taking the commutator -2]- —sHy introduces extra factors of e , and I need only to apply proposition 3.4 to, at most, terms with third-order commutators, no further problems arise. —tH ne Lemma 3.9. V € C™; then the integral kernel of e 0 is in C™ +2—2~8 for e > 0 separately in each space variable. —tHy —tHy Proof. The integral kernel is <by +e 6y> = <e by ,» by>. To show continuity properties, | use ordinary Sobolev spaces on bounded subsets (2 of R™. AsV Ec™ , D@ Vv € LQ) for la] < m, and so V € H™Q). —tH It is well known that e 0 takes L?(R™) into L®(R™) (see theorem B.11 , —tH, © —tH of (14). Forf €L4g =e CF EL°(R™ and Hye %f € LOR") by the semigroup property. As V € C™, vg € L™(Q). Thus, —Ag € L™(Q) and min(z,m) —e by Weyl’s lemma, g € H*(Q). To proceed, one notices Vg € H (Q) for +2. all = > 0, and thus, g € H™ a8 (2). This holds for all Q bounded and all —tH —(m+2-=) —t_g/ f € L?. Thus, e Oe ops “s , and so as 6, € H *~ (Q), —tHy m+2e” e oy € H (2). The Sobolev embedding theorem then gives —tH +> —0 —tH e 9 bx € c™™?—2~ 0 for all fg > 0. So I have <édy,e 0 6y> is in m+2—3—£, ; . ; Cc 2 with respect to x for fixed y. The symmetry also gives <e by , by> = <dy,e 6y> is in C™ w.r.t. y for fixed x. QO To complete the proof of theorem i, integrals of the form K -tu, (Xq » Xqg) dXg need to be evaluated for various Ty, . With lemma Rr Txe 9° 0 XQ -22- 3.9 and Fubini’s theorem, an interchange of integrals produces K 3 = ’ bs et (x,y) Tx K tH (x,y) where Tx acts on the x coordinate of K _,,.. This holds provided Tx does e not involve taking too many derivatives and is the reason m is as defined in (1.1). As Tx is explicitly known, it is easy to identify many vanishing terms. In particular, the commutator terms are: (Yx, > Hl = —iB*&x—x9)) VV + B* . B* + 21 B*V - V + B*(x—xp) - B*V and ['Yx, » Hol » Hol = —iB™(x—xq) - WAV) + 2iB*VV - Vv + B*(x—xq) - B*VV + 2B*V - B*V . x: . * : 37 _ 9? i Note that as B” is skew-symmetric, B™V - V is Bi xox) » which Ox,Ox j vanishes on smooth functions. Computing, one sees that the first non-trivial term, i.e., the term not obviously vanishing with the lowest weighted canonical order, is 2iB*VV -VK —tHy ’ which has w.c.o. e (3 — 3 , 0) and so fe*vv - VK tH Xx) dx e feet for all «© > 0Oasti0O by lemma 3.4. Thus theorem 1 follows. ~23- 84. The pseudodifferential operator for the heat kernel. There are two possible ways to proceed from this point. As Robert (11) already has explicit symbol computations for HZ , for Re z < 0, I could 0 —tHy use the inverse Mellin transform to get asymptotics for tr Tx9@ in the —tHy same manner as Helffer and Robert (7) did for tr e The Mellin transform takes et to T(s)t © , and thus the inverse Mellin transform allows —tHg z one to compute tr Tx9@ from tr Tx Ho ; provided the integrals converge absolutely so that the order of integrations can be interchanged. However, it is also possible to compute the symbol for the heat kernel, —tH e 0 , directly, in a similar manner as that of Schrader and Taylor (12). This is the method I will use. One assumes that there is a symbol for the heat kernel and one computes the transport equation from the relations such a symbol must satisfy. Next one needs to solve this transport equation. The standard technique is to guess a first approximation and then to iterate by successive applications to the transport equations; thus, one obtains a formal series solution. The hope is that these approximations are better and better in the sense that they lie in more and more negative symbol classes. In particular, one wants the successive expansions to differ from the actual symbol for the heat kernel by symbols in these progressively smaller symbol classes. That is the difference of the formal expansion, and the symbol for the heat kernel is in g-O = a sd log ® J functions with compact support. As a final introductory remark, ~ will be This symbol class in particular contains smooth used in this section to mean that the symbols are equivalent up to symbols in 57°? ~Dhe— To compute the symbol for the heat kernel, one needs the transport equation. This is the equation satisfied by such a symbol. Suppose one has a symbol a , which satisfies (e f(x) = J e a(t,x,¢) f(€) d€ for any f € fF. —tH As Ho is the generator for the semigroup e 0 ; one has 3 —tH (é + Ho] e = 0. This implies the following transport equation: da ~ 5¢ + o(Hp) our: 0. (4.1) One next assumes a ~ 7 aj; with a; € gk log ® Let J 3 se = —(?+V_1)ag with initial condition aq(05x,€) = 1, This choice is taken as this leads to symbols whose trace can be easily computed. The alternative 3 choice = = —(€?+V)ag leads to evaluating Je not computable in detail. —tV dx, which in general is HP +Va) 5 tAC€) Thus, ag(tsx,€) =e (4.4) defining A. This choice of initial conditions implies a (0;x,€) = 0 for all other —-0-H js, ase 0 I. To continue by induction, let oa, a! = —(7?4+V a, + DY Vi_., ai ot 2k" J acj 2k—q “i-4 + 2i€ - Vaj_, —A 8 j ok Define by 2; the latter quantities, so Q; = D> Vox—q 2j—q ~ 2° V8g_ek 7 Aas oy: (4.5) q>0 J —— . . Thus, = Aa; + a; t—s)A(x,€) t _¢ or a (t,x,€) = l, e Q j(s,x,€) ds. Remark. This choice of a recursion relation is to easily allow a proof that each a. € 5 Ik log ® . j In particular, I want to control the additional powers of € and x by related powers of t. Thus, one has: Lemma 4.6. For j, where b ,(t,x,8) = b(t re w, 7) 1 for w = EL = tx), t/2k and 0b, is smooth in all arguments and a polynomial in t , W, and 77. Proof. For j = 0, it is clearly true with bg = 1. One proceeds by induction. j = tk ha, x. & —tACx,€) Then, using (4.5), one has = 2k i¢. Q,=¢t ( XS Vig Bjig t* + 2E- VO)_y t q>0 ~ 2 mw ic2k . 2k —2t€-W yb; te — Ab, 4 t™ + Je2k j-2k ~ ae 2 2° _—— -26- j-2k +t AV, bt 2k-q 4 ~ i ~ 3 ~ ~ ~ 2i t* € VV OK b k A b jak + 2t VV Vb j_ok j — 2 2 Lt One sees that Q, is smooth (except at 0) and is a polynomial in ti/2k »W,and 7 , as the Vo's are radially homogeneous of degree @. t Thus aj = | eft —s)A 2 (s) ds ¢) J t Ja —(t—s)A A =f sk Gys)e e § ds 0 . . + tL _ W/2k(, —J/2k I, 2k i oF ds} eth ; ~ . . . ~ — if: t jy ~ As &; is a polynomial in 4i/2k ,b;=¢t j/2k i] s** Q ; ds is also such; and so the lemma holds. Oo I still need to show that the aj's are in sik log ® , to prove this note that there is a C > 0, so that A(x,€) > Cl<x>* + <¢>)?. Thus terms such as tA w? e tA and 7 e ” are bounded. Proposition 4.7. aj E sik log ® , for any t < T uniformly, for any fixed T. -27- Proof. Using Lemma 4.6, lajl = it3/2k be Nice t3/2k g—tA/2 | ae as bj is a polynomial in w and 7 and the other variables are over compact domains. Again, using the fact that A > Co? , we have ,3/2k e—tA/2 <C, g@i/k —j/k log ® A similar argument works for all derivatives of aj ; and thus aj Es uniformly for t < T. oO Now to see that these symbols are a good approximation to the heat g kernel, let Ap = a. and W, be the corresponding operator. It remains to £ jo ! € —tHy show that e _ Wey are in progressively smaller classes as @ tends to infinity. H At t = 0 by construction, e 0 _ W (0) = 0 for each &. Further, (2 +H) wet =J Dg el® € Fe) dx for f € fF, where Do -2 VoK—a aj + b-g< isl &-k<j<b b-n< isl (—£42) log ® So Do Es , by proposition 4.7. —tH | (—£4.) log ® It remains to show that o(e 0) _ Ap € S a 6 , which completes the proof of the asymptotic symbol expansion for the heat kernel. Proposition 4.8. Let v(t,x) satisfy (2 + Ho] v = g(t,x) with initial condition ~28- v(0,x) = h(x). Then sup Iv Ills < Cilllhils + Co sup lig(t, - DIllg - Proof. Duhamel’s principle implies —tHy t v(t,x) =e h(x) + | e g(s,x) ds. 0 Thus, by proposition 3.2, sup lv Ills < Cy [lh llls + Co sup le lls - —tH And so I have a symbol expansion for e 0 . -29- §5. Asymptotic Estimates. In this section the proof of theorem 2 will be completed. Recall from section 3 that it remains to evaluate J K “tH (xo XQ) dxg for various e xo 2 operators T x9 . In particular, let Ty = . B* . B* and x) 7 GED (20 B* VV . Vv + 2B* V7 - B*Y)Qi B*(x —x9) -V+tq IB*(x —x9) 17) . These are the terms that do not trivially vanish in the commutators. Using the explicit symbols for the heat kernel constructed in Section 4, these integrals can be evaluated explicitly. Some symmetry arguments show that there are more vanishing terms, but more importantly, the explicit constants can be determined. These perturbation terms are much easier to —tH evaluate than constructing the symbol for the heat kernel for e and then computing the trace of the difference between the symbols. The following proposition insures that the above integrals exist. Proposition 5.1 (Robert, proposition 3.2 (11)). Let 4 be an order function for ® and N be a non-negative integer. Suppose that “4 > N log ©® and that Ji<e>? 7° oN e MP dé = O(1) on compact sets of x for 0 < = < 1. Let Te ops". Then (1} T is an integral operator with kernel K+(x,y) € COR™ x R®). Moreover for all ja! < N, [8] < N, DE pe K-(x,y) is continuous and satisfies -30- a nf _ Cc IDE Dy KyGuy)! < ER CHTRCS) VT ee for all a (x,y) € R®" and —1/2 V(x) = |! oN 2H “ R® (2) If ON € L7(R"), then T is a nuclear operator and Tr T = J K+(x,x) dx. RD In particular, this shows that T is trace class when (—2(i+1)—=2) log ® nN wre &™" — ops for any « > 0. Further, for a class of T € OPS oma . 2 operators Zx0 , one has i} Kz... *0 ; Xq) dx < o if [Yy,z,, (x9)] EL’, that d is, if 2x5 e ops® and if there is a positive constant M so that Hex! ag As < M independent of x, . Remark. I do not need to worry about using an asymptotic expansion for the N (—+2) log ® symbol of the heat kernel. For ifd =a — } aj,d €S a j=l , for N large, and as d(Q,x,£) = O, one can find uniform constants so that ae pe di < ct™ @~M— Il tor M > 2n and m < —M +B — 2. This implies tr d = O(t™), and taking N sufficiently large implies that the results are not affected. -~31- Lemma 5.2 * pe 2—2.+)) K a Gq, Xq) dx, = BEB PB TK J Tye tHg “0 0 0 4 0 €,—8 +3) + oft ° “ “) as tio, n—t where Cy = vol(S_) ra +24) —ie_ 0 4k 2 k In (V,(o)) 72k —~ 2? _L i eg. : _ and £y = 3 aK max({j : vj ¥ 0, j < 2k} U (k} U (2k—1}). Proof. As T x0 is a constant, one needs to evaluate j a(t » xX , €) dxdé. R20 For j = 0, one has —t(e?+V | ° (€° +V.,(x)) axdé R22 _ —te? —tV>,(x) Je dé Je dx If VK is homogeneous over all of R" one has k _tr? —tV,,@r? = vol(S"—) f ni) e " dn - J e 2k r"—! dr do Rt Rt xs"! where 7 = |€|, r = |x|, and o are the angle variables from x. However, (1.2) requires only homogeneity outside a compact set %. Thus assuming % is contained in Bp ; the ball centered at 0 with radius R, one has 2 fag dxde = vol(S"—) fo 78 te an . Rt -32- x + R —tVv.,(@)r2k _ | e an(e}e r"—! drdo| . R*xs”™ Note that the integral over Bp is bounded independent of t. Integrating this, using oo) m fot eT" ay = 1 r@th with n + 1,1,m > 0, from (5), 0 mr us”) . —_t2 0 n/2t1+2) 2 k _ 2k * st? (V,,(o))% This gives the leading behavior and determines Cg . It remains to verify that (“0 —5a +e) ; all of the remaining terms are at least O . —t(e? +V_ 1 (x)) Recall from lemma 4.6 that a. = t3/2k bie 2k j j , where bj is a polynomial in t/* , t4/?Kixj, and tle. f a(x) dxdt = oft™? as —1/2Kk —1 , or t . /2 each extra factor of |x| or | €| introduces an extra t ; however, these are cancelled by the pairing of e/?2K with ix| and t?/2 with | €| . Thus 2+ —284 +) multiplying Ja(x0) dxdt by +t? produces terms of order oft x 2 ) Among terms other than ag which has already been considered, the one with the lowest order in t is ag, —2)ek . —33- (a,j) For the T x9 similar arguments are needed; however, the symbols for (2,5) T," contain elements which vanish when integrated over €. Indeed, these 0 terms are expected to vanish for the more general case without homogeneous decomposition. However, a proof just using operator norms is not powerful enough to identify these terms, and although there are techniques to allow for non-smoothness of symbols in the x variable, these lead to additional (2,j) complications. The symbol for T xo is Oy [—2B*vv - ¢ — 2B*¢ - B*€) o (—B*(x—x 9)» € + ZIB%(K—xQ)I7) 0---0 (—B*(x—xq) - € + 4[BMx—x9)) — j copies (2,5) The terms for T x9 © ay which do not have (x — Xq) remaining are t?) 4. ax pe * | : Gap 2i B*VV - VGB (x —Xq)-VC-- (—B™(x—xq) - (€—iV)ay)---) (5.3) j—l1 copies and t3rs i * . * i * — . eee — 2 coorennd . awn f eae GED! 4i B™e - B VGB (x 0) v( vf B™(x—Xp) - (€—iV)ag)---) (5.4) j—l copies and th position Yo BV BYV 12 SS GB —x_) © VO--(SB*(x —xq) 1) (s+ J)! 1Ek<i ~34- jth position an (B*(x—xg) - VG-- (CB*x—xq) - €-iV)ay)) + GB* x —x9) «VO GIB —¥9)F o>) . (5.5) j—l copies Using these symbol expansion, one obtains: Lemma 5.6. 3,45 2044 j O(t? 2" 20k) for j >1 K (2.3) (Xg XQ) dxo = ln, ln, t R" T,. e," ca 4 otfoee) tor j = 00, where £y is as in lemma 5.2 and C, is defined in the proof. Proof. From (5.3)-(5.5) one can read off the |x| or |€| powers in the terms composed with ap - Any x derivatives of ap will just increase the t power by 2K for that term. (5.3) has a VV and a € remaining for j > 1, which reduces the t power by j—-(2 —2 £_ ny 1) 2k—1 + i= 3 — 3 . Thus, these terms are of 22 Fak aK Ti }. Nie When j= 0, the term becomes a —2 B*VV . ga p[x.£}a(e—iV ap) . On the behavior. I show that these (eran e +) surface, this appears to cause O terms, in fact, vanish. The feature that causes many of these terms to vanish is an integration ~35- over all angles of functions independent of angular change. In particular, a g(X,€) for € < k are independent of the angular variable & . For t a, , this is clear, and for 2 < k, b, = 4 E72 J Vv (x)s¢/2k —! as, Thus, 0 £ 0 2k—€ the integral over the € angular coordinates of i} B*VV .¢ a p(x,€) a{&) = Q, lel C, = —§ J B*VV,, + (€(—2i€ - Vag)—iVag) dxd€ , — i n—1p/n 1 B*VV2,(0) W(DVxlo)de or Cc, = Dk vol(S WE+2—-p | a ai S) (Vo,Cor))* « B*VV.,(0) -W(DVx,(o) -w)dwde 1 1 n-l xs (Volo i n n 1 + @& PE+D PGE +2-p f . In conclusion, (5.4) and (5.5) add just |¢{* terms, which vanish, and so the result holds. oO To complete the proof of theorem 2, one combines lemmas 5.2 and 5.6 to obtain —tH —? 1 2 i teeth — 90) = BEBE GPO TY Loom FH) wy 3 + of ota re) —~2 22 Boy, : _ where £9 = 3 5k max({j : Vj * 0, j < 2k} U (2k—1)}). ~36- $6. Some Further Remarks. For potentials without the same polynomial growth in all directions, similar theorems will hold. However, the growth of the potential in directions not parallel to the magnetic field must be at least quadratic in the distance; —tH —tH and tre 0 will differ. A further —tHg otherwise, the leading orders of tr e requirement is that Hy must have a discrete spectrum and e must be trace class. The only change necessary for these results is to use a new weight n n ®o=-14+3 ki + |€|?, where V~>> oki . i=l i=l]? In theorem 1, one sees the weakness of the operator norm method, as the more precise pseudodifferential computations show that the term B*VV .V does’ not contribute a leading term. The pseudodifferential operator computations imply that for a broad class of potentials, one should expect two orders in t faster decay in the perturbed term with a magnetic field perturbation. The gauge invariant techniques in this paper will also extend to the non- uniform magnetic field case. These techniques imply that [2A -D + A.A, Hol and higher order commutators will determine the behavior of the asymptotics. For the compact manifold case, the weighted Sobolev spaces need not be used; however, the gauge invariance will play a similar role. This gauge invariance should allow the canonical order techniques to be applied to compute the perturbative effect of a magnetic field on the asymptotics for the trace of et& or similar operators on a compact manifold. (1) [2] [3] [4] (5) [6] [7] [8] [9] [10] [11] (12] {13] {14] [15] -37- References Adams, R., Sobolev Spaces, Academic Press, New York 1975. Beals, R., A General Calculus of Pseudodifferential Operators, Duke M.J., 42(1975), 1-42. Cycon, H. L., Froese, R. G., Kirsch, W., and Simon, B., Schrodinger Operators with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin, to appear. Davies, I. M., The propagator for a charged particle in a constant magnetic field and with a quadratic potential, J. Phys. A.: Math. Gen., 18 (1985), 2737-2741. Duistermaat, J. and Guillemin, V., The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics, Invent. Math., 29 (1975), 39-79. Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, 4t* Ed., Academic Press, New York, 1965. Hellfer, B: and Robert, D., Properties Asymptotiques du Spectre D’Operateurs Pseudodifferentiles Sur R", Comm. P.D.E., 7 (1982), 795-882. Iwatsuka, A., Magnetic Schrodinger operators with compact resolvent, J. Math. Kyoto Unov., 26 (1986), 357-374. Leinfelder, H., Gauge Invariance of Schrédinger Operators and Related Spectral Properties, J. Operator Theory, 9 (1983), 163-179. Reed, M. and Simon, B., Methods of Modern Mathematical Physics: Fourierx Analysis and Self-adjointness, vol. II, Academic Press, New York 1975. Robert, D., Properties Spectrales D’Operateurs Pseudodifferentiels, Comm. P.D.E., 3 (1978), 755-826. Schrader, R. and Taylor, M., Small A Asymptotics for Quantum Partition Functions Associated to Particles in External Yang-Mills Potentials, Comm. Math. Phys., 92 (1984), 555-594. Simon, B., Maximal and Minimal Schrodinger Forms, J. Operator Theory, 1 (1979), 37-47. Simon, B., Schrédinger Semigroups, Bulletin A.M.S.., 7 (1982), 447-526. Taylor, M., Pseudodifferential Operators, Princeton University Press, Princeton, 1981.