Absence of a Scott Correction for the Total Binding Energy of Noninteracting Fermions in a Smooth Potential Well Citation Huxtable, Barton Duane (1988) Absence of a Scott Correction for the Total Binding Energy of Noninteracting Fermions in a Smooth Potential Well. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/87FQ-5X29. https://resolver.caltech.edu/CaltechETD:etd-09062005-101909 Abstract It is shown, for V in a particular class of smooth functions, that the total binding energy, E(Z), of Z noninteracting Fermions in the potential well Z 4/3 V(Z 1/3 x), obeys E(Z) = c TF (V)Z 7/3 + 0(Z 5/3 ) as Z → ∞. Here c TF (V) is the coefficient predicted by the Thomas-Fermi theory. This result is consistent with the conjectured Scott correction, which occurs at order Z 2 , to the total binding energy of an atom of atomic number Z. This correction is thought to arise only because V(x) ~ - |x| 1 near x - 0 in the atomic problem, and so V is not a smooth function. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Atomic physics; quantum mechanics; Scott correction; semi-classical approximation; Thomas-Fermi theory Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Simon, Barry M. Thesis Committee: Simon, Barry M. (chair) Barnes, Charles A. Cross, Michael Clifford Koonin, Steven E. Defense Date: 16 June 1987 Non-Caltech Author Email: bhuxtable (AT) usersystems.com Funders: Funding Agency Grant Number NSF UNSPECIFIED Earle C. Anthony Fellowship UNSPECIFIED Caltech UNSPECIFIED Record Number: CaltechETD:etd-09062005-101909 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09062005-101909 DOI: 10.7907/87FQ-5X29 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3349 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 06 Sep 2005 Last Modified: 02 Jul 2025 23:20 Thesis Files Preview PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

ABSENCE OF A SCOTT CORRECTION FOR THE TOTAL BINDING ENERGY OF NONINTERACTING FERMIONS IN A SMOOTH POTENTIAL WELL Thesis by Barton Duane Huxtable In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1988 (Submitted June 16, 1987) - 11- This thesis is dedicated to the memory of my father, Thomas Richard Huxtable, Jr., M.D. There are questions which we could never get over if we were not delivered from them by the operation of nature. Reflections on Sin, Pain, Hope, and the True Way Franz Kafka -111- ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Barry Simon, for suggesting the problem in this thesis, and for his guidance and encouragement during its solution. I would also like to express my gratitude to Dr. Didier Robert for correspondence that proved crucial in this work, to Dr. John Morgan III for several encouraging conversations, and to Dr. Robert N. Hill, whose tutelage has been a continuing inspiration. Many discussions with Kris Odencrantz and Askell Hardarson were also helpful in completing this work. The support of the National Science Foundation in the form of a Graduate Fellowship is sincerely appreciated. I am grateful for further support from the California Institute of Technology in the form of the Earle C. Anthony Fellowship, Special Institute Fellowships and Graduate Teaching Assistantships. I wish to thank all my friends at Caltech, especially Gary Kruppa, Joe Rushanan, and Jamil Tahir-Kheli, for providing amusement and distraction these last several years. Finally, I wish to thank my mother, Carole, without whose support and encouragement this work would never have been completed, and my wife, Jacqueline, for helping me (and tolerating me) through this last year. -IV- ABSTRACT It is shown, for V in a particular class of smooth functions, that the total binding energy, E(Z), of Z noninteracting Fermions in the potential well Z4/3V(Z1/3x), obeys E(Z) = ctf(V)Z773 +0(Z573) as Z → ∞. Here cτp(V) is the coefficient predicted by the Thomas-Fermi theory. This result is 2 consistent with the conjectured Scott correction, which occurs at order Z , to the total binding energy of an atom of atomic number Z. This correction is thought to ariee only because V(χ) ~ —∣x∣~3 near x — 0 in the atomic problem, and so V is not a smooth function. -v- TABLE OF CONTENTS Dedication...................................................................................................................................ii Acknowledgements................................................................................................................. iii Abstract.................................................................................................................................... iv Chapter I: Introduction.......................................................................................................... 1 Chapter Π: A Fundamental Result..................................................................................... 6 Chapter ΠΙ: Spectral Asymptotics for a Class of Potentials.................................. 14 Chapter IV: Conclusions..................................................................................................... 25 Appendix A: Mathematical Detail.................................................................................... 30 Appendix B: The Method of Stationary Phase........................................................... 44 References................................................................................................................................48 -1- CHAPTER I Introduction Since the advent of the quantum theory, there have been efforts made to apply the theory to increasingly more complicated physical systems. This has, more than often not, made necessary invention the of various approximation schemes. One of the earliest of these is now generally known as the Thomas-Fermi theory, and one of its first uses was in the study of large atoms [24]. For an atom with Z electrons, an infinite mass, point nucleus of charge Z, and neglecting we have H(Z) - ]Γ (-Δi - i) + ∑ relativistic effects, the quantum mechanical Hamiltonian or Schrodinger operator (1.1) i=l We consider this ∣xi-Xj∣^1. l≤i<j≤Z Hamiltonian over antisymmetric wave functions in z Λ L^(R3,∙C^) and denote by E(Z) the ground state energy of H(Z). (Our i=l units are such that m==^,e==λ = l, so our energy unit below is ⅛½- ≈ 54.4 eV.) Milne [14] used the Thomas-Fermi theory to show that A E(Z) ~ cZ7/3 with c = cMilne = - .308. (1.2) (This value of c was a little low due to inaccuracies in the numerical solution of the Thomas-Fermi differential equation ([24], Equation 1.3). The correct value is Cyp = —.3844.) Using spectroscopic data to find E(Z) for -2- Z = 2,3,...,9, Young [25] found that the observed values obeyed -r7, , √7∕3±1∕1OOO E(Z) =“ cγou∏g . f°r 2 ≤ Z ≤ 9 and cYoung = ~∙2θ8∙ It was suspected by Young that the discrepancy between Cγouπg and ≈jv{jjπe was due to the numerical difficulties in the evaluation of Cγθung. (However, the correct value c-γp makes the discrepancy worse.) The first improvement on the Thomas-Fermi model of the atom was proposed by Dirac [5], who took into account the effect of exchange. As noted in [19], this adds to (1.2) a term —.111 Z 5/3 , which does not improve the agreement with the observed values. Equation (1.2) was cast further into doubt by Foldy [6], who found, from the then recent Hartree calculations on the atom for several Z between 10 and 90, that apparently, E(Z) “ -∣ Z12/5 for 10 5 Z 5 90. One is surprised by this because (1.2) was derived from the Thomas-Fermi theory, in which the statistical assumptions should hold better for large Z. This situation was finally resolved by Scott [19], who proposed the following leading order behavior: (1.3) E(Z) - ctf Z7/3 + iz2 -.ιnz5z3. The Z2 term is called the Scott correction and arises as a result of the inability of the Thomas-Fermi theory to describe the electrons near the nucleus, where the nuclear Coulomb potential varies rapidly. The argument essentially consists of treating the inner electrons as if they were in Bohr orbits around a nucleus of charge Z and treating the outer electrons with the usual Thomas-Fermi theory. (This argument is done very nicely in [18].) This -3derivation suggests that the Scott correction is independent of the presence of electron-electron repulsion, since the inner electrons that contribute to the Scott correction are insensitive to the presence of the other electrons when Z is large. However, electron-electron repulsion does contribute to the and Z3 Z terms in (1.3). In [20] it is shown that numerical fits to Hartree-Fock calculations for various Z are in exceedingly good agreement with (1.3). (The Z 5/3 coefficient is actually —.133, because of corrections to the Thomas-Fermi theory other than exchange effects as discussed in [1].) Lieb and Simon [13] give a mathematically rigorous justification of (1.2) (with c = Cyp) starting from (1.1), in effect proving that the Thomas-Fermi theory gives the correct leading-order behavior. Their proof extends easily to more general Schrödinger operators of the form Z (1.4) H(Z) = 22 (-Δi + Z4/3 V(Z173xi)) + 22 i-1 ∣xi~ xj∣'1 l≤i<j≤Z and shows that E(Z) ≈ cτp(V) Z7/3 + o(Z773) as Z - +∞. (V(x) = — ∣x∣'1 in (1.4) gives (1.1).) Here c-pp(V) is the coefficient one obtains by applying the Thomas-Fermi theory to (1.4) and computing E(Z). As noted in [13], the proof used cannot be improved so that we could see the Scott term. The proof basically gives upper and lower bounds to E(Z), and the method of constructing these bounds makes them differ at order 0(Z ). Proving that the Scott correction in the atomic binding energy is correct has remained a challenging open problem. It was shown by Lieb [12] that a modification of the Thomas-Fermi theory, known as the Thomas-Fermi-von Weizsäcker theory, does produce a Z correction in the atomic binding -4- energy. However, unlike the case of the Thomas-Fermi theory, there is no theorem that connects the Thomas-Fermi-von Weizsäcker theory to the real quantum problem. There is probably no such theorem because one cannot simultaneously get the Z correction conjectured by Scott [19] and the electron density conjectured by Lieb [12]. Thirring [23] has produced a lower bound, which says E(Z) ≥ cψp Z7Z3 (1 +0(Z^2z33)) for the atomic problem. Siedentop and Weikard [22] have recently proven the upper bound E(Z) ≤ cτpZ7z3 + ∣Z2 + o(Z2). The problem treated in this thesis was originally motivated by the results of Bander [1], where it appeared, at least formally, that if V in (1.4) had no singularities and was smooth, then E(Z) would be of the : -m E(Z) = ctf(V)Z773 + 0'Z5z3), (1.5) and we would have no “Scott correction” or Z* contribution. In this thesis, we treat the case of noninteracting particles; that is, we drop the ∣×i — xj j 3 terms in (1.4). Also, we consider smooth potentials in the class V, namely, those V ∈ Coo(R3), which obey c∣x∣2 ≤ V(x) ≤ C∣x∣2, ∣VV(x)∣ ≤ c'∣x∣ for some c, C, c' > 0, and all higher derivatives of V are bounded everywhere in x. The main result of this thesis (Theorem 7 at the beginning of Chapter IV) is that, for the Hamiltonian H(Z) = (-Δi + Z4/3 V(Z173x∙ )), i=l the ground state energy E(Z) is as in (1.5) when the Fermi level is not at a critical value of the potential. (The Fermi level is the energy of the highest occupied single particle state.) This result is of interest because it suggests that there would be no Scott correction in the atomic binding energy, except for the fact that the nuclear potential seen by the electrons has a singularity, which is consistent with Scott’s intuitive argument. In Chapter II we start by introducing some notation and then prove a fundamental result (Theorem 1) that tells us, in effect, the “right way” to approach our problem. Two examples are discussed along the way. In Chapter III we use some recent mathematical methods to prove some facts about two functions related to the spectrum of —Δ + Z4Z3 V (Z^Z3 x ). The proofs are outlined in Chapter III, leaving many of the mathematical details to the Appendices. Finally, in Chapter IV we put the results of Chapters II and III together to obtain our desired result. We then discuss the reason for not allowing the Fermi level to be at a critical value of the potential, and possible extensions to smooth potentials other than those in the class V. Finally, a formal calculation is given, suggesting what the Scott correction might be for certain singular potentials. -6- CHAPTER H A Fundamental Result As motivated in the Introduction, we are interested in the ground-state energy of the quantum mechanical Hamiltonian Z5~" (-Δ1 + Z473V(Z173Xi)) H(Z) = (2.1) i=l Z∕ 2 3 Q A L (R ; C ). We choose units so i=l . This ground-state energy is given by over antisymmetric wave functions in that λ = 1 and m = E(Z) = Z ∑ ej(Z)’ J=l where βj(Z) = jth eigenvalue (counting multiplicity) of — Δ -+-Z4Z3 V(Z^z3x) over IA(1R3j Cq). By rescaling energy by Z“4Z3, and distance by Z'^Z3, and defining h = Z 1 /3 , we obtain (tolerating a common abuse of notation) Z'473E(Z) = E(h), where (2.2) E(h) = h’3 Vj ej(h),∙ j“l (2.3) ej(h) = jth eigenvalue (counting multiplicity) of — hi^∆ -+- V !,χ) over L2(1R3! Cq). -1- It must be stressed that h = Z -1/3 is not related to ħ, which is implicit in (2.1). We use this notation because it is traditional in the work of Chapter III and because h → 0 is suggestive of a semiclassical connection, which will be seen later. Whereas before, we were interested in the asymptotic form of E(Z) as Z → oo, with this change of variables, we are now interested in the asymptotic form of E(h) as h -→ 0 . From the discussion in the Introduction, we anticipate that (2.4) where c∣ E(h) = c1h"3 + c2h'2 + 0(h^1) as h → 0 c2 are constants depending on the potential V. It is the primary point of this thesis to show that for a class of smooth potentials there is no “Scott correction;” i.e., c2 = 0 in (2.4). Now we must introduce some notation. We only consider potentials V, whose positive and negative parts, V+ and V__ , satisfy V_j_ ∈ L 2(R 3)∣oc and V_€L2(R3) -)- Loo(R3). For such potentials, the Hamiltonian H = Hh = -h2∆ + V is self-adjoint. (See [16], sections X.2 and X.4.) Let βj(h) be the eigenvalues (if any) of H∣1 as in (2.3). Make the definitions (2.5) Nh(e) = #{j ; βj(h) ≤ e} ≡ the number of eigenvalues of H∣1 that are ≤ e; ∫∫ d3xd3p = -^ ∫ (e-V(xV372d3x. P2+V(x)≤e 6π V(x)≤e We further restrict the potentials under consideration by requiring that there be ∕zθ ∈ R and € > 0 so that Nc(∕Zq) = 1 and Nc(∕2q + ∈) < +»· This -8- o has at least h' (=Z) eigenvalues at the restriction on V says that bottom of its spectrum. (This is necessary for (2.2) to make sense, and physically it means that V is a deep enough well for (2.1) to bind Z particles.) To see this, note that h^ Nc(e) is the semiclassical estimate for N^(e), and it is known ([17], Section XIII.15) that lim h3N.(e) = Nc(e) hW h c for e ≤ μ,Q ÷ e. Since Nh(μθ+e) = h^3Nc(e) (1 +o(h)) as h → 0 and Nc(∕zθ+e) > Nc(∕χθ) = 1, we have "+^ e) > h 3 N∣1 for h sufficiently small. Next, define (2.7) ∕Zj1 = min{e : N^(e) ≥ h } ≡ Fermi level that results when the lowest h^3 = Z energy states in the potential well are occupied, and *h(e) = J!oo Nh(e?) de'∙ (2.8) The need for Ψ∣1 will become apparent momentarily. Since N∣1(e) is right continuous and non-decreasing in e, it defines a measure on ]R and we have h^3 E(h).∙ = ej-(h) j-1 ∫ e dNh(e) - (W ~ h^3¼ (-oo√zh] (the second term may be thought of as a correction to the first term, which over-counts if Nj1CtZj1) > h' ; this may happen if the Fermi level μh has a multiplicity greater than one) - h'3⅛ - !Z ni><*> dc -9- - h^3¾ - *h'¾> after an integration by parts. In deriving this result, we had in mind that h^3 = Z = integer number of particles, but (2.9) E(h) = h-3zxh - *h⅛ makes sense for all h > 0, and we will henceforth take (2.9) as our definition of E(h) for all h > 0. To understand the motivation for the following theorem, it should be noted that N∣1(e) — h -3 Nc(e) may oscillate with increasing magnitude ~ h^-2 as h -* 0. (This will be the case in the two examples to be discussed after the theorem.) It might be expected that this behavior carries over to E(h); that is, E(h) — c∣h -3 might oscillate with magnitude ~h -2 as h → 0. Hence, there would be no C2h term in (2.4), which is the term of primary interest to us! The following theorem shows that the asymptotic form of E(h) may be smoother than Nj1 as h — 0, and motivates the study of the asymptotic form of Ψj1 for small h. First, define n(e,h) so that Nh(e) - h^3Nc(e) + h'2n'e,h). (2.10) This function n(e,h) describes the error made in approximating N^(e) by h^3Nc(e). Theorem L· Suppose there are χzθ e R and e, hθ, nιπax > 0 such that: (2.11) Nc0uθ) = 1, (2.12) Nc(∕Zq÷∈) < +∞, and (2.13) ∣n(e,h)∣ < nmax for ∣e - χzθ∣ ≤ e, h c (0, hθ]. -10- (2.14) + 0(h^b as h → 0. Then E(h) = μgh~3 — The following two examples illustrate Theorem 1, which is then proven next. The examples also show that Theorem 1 is optimal in the sense that one does not generally have smoother asymptotics on Nj1 as input, nor can one generally conclude smoother asymptotics on E(h). Example 1 (“Bohr atom”): Let V(x) = — (x∣^^. The eigenvalues of — h2∆ + V are then known to be n = 1, 2, 3, ... £ = 0, 1, ..., n-1 m = -£, -£ + 1, ..., £ s = 1, 2, —, q 4h2n2 ' en⅛s^^ By direct computation, for e < 0 = Σ n.l,2,3,∙ ∙ ∙ n≤—⅛= (u] 5n-1□ Σe Σq 1 = Σ n=l s-Ι € =0 m==-£ 2ħT-β — where V q(⅜[≈->J3 + ⅜M2 + l[υj), 1 2h4-"e and fx] greatest integer ≤ x. Define δ = δ (e,h) V— [υ}j then one finds Nh(e) = h^3Nc(e) + h^2n(e,h), where Nc(e) = ^(-e)'3/~ and n(e,h) = q((½- δ) + (i—tf+δ2)^ψ= + ( — 7+¾--⅞-)h2l. Note that, since 0 ≤ 5(e,h) < 1, ∣n(e,h)∣ is bounded for 6 2 3 } h c (0,hθ], e < eθ where hθ and — eθ are any positive numbers. By similar calculation, one finds *h'e' = [u] ∑ m2(e - ⅛) n=l -11- q 12h34-e -‰ + h-¼(e,h), where ∣φ(e,h)∣ is bounded for h € (0,hθ], e < eθ . The Theorem then gives, with μθ = -(⅛)273, E(h) = -∙5---ξq (2.15) h^3 + ∣h-2 + 0(h'1). With q = 2 for electrons, we get the usual “Scott correction” in these units. An exact calculation of E(h) shows that the term 0(h^b in (2.15) oscillates with amplitude ~ h^* as h — 0 (see “model (a)” in I21]). In this sense, (2.14) gives the best possible asymptotics for E(h). Example 2: 2 Let V(x) = ∣x∣ . As in example 1, the eigenvalues are known explicitly: etans^ = h(2^+2m+2n+3)5 £,m,n ∈ {0,l,2,...}, s = l,2,...,q. Direct calculation gives Nh(e) = h^3Nc(e) + h^2n(e,h), and <r∣1(e) = h^3 Nc(e) = ∣e3 e^ + h^^ ≠(e,h). Again, ∣n(e,h)∣, ∣⅛(e,h)∣ are each bounded for h ∈ (0, hθ], e ∈ [0,eθ], where hθ, eθ are any positive numbers. The theorem gives (2.16) E(h) = I (ή)173 h'3 + 0⅛'1). In this case we have no “Scott term.” A direct calculation of E(h) shows that (2.16) gives the best possible asymptotics for E(h). Proof of Theorem 1: First define μ∣(h) so that A⅛ = Aîq + hμ∣(h). Using -12- this with (2.8), (2.9) and (2.10), (2.17) E(h) ≈ h'3M0 + h^2μ1 (h) - Ψh(∕zθ) - ∫⅛+hz21 Nh(e) de = h 3μ0 — + h'3 J2o+hχ2l∏ — Nc(e))de — h^2 J^+hzzιn(e,h) de. We show that these last two terms are 0(h"^) as h → 0 to obtain the desired result. To this end, we first show that A∣(h) is bounded for h sufficiently small. From (2.6) we see that Nc(e) is continuous and strictly increasing for those e such that 0 < Nc(e) < +ββ. From our hypotheses, 1 < Nc(μθ-)-e) < +∞, so we may choose δ ∈ (0,g) sufficiently small so that Ν^(ί 1 — δ,1 +i]) C [∕2q-∣,∕2q + ∣]. Let hj = min{hθ, ∏^}∙ Recalling (2.7) and (2.9), we have A⅛ = min{e : Nc(e) + hn(e,h) ≥ 1}, which gives, for h ∈ (0, h∣], (2.18) √j1 ∈ Nc ([1 hnmax, 1 + hnmax]). Taking the derivative in (2.6), we find that Nς(e) exists and is finite, continuous and nondecreasing for —∞ <e < μθ + ∣∙ Also, Nς(e) > 0 when Nc(e) > 0. Using this with (2.18) yields -1 (1.5)) ph-^0∣ ≤ h⅛ax nun i or (2.19) Ai1(h)J ≤ ∏mtx Nc(Nc'1(l-5)) for h ∈ (0, h]. -13- For the last term in (2.17) we have the bound lh'2Ch"1 n(e>h) de∣ (2.20) ≤ h^1 p1 (h)∣ nmax for h ∈ (O,hjJ, and so this term is 0(h^-lx) as h → 0. The second-to-last term in (2.17) is bounded by (2.21) ι∙Ao+h^ι,, .τ c(e))de , XX j = ,-3∣∙h∕z, h'-33∫μθ 1d -N h-3∫c∕1 (Nc(,uθ) _Nc(μθ+e))de ≤ h'3(∫θlzxι1 e de; max Nς(∕x∩+e) ∣e∣≤hμ1∣ c θ ≤ h^1 ∣μ1(h)∣2 Nc(μ0 + ⅜) for h ∈ (0,h1 Equations (2.17), (2.19), (2.20) and (2.21) give the desired result (2.14) for h ∈ (0,h1]. -14- CHAPTER III Spectral Asymptotics for a Class of Potentials This chapter is devoted to determining the asymptotic form of the functions N∣1 and Φj1 as h — 0 for a class of potentials, T, to be described were introduced in Chapter II. Our goal shortly. The functions N∣1 and here is to prove that prove that satisfies the hypotheses of Theorem 1, and then to is of such a form that E(h) has no “Scott correction” for potentials in class V*. An outline of the arguments will be given in this chapter, leaving the details of the proofs to Appendices A and B. Also in this chapter we shall let q ≡ “number of spin states of the fermions” assume the value q = 1 . This is because q appears only as a multiplicative factor in N∣1 and Ψ∣1. It should be noted that E(h) does not depend on q in this simple manner. The methods used in this Chapter are those of Chazarain [4] and Helffer and Robert [7]. The new content here is in evaluating the “second term” in the small h asymptotics and the straightforward extension of their methods First we shall define the class of potentials to be considered (see [4]). Let V be the set of all real valued, Coo-functions V on lRn which satisfy as ∣x∣ — +βo : '3“V(x) 0(∣x[2 ∙3χV(x) 0(1) lαl) for0≤∣a∣≤2 (3.1) and for ∣a∣ ≥ 2 -15- (3.2) V(x) ≥ c∣x∣2for some c > 0. For our purpose of outlining the arguments in this chapter, (3.1) will not appear in what follows. It, along with (3.2), will be used in Appendix A to produce bounds on the phase, S, and amplitudes, aj, in a geometric optics expansion, which will be introduced later in this chapter. The primary consequence of (3.2) to be used in this chapter is that N∣1 is a tempered distribution. Finally, in this chapter we will work with potentials V (x) for X ∈Rn. Although we are primarily interested in the n = 3 dimensional case, the case of more general n ≥ 3 is no more difficult. For all V ∈ V, H = — h Δ+V is self-adjoint and has a spectrum composed of eigenvalues {e∙(h)}°^, that obey ei(h) → +∞ as j → oo for each fixed h >0. We may apply the Cwikel-Lieb-Rosenbljum bound (see [17], Section XΠI.3) to Nj1(e), defined in (2.5), to obtain (3.3) Nh(e) ≤ cnh-n J (e-V(x))n72 dnx. n jV(x)≤e Equations (3.2) and (3.3) then show that N∣j(e) ≤ c'h^πe11. Thus, we may think of Nj1 ∈ as defined by (Nτh,f) = ∫X Nh(e) f(e) de for all f ∈ J. The derivative is ti.en in f' and may be written as a sum of i-f unctions (3.4) oo N^(e) = y 5(β-ej(h)). j-1 The function is often called the density of states for the quantum Hamiltonian H. Furthermore, the Fourier transform is in J,'. We thus see -16- that the trace of the (quantum mechanical) propagator is a tempered distribution, because formally, (3.5) ⅛(t) = g-ιtej(h) ∫°^ e^ite Nfr (e) de ≈ j=l tre-it∏. To get at NJ1, which will ultimately enable us to get information on Nj1 and Ψj1> we shall consider a convolution * θ∣1. Since we are interested in the asymptotics as h → 0, it would be nice if 0^ — d-function as h → 0. In particular, we put (3.6) ⅛(.) - j⅛ i(-l), where p Ç. C“(R) will be chosen later. We require p(0) == 1, which makes have unit weight; that is, (3.7) ∫^o 0h(t) dt ≈ p(0) = 1. For technical reasons in the proof of Proposition 5, we require, as in [7], Section 5, that ρ is even, p ≥ 0 and there is some 5θ > 0 so that (3.8) p(σ) > 0 for σ∈[-iθ, 5θ]. (To construct such a p, start with Pγ ∈ Cq°(IR) with Pj even, 0 and x,1(0) >0, then put pit) - ⅜-^⅜-)∙> Since a convolution is equal to the inverse Fourier transform of a product of Fourier transforms, we have -17- (3.9) (¾.⅛)(λ) - ⅛L~ eitλ ⅛(t) Sh(t) dt — y-∫°° eltλp(th)tr e_ltHdt 27Γj -oo = ⅛ ro° χ,(t) eitλ7h tr e‘itH/h dt. 2πh J-oo (Chazarain’s [4] notation Iτ(h) = (2τrh)(N^*θ∣1)(- τ) in the notation used here.) To make further progress, we must determine the propagator which is the unitary group that solves the Schrödinger U(t) == equation. That is, if ih3ttf(t,x) = Htf(t,x), (3.10) then tf(t,x) ≈ e'itH/htf(0,x) - U(t)0(O,x). To this end, consider the effect of applying the propagator to a plane wave of momentum T}. Make the definition A. (t.x.π)βistt∙x'wh _ e-‘tH/h β"7'xzt, (3.11) Clearly, there is some freedom in choosing A∣1 and S, and the choice we make is that S does not depend on h. Putting (3.11) into the Schrödinger equation (3.10) gives (3.12) 3tS + ∣3xS∣2 + V(x) = 0, (3.13) 3tAh + 23xS∙3xAh + (ΔS)Ah = ihΔAh, S(0,x,η) = η-×-, Ah(0,x,7) = 1∙ The first of these (3.12) is just the Hamilton-Jacobi equation from classical mechanics. The second (3.13) is usually solved by making the expansion -18- N hj aj(t,× 5 77 ), A(hN)(t,x,r?) = (3.14) j-0 where the amplitudes a- satisfy the transport equations (which derive from u (3.13)): (3.15) 3t.j + ¾S ■ 3xaj + (ΔS)aj -{ _ jζ° }i ao(O,x,77) = 1, βj(O,x,r7) = 0 for j ≥ 1. Formally, we expect -» A∣1 as N — <×> for small h. In Appendix A we will construct Aj1^ by solving the transport equations (3.15) for small times t and then will study how well the operator E(t) defined by (3.16) (E(t)f)(x) = (2Th)’nJjAj1N)(t,x,r?)el(S(t,x’7?)‘’7'y)/h f(y)dy dη approximates the operator U(t). This method of constructing an approximation E(t) to U(t) is known as the geometric optics expansion. Merely to outline the arguments, in this chapter we will assume we can solve (3.12) and (3.13) in some neighborhood of t = 0 and will write formally (3.17) (U(t)f)(x) = (2τh)∙1,∫∫Ah(t,x,T7) i(S(t,x,77)-77 y )∕h f (y)dy dη. A way to think of (3.17) is that the integral over y decomposes f into its plane wave constituents e*^ χy,*ξ and the integral over TJ superposes the solutions (3.11) for the time evolution of the plane waves. Returning to our discussion of the convolution which is a smooth approximation to N^, we obtain from (3.9) and (3.17) (3.18) (N'h*θh)(λ) = (2τh)-π-1∫∫ ∫ oo p(t)Ah(t,x,r7). K2n ^°° i(S(t,x,7)-×-7+λt)∕h dtdx dT]. -19- The small h asymptotics of the integral in (3.18) may be determined by the method of stationary phase. The phase function ≠(t,x,η,λ) = S(t,x,r)) - x-r? + λt (3.19) is stationary at the points (t,x,7)), for which (3.20a) 3t≠ = 3tS (3.20b) 3x≠ = 3xS - η =0; (3.20c) ∂ηφ = 3ηS - x =0. + X =0; Equations (3.20a) and (3.20b) with (3.12) say that ∣η∣2 + V(x) - X. (3.21) Equation (3.20c) implies that (t,x,77) are the period, initial position, and initial momentum, respectively, of a closed periodic classical trajectory. (This point will be made clearer in Appendix A.) Note that from (3.12) all trajectories of period zero {(t,x,77) : t =0, (x,7?) ∈ R2π} satisfy (3.20c). At this point we exercise our freedom in choosing p ∈ Cq°(1R). Make the definitions : Xτ = set of periods of periodic classical trajectories of energy t; (3.22) £, 1 = U Iτ, for an interval ICR. τ ∈I (Note that our JLτ = JL.τ in [4].) In the hypotheses of the next theorem these definitions will appear, because what is desired is to take p(t) = 0 outside a small neighborhood of t = 0, and this neighborhood shall not include any periods, other than zero, of closed periodic classical trajectories -20- with energy τ near X. (That this may be done is left to Appendix A.) This condition on p means that in the integral (3.18) we need only concern ourselves with those points of stationary phase with t = 0. This set of points of stationary phase lies on the compact (2n— 1) - dimensional manifold (3.23) Wχ = {(t,x,η) ∈ 1R2π+1 : t = 0, ∣77∣2 + V(x) = X}. Compactness of Wχ follows from (3.2). If we further assume that X is not a critical value of the potential V, then the phase φ in (3.19) will be nondegenerate over the manifold Wχ. That 2 is, the matrix 3(t x η) Φ will not have determinant zero anywhere on Wχ . We can now use the method of stationary phase (discussed in Appendix B) to calculate the small h asymptotic behavior of the integral (3.18). We obtain: Theorem 2: Suppose (Xθ- eQ>^Q÷εθ) is a bounded open interval such that V has no critical values in [Xθ — £q,Xq +eθι and that (3.24) supp p ∩ I , Λq ♦ ) = {0}. Then (3.25) (NJ1*0h)(λ) - (2πh)^n vol(Wχ) + 0(h2^n) as h — 0 for X ∈ (Χθ—£θ,Χθ+£θ). Moreover, the term 0(h2~n) is uniform for X ∈ (λθ-£q,Xq+£q). (θ∣1 is related to ρ in (3.6) and Wχ is given by (3.23).) The first term in (3.25) appears in [7], Theorem 3, and for one value of X (instead of X in an interval) in [4], Theorem 2. Although not part of our argument here, it should be mentioned that the effect of the presence or absence of periodic closed classical trajectories of periods other than zero on Nja or N∣1 has been well observed. (See [3], [4], [7], -21- [10], [11], and [15].) They determine the detailed structure of N^(e), that is, the oscillatory behavior of the function n(e,h) appearing in Theorem 1. Here we want only to obtain that n(e,h) is bounded as h — 0, and it will not be necessary to consider the periodic closed classical trajectories of periods other than zero. Now we move on to computing ^*0^ and Since ⅝<λ> -fLNh(e) de' we have (Nh*0h)(λ) = ∫^oc (N'h*βh)(τ) dr. (3.26) Using (3.18) in (3.26) and recalling (3.19) then gives (3.27) (Nh*θh)(λ) =(2τh)-n-1Jλoo ∫∫ Jθ^p(t)Ah(t,x,T7)elφct,x,z7,τvhdtdxd77dτ. ^°° lR2n "°c Apply the method of stationary phase (modulo a technical point to be mentioned shortly) to the integration over (t,τ) in (3.27). The phase is stationary for (3.28a) 3t≠ = 3tS+τ= 0; (3.29b) σrΦ = t = 0. Equations (3.28a and b) combined with (3.12) give (3.29) {(t,τ) : t =0, τ =∣r7∣2 + V(x)} as the stationary point of the phase φ. The method of stationary phase then yields: Theorem 3: Under the same hypotheses as Theorem 2, (3.30) (Nh*θh)(λ) = (2ττh)'n ∫∫ dx dr? + 0(h1"π) ∣r7∣2+V(x)≤λ as h -> 0 for λ ∈ (Xθ-éqAq+éq) ∙ Moreover, the term 0(h^^π) is uniform in X. (The result (3.30) appears in [7], Proposition 5.1, for a single value of X.) -22- For Ψh*θh. note that from (2.5) and (3.26), (*h*θh)(λ) = ∫*oo (λ-τ)(N'h*βh)(τ) dτ, (3.31) which gives, using (3.18), (Ψh*0h)(λ) = (2τrhΓn∙1Jλoo (3.32) JJ ^°° r2∏ ∫ζo (λ-τ)p(t) -o° • Ah(t,x,î7)el0(t’X,T7’T)/h dt dx d?7 dr. Applying the method of stationary phase as in the case of * θ^1, but calculating to higher order in h, yields: Theorem 4: Under the same hypotheses ∫∫ (λ-∣η∣2-V(x)) dxdη +0(h2-n) as Theorem 2, for λ ∈ (λθ-eQ>∖)÷e(p, (3.33) (Ψh*θh)(λ) = (2τh)"n ∣77∣2+V(x)≤λ as h → 0. Several technical points that will be dealt with in Appendix A should be pointed out. As has already been indicated, we will be using A^θ in (3.14) in place of A∣j in the integrals (3.18), (3.27) and (3.32). This error will be shown to contribute at an order of h higher than we are interested in for N sufficiently large. Also, the convergence of these three integrals needs attention. In the integrals (3.27) and (3.32) there is a “sharp edge” on the (t,τ) region of integration at τ = λ . The method of stationary phase that is discussed in -23- Appendix B cannot be applied in such situations because everything must be smooth. We will also have to consider the region of negative τ separately. To remedy this, we take three smooth functions to make these cuts. For λ ∈ (λθ — e0 « λ0 +€0 ) i^ixed> choose χ1, X2. X3 e C0°(F) so that supp χ1 C ( —∞. —1] supp χ2 C[— 2,λ-1] z 2 supp χ3 C [λ — e, +oc) (3.34) X1(τ)+χ2(τ)+χ3(τ) ≡ 1 where € > 0 is sufficiently for τ ∈ F, small so that λ-e > ∖θ-εθ, where (λθ-e0,∖)+e(P ιs as *n theorem 2. We put (3.34) in the integrands of (3.27) and (3.32). The Xj terms will be treated separately in Appendix A and will -l-oo contribute 0(h ) as h -→ 0 . The χ2 terms are treated using the method of stationary phase (now the integrands are smooth) as we have just described. The χ3 terms will be computed by integrating the result of Theorem 2 for τ ∈ [λ— e,λ]. In the sum of these terms, the χ-functions will drop out and we obtain Theorems 3 and 4. This is the reason we need Theorem 2, which is shown by a different application of the method of stationary phase than is used in Theorems 3 and 4. At this point we need to recover the small h asymptotic form of Njj and Ψ∣1, which are of primary interest to us, from the small h asymptotic forms of N∣1 * θh and * θ∣j, which we were able to compute. The following result makes this connection. Proposition 5: Under the hypotheses of Theorem 2, we have (3.35) Nh(λ) - (Nh*θh)(λ) = 0(h1^n) -24- as h — 0, uniformly for X ∈(λθ — y ,λθ). Also, Ψh(λ) - (Φh*θh)(λ) = 0(h2^n) for λ ∈ (∖θ-½,Xθ+½). (3.36) Combining Theorems 3 and 4 with Proposition 5 gives the desired result of this chapter: Theorem 6: Suppose V ∈ V and that Xθ is not critical for V. Let eθ > 0 so that V has no critical values in [Xθ —eθ,Xθ + cθj. Then Nh(λ) = (2τrhfπ (3.37) as h - 0 and 0(h^^n) is ∫J dx dη + 0(h1^n) ∣T7∣2+V(x)≤λ uniform for X ∈ (λ∩ — ,λθ + ⅛). Also, for X ∈ (λθ-⅛,χ0+^), (3.38) Ψh(λ) = (2πh)'n ∫∫ (λ-∣η∣2-V(x)) dxd?7 + 0(h2^n). ∣U∣2+V(x)≤X In Chapter IV we conclude that if the semiclassical Fermi level, z√θ, defined in (2.il), is not critical for V ∈ V^, then E(h) in (2.2) has no “Scott correction.” -25- CHAPTER IV Conclusions In this final chapter we present our main result as Theorem 7. We then discuss extending this result to a larger class of potentials and the difficulty encountered if the potential is singular. Theorem 7: Let the number of dimensions n = 3, V ∈ V^ and ∕2θ be as in (2.11). If χxθ is not a critical value of V, then for E(h) in (2.2) E(h) = c∣h"3 -(- 0(h^l) as h -→ 0, (4.1) where the constant c^ is as predicted by the Thomas-Fermi model. In terms of Z = h , this says that the ground state energy of the Hamiltonian H(Z) in (2.1) is given by (4.2) E(Z) = c1Z7z3 + 0(Z573) as Z----- H∞, and so there is no “Scott term.” Proof : This is an immediate corollary to Theorems 1 and 6 with (4.3) c1 = uq - (2τr)^3q ∫∫ (χz0-∣Z7)2-V(x)) d3x d3Z). ∣T7∣2+V(x)≤∕z0 That c∣ is as in the Thomas-Fermi theory is all that needs comment. The Thomas-Fermi energy functional for the Hamiltonian (2.1) is (after the changes of variables made in Chapter II) (4.4) where h (4.5) δ(pjh) = h^3{∣7 ∫P5z3(x) d3x + ∫p(x) V(χ)d3x}, p(x) is the particle density that must satisfy J p(x) d3x = 1 -26aπd 7 = Minimizing (4.4) with the constraint (4.5) gives p(x) = -⅝ (λ - V(x))¾q, 6π τ (4.6) where X is a Lagrange multiplier. From (4.5) we must have λ = χxθ. Using (4.6) in (4.4) then gives (4.7) S(p;h) .-3 {^0 15τC ∫ (μθ-V(x))572 d3x}. V(χ)≤μ0 Thus, (4.7) and (4.1) agree to leading order if we do the integral over 7 in (4.3). This completes the proof of Theorem 7. A few words are in order on the physical meaning of the hypothesis that the semiclassical Fermi level, μθ, is not critical for V. If μ,Q were critical for V, then the phase φ in (3.19) would be degenerate at some points on the manifold Wji,t in (3.23). In the evaluation by the method of stationary phase of Nfo * θ^1, the successive asymptotic terms would decrease by h^ for some δ anc^ would finally give <1. This would carry over to N∣1 and E(h) = cτph-3 + 0(h-3+2^). This last term means that E(h) may oscillate more widely about Cqψh than if ∕2θ were not critical for V. It should be possible to extend the results of Chapter III, and hence Theorem 7, to potentials that do not grow at infinity, by using the functional calculus developed by Helffer and Robert [8]. They extend Theorem 3 to a broad class of smooth potentials, which includes, for example V(x) = 1 + ∣x∣2)d, d < 0, They are able to do this by using their functional calculus to smoothly truncate the operator H below the continuous spectrum proceeding similarly to Chapter III. For potentials that have singularities, for example, (4.8) V(x) = —∣x∣^a, 0 < a < 2, and then by -27- one has the immediate difficulty that V is no longer smooth or bounded. The assumption that V is smooth and bounded is often used in Appendix A in the proofs of the results of Chapter III. Worse still, formal calculations of N*1 and in terms of V, such as those in [1], give asymptotic series that decrease in powers of h . That is, we might compute something like Nh(λ) = h'3A(λ) + h^1B(λ) + ... n Φh(λ) ≈ h^jC(λ) + h'1D(λ) + .... (4.9) Thus, if V is such that for some δ > 0, E(h) = cψp h^3 + c2h~1+i + ■■·, (4.10) then D(μθ) in (4.9) is found to be infinite. This is not surprising, but knowing D(μθ) is infinite does not then tell us what δ should be in (4.10). Here lies the main difficulty in finding, even from a formal calculation, the first correction to E(h) if it is less that two orders in h down from the leading term, which may be the case for a singular potential (e.g., V(x) == — ∣x∣"1 in Example 1 of Chapter II). We now present a very rough calculation that perhaps sheds some light on the origin of the Scott correction in the case of singular potentials. Let V be of the form given in (4.8). Then the eigenvalues ®j(h) of H = — hx-Δ 4- V are given very approximately by inverting h-3Nc(ej(h)) - j, j = 1,2,3,..., which gives, using (4.8) and (2.6) (4.11) where b = ’ ej(h)------ h^3bj^b, . We then have -28-3 j-b ej(h) ~ -h -3b E(h) = (4.12) j-1 j-1 -⅛h'3 - 7(b)h^3b + 0(1), 0<a<∣orb<l, = < —h^3(log h'3+7) + 0(1), a = ! or b “ 1 -f(b)h^3b + b4yh'3 + 0(1), ∣<a<2orb>l where Ç is the zeta function, 7 = .5772 is Euler’s constant, and m ™ - m1⅛ ( Σ n' - Jo "^ d")∙ n≈l Note that for a = 1 (Coulomb potential) we have, from (4.12), E(h) ~ (-3h^3 + 2.45h^2 + ∙∙∙), which is of the right form, although the ratio of the coefficients is not quite right. The point in computing (4.12) is to point out that what we would naturally call the “Scott corrections” are the h^ terms. This suggests that the Scott correction for potentials of the form (4.8) is of the same order in h as that of the energies in (4.11). If V (x) were of the form (4.8) only for ∣x∣ near zero, ot_ then we would still expect the Scott correction to occur at order h~j , because the lowest states would still scale as in (4.11), and they would be the dominant terms in the sum (4.12). This is just an extension of Scott’s intuition to the more general singularity considered here. If this idea is true, namely, that the Scott correction depends on h (or Z) in the same manner as the ground state in a singular potential, then it is an accident of the particular form of the atomic potential singularity (it is Coulomb) that one obtains in (1.3) what looks like an asymptotic series in h or Z . That is, -29the Scott correction might occur at a different order than 2 η if the nuclear potential singularity were other than Coulomb. Finally, we mention that it may be possible to find a way to combine the methods of Siedentop and Weikard [22] with the methods of this thesis to produce an upper bound for the multinucleus or molecular Scott correction problem. It is tempting to try to cut the problem into two parts, treat the nearly spherically symmetric nuclear Coulomb singularities with the ideas in [22], and then treat the electrons in the unsymmetric but smooth potential away from the nuclei using the ideas in this thesis. But it is not at all clear how to make this “cut.” -30- APPENDIX A Mathematical Detail The purpose of this appendix is to review the results of Chazarain [4] and Helffer-Robert [7], and to extend their methods to compute the leading small h asymptotic form of Ψ∣1 to two orders. A discussion of the method of stationary phase follows in Appendix B. Some preliminaries Let 3> = {⅛) ∈ L2iΔ⅛)∈L2 and ∣x∣2tf>∈L2}. For any potential V ∈ V, the operator H = -h2∆ + V 2 is self-adjoint on 3), positive, and is an isomorphism from 3) to L . The spectrum of H consists of a discrete set of positive eigenvalues of finite multiplicity {ej(h)}. Let {φ∙} be the corresponding set of orthonormal ∈ i(Fn) all j. (For these results, see [2], eigenfunctions of H; then Section 7. Use <p(x,∈) ≡ Ï, Φ(x,∈) = (1 + ∣x∣2+∣∈∣2)iz2 as weight functions.) The unitary group generated by H is U(t) — e-ancj is a bounded 2 operator on L . The unitary group solves the Schrödinger equation (3.10), and we may write the explicit formula .. OO (U(t)tf)(x) = e 1 Sj <1 ∖ /1 ≠j(x) ∫ ≠j(y) ⅛∣(y) dy j=l for ip ∈ L2. The distribution tr e-= tr U (t) is defined as follows for θ ∈ Ï(IR): -31- <tr U(t), θ(t)> = tr U0, where Uθ = θ(t) U(t) dt. To see that the trace is finite, note that ,r, ,, > (Uθ⅞0)(x) ° , ∖g κ —ite,(h)∕h r -----≈ Jrooo θ(t) X e 0j(x) ∫ ≠j(y)tf(y)dydt j=l co = ∫( θ(^j) ≠j(χ) 0j(y)) tf(y) dy. j=ι because the suτns converge absolutely. Thus, oo true= ∑δ'hj j=l is finite since 0 ∈ J. Moreover, (A.1) (tr U(t), Θ(t)> = <NJ1(e), 0(g)> • from which we see that ≈<⅛(∏).θW>, = tr e-it^∕h, w∣lich we used iπ (3 9). Solutions to the Hamilton-Jacobi equation and the transport equations We will later need an estimate for 3∣S, where S is the solution of the Hamilton-Jacobi equation (3.12). From the general theory of first-order partial differential equations, we can solve (3.12) for S, once we know the solutions (x(t,y,77),∈(t,y,∏)) of the characteristic equations (which are just Hamilton's equations of motion): 3tx = 3f(∣ζ∣2+V(x)) = (A.2) 3√ = -3χ(∣ef+V(x)) = 2∈, —3xV(χ), ∈(O,y,77) = η. Since the energy is constant (A.3) x(0,y,η) = y ; ∣∈∣2 + V(x) = ∣77∣2 ÷ V(y) -32- aπd V(χ) — +oo as ∣x∣ → 4^oo, the solution (x,£) of (A.2) exists and remains bounded for all t ∈ R. From the equations (A.2), and making use of the hypotheses on V in (3.1) and (3.2), it is shown in [4], Section IV, that (3.12) has a Co° solution S(t,x,Γ7), (t,x,T7) ∈ [—T,T] × R2n for some T > 0, and we have 3xS(t,x,T7) = ∈(t,x,77)j (A.4) 9ηS(t,x,77) = y(t,x,Z7). It is shown that ∣3tS(t,x,T7)∣ ≥ C λ2(x,∏) (A.5) for C > o independent of ∣t∣ ≤ T and where λ(χ,?7) = (1 4- ∣x∣2 + ∣rz)2∙ Also, one has 3p 3* 3^ S(t,x,η) = O(λp(x,∏)) (A.6) for p 4- ∣α∣ + ∣∕3∣ ≥ 1 . The transport equations (3.15) are solved for ∣t∣ ≤ T in [4], Section V, by a clever change of variables and by integrating along the characteristics given by (A.2). It is then shown that (A.7) 3p 3≈ 3J aj(t,x,77) = O(λp(x,r7)) for p 4- ∣α∣ 4- ∣∕3∣ ≥ 0 and ∣t∣ ≤ T. Error made in using E(t) in place of U(t) For some integer N, which may be chosen at our convenience, define E(t) for ∣t∣ ≤ T by the expression (3.16) for all f ∈ J,(Rn). Let F (t) = E (t) — L, (t). As discussed in [4], Section VI, sup flεit‰L2,L2) ∣t∣≤τ 0(1); sup ∣p,tW) ∣t∣≤τ 0(h N+2∙ -33- and so E(t) and F(t) may be extended to bounded operators over all of L2(Fπ). For β ∈ Cq°((-T,T)), define the bounded operator Eθ = ∫Tr θ(t) E(t) dt. ([4], Proposition 7.1), and we may The kernel of Eθ is then in J(Fn×Fn) then define tr E(t) as a distribution over Cq°((-T,T)) by <trE(t), θ(t)> = tr Eθ . Define Fθ and tr F(t) in a similar manner. One then has Proposition : ([4], Proposition 9.1) For all integers M, one may choose the integer N in (3.14) so that there is a constant C such that (A.8) sup ∣θij∖t)∣ 0≤j<2n ∣<trF(t), θ(t)>∣ ≤ C hM for all θ ∈ C^o((-T,T)'). This proposition tells us the error made in replacing U(t) by E(t) in the calculation of Nj1*θj1 and ^h*θh' Proof of Theorem 2: From (3.9) and (A.1) (N'h*θh) (λ) = ⅛ <tr U(t), p(t) eitvh> and so, from (A.8), (A.9) (Nh*0h)(X) ~ 2⅛.(tr E(t), ∕>(t) eitλ7h) ≤ ChM —1 sup O≤j≤2n (p(t)e ~itλ7h) t The sup above is bounded since p is fixed and λ is in a bounded interval. By choosing N large enough in (3.16), we may then use E(t) in place of U(t) in our calculation of N∣1*θh, because the error committed in this replacement is at a higher order in h than we are interested in. -34- Now ⅛<tr E(t), p(t) eitλ7h (A.10) = (2τrh)~n-1 J∫Jp(t)Aw∖t,x,r7)eκsct>x^-χ∙77+λtyhdtdxdr7 which, as just noted, is a good approximation to (3.18). The phase in this integral is stationary at those points (t,x,77) satisfying (3.20). From (A.2) and (A.4), the graph of the classical Hamiltonian flow for ∣t∣ ≤ T is So (3.20b and c) tell us that t must be a period of a periodic classical trajectory. The condition (3.24) tells us that the only such period we need to be concerned with in evaluating (A.10) is t = 0. The point (x,T)) must also be on the λ energy surface (3.21), and so we have a manifold of points of stationary phase given in (3.23). At a given point (O,x,77) ∈ W^, we have, for the phase (3.19) 2!7∙9xV -3xV -27? -3xV 0 0 -27? 0 0 The subspace perpendicular to at (O,x,77) is span{(1,0,0), (0,3xV, 27?)}. On this subspace φ" is the 2 × 2 matrix Φ" ! and -35- det φ" ± = - ( ∣3xVj2 + 4∣∏∣2)2 . This may be zero only if 7? = 0 and 3xV = 0, and so, by (3.21), X would have to be critical for V. Thus, φ"is nondegenerate for all (t,x,77) ∈ and X ∈ [Xq—£q,Xq+£q]. Also, both jdet φ', ∣ ∣-^2 anj JJ≠z,j_~^|| are bounded uniformly for (t,x,77) ∈ Wχ and X ∈ [ λθ—εθ,Χθ+εθ]. To evaluate (A.10), it is convenient to work in coordinates new (t,τ,ω) ∈ FXRXR^π'∖ where t is the same as before, τ = ∣η∣2 -)- V(x) and ω denotes the remaining (2n—1) coordinates. partition of unity over (It may be necessary to make a U W> and choose the coordinates ω in λ∈ [Xo-^o,λc+^cl each partition, in· which case we proceed in each partition as in the following, and then sum over the partition.) In these coordinates, Wχ={(t,τ,ω) : t = 0,τ = X, all ω}, and on Wχ 3,,S *"± = -1 0 The integral (A.10) may then be evaluated using the results of Appendix B (2πh)^n'1∫∫Jp(t)Aw∖t,τ,ω) e lφit>τ-^A) 3(x,77) dt dτ dω 3(τ,ω) = (2ττh)^n J{p(0)Aiw(0,X,ω) 3(x,77) 3(τ,ω) = λ} dω + 0(h2'n), where the term 0(h2^π) is uniform for X ∈ ^Q~e0,^0~e0'' ^^o see 0(h^^n) contribution is zero, apply the result from Appendix B and use (3.7), (3.15) and note that the resulting integrand is odd in z?. Finally, for the leading term, we have -36- 3(x,77) du) j 3(τ,ω) τ=λ = vol(W∖ This completes the proof of Theorem 2. Proof of Theorem 3: We start with (3.26) and use our partition of unity satisfying (3.34). Thus, (A.11) (Nh*θh)(λ) = ∫ 1 X1(' j -oo i + rλ-e∕2 JJ -oo + fλ jχ→ x< We now show that the first integral is 0(h+oo), evaluate the second by the method of stationary phase, and use Theorem 2 to evaluate the third integral. Concerning the first integral in (A.11), we note <W<τ> = ⅛ ∑ ■ j=l where we have used (3.4) and (3.6). Since p E J(Rn), for each integer M ≥ 0 there is a constant such that ≤ CMhM ∣ej(h) -τ∣-m for τ ∈ (—00,—1]. From (3.3) we have ΓΠ,' ≤ Γ ,'k'∏πj' ;~1 , ej(h)" C'h^ and so -37- ∣ej(h)- ri“M ≤ ∣-t∣~mz2 ∣ej(h) + II-M/2 (σ)Μ/2π h-M∕2,τj-M∕2 j-M∕2n. Thus, for each integer M ≥ 2n + 1 there is a constant C∣v∣ such that oo —M∕2ni lτ,-M∕2 l(Nh*8h)(T)l ≤ ⅛ =MhM/2( Σ j j-1 We finally conclude that, for each integer M ≥ 2n + 1, there is a C'm such that ft, Xι^XN∙hwh)(τ,at ≤ ¾h M/2—1 and so see that the first term in (A.11) is 0(h +∞ ' ). Note at this point, for use in the proof of Theorem 4, that the same argument gives (A.12) ∣rt ⅛ τ)(λ-τ)(Njl*θh)(τ)dt c'ttf ‰,M∕2—1 ≤ cm h For the second integral of (A.11), we use (A.9), and up to negligibly high order in h, we write (2τrh) ~n ~1 ∫∫∫∫ χ2(τ)p(t) N∣1 (t,x,7)el(S(t’X,T7)~x 'η +Tt)/h dt dτ dx dη. First we show that the integration over distant regions of (x,∏) contributes 0(h+oo). By (A.5) there is an R > 0 so that for λ(x,77) ≥ R and τ ∈ [—2,λ-∣j. ∣3tS-f-^rj ≥ i Choose some K∣, ∕C2 ∈ C00(F) so that supp K∣ C ΓR,+oo) supp K,~t c (-∞,2R] A1 +K2 = 1. -38- Since —ih(3tS+τ)^13tei^τzh = repeated integrations by parts gives (A.13) (2τrh) "n ~1 ∫∫∫∫ χ2(τ) p(t) A(hN) (t,x,r7) K1(λ(x,∏) ) e^^'r)/hdt dτ dx dr? = (2πh) ~n ~1(ih)k ∫J∫∫ {[3t0tS +τ) ^1]k p(t) A^ (t,x,77)} • χ2(τ) κ1(λ(x,T7) ) ei0(t’X)7?’T)/h dt dτ dx dr?. is bounded via (A.6) and (A.7) The term in {■··} times (the integration over t then gives a convergent integral in (x,r?) as in the argument showing that Eg has kernel in ∕(FnXEn)), and so (A.13) is 0(hk π ^) for any integer k ≥ 0. Thus, we need only compute (A.14) (2τrh) ~ n “1 ∫J∫∫ X2(τ)p(t) AjjV)(t,x,r?) κ√λ(x,r7) ) e1^t∙x>^>r^hdt dτ dx d7?i The phase φ is stationary in (t,τ) at the point (3.29), at which 277∙a>V 1 1 0 = 3(t,τ)* Thus, del ¾τ, φ ___1/9 4 )] π ^ ≡≡ 1 and 3-t;T) φ -1 is bounded uniformly over the (x,r?) region of integration. Evaluating (A.14) by the method of stationary phase then gives the result (2πh) (A.15) n ∫∫ χ2(∣r7∣2 -f-V(x)) dx dr? + 0(h1 π). Using Theorem 2 for the third term in (A.11) gives the result (A.16) (2ττh)~nJ^_ζ χ3(τ) vol(Wτ)dτ -r 0(h2-π) = (2τrh) n J∫ ∣r7∣2+V(χ)≤λ χ3(∣r7∣2 + V(x)) dx dr? + 0(h2 n). -39- Combining (A.15) and (A.16) proves Theorem 3. Proof of Theorem 4: Start with (3.31) and proceed similarly to the proof of Theorem 3, except that we evaluate to the first two orders in h. (*h*θh)(λ) = f~L xi(T)(x-T^Nh*eh)(T) dτ + i-277z2 ⅛τ×λ-τ)^h*θh)(τ) dτ + Jx-€ T)(Nh*0h)(T) dτ' The first term on the right side above is 0(h -†-OC ). The second term is (2τrhΓn ∫∫ {χ2(p7∣2+V(x)) (λ-∣τ7∣2-V(χ))}∣^^^dxdr? (A.17) + 0(h2^π). The 0(h^ n) term is zero as in the proof of Theorem 2. Using Theorem 2 on the third term gives the result (2τh)-n j£_£ X3(τ) (λ-τ) vol(Wτ)dτ + 0(h2~n) (A.1S) = (2τrh)~n ∫∫ X3(∣Π∣2+V(x))(λ-∣η∣2-V(x)) dx dη + 0(h2^π). ∣η∣2+V(x)≤λ Add (A.17) and (A.18) to complete the proof of Theorem 4. Proof of Proposition 5: We shall show Lemma 5,1 : Under the hypothesis of Theorem 2, there exists a constant Co > 0 such that (A.19) ∣Nh(λ+τh)-Nh(λ)∣ ≤ C0h1-πT-∣τ∣,'n^1 for λ ∈ (λ0 -yΛ0 + y), h ∈ (0,lj and τ ∈ R . -40- Giveπ this result, consider for λ ∈ [λθ — y,λθ + yj. (Nh*θh) (λ) - Nh(λ) = ∫^c(Nh(σ)-Nh(λ))θh(λ-σ)dσ . Make the change of variable σ = λ+τh and use (A.19) to get (A.20) ∣(Nh*θh)(λ) - Nh(X)∣ ≤ Cθh1~n∫^o(l+∣τ∣)n + 1θh(-τh)hdτ = Cθh1~n i ∫^o(l+∣τ∣)n + 1p(τ)dτ, where we have used (3.6) and p ≥ 0. Since p ∈ J(R), equation (A.20) gives (3.35). From the definition of Φj1 in (2.8), we have *h(λ+τh) - Ψh(λ) = τhNh(λ) + ∫Jj+ τh (Nh(∕z) - Nh(λ) ) dμ . Make the change of variable μ, = λ + hυ and use (A.19) to get (A.21) ∣ψh(λ + τh) - Ψh(λ) - τhNh(λ)∣ ≤ Cθh2~π 1 Now, using (3.6 and 7) and —ip'(0) = ÷l- ro° 27Γ j-oc (A.22) cl+J∑Γ^ ~-1. τρ(τ)dτ, (Ψh*θh)(λ) - Ψh(λ) + ihp'(0)Nh(λ) = ⅛ O⅝(λ + τh) - *h(λ) - rhNh(λ))p(τ)dτ . Equation (3.36) follows from (A.21), (A.22) and the fact that p'(0) = 0. Given Lemma 5.1, this completes the proof of Proposition 5. Proof of Lemma 5.1 : Suppose τ ≥ 0. We have three cases. Case 1: Suppose hτ ≤ 5θ, where 5θ is from (3.8). Then Nh(λ+τh) - Nh(λ) - ∩+τh NJ1(h)d∕2 ≤ ci∫3(^) N^(∕x)dA 2πC1h(NJ1*θh)(λ), -41- where C∣ > 0 so that g- ≤ p(σ) for σ ∈ [—Theorem 2 then says that there is a C-, >0 such that ∣Nh(λ+τh) - Nh(λ)∣ ≤ C2h1-n for λ ∈ (λθ-e0’^0+€(P and hτ ≤ 50, Case 2: Suppose hτ ≥ y . Then, using (3.3), ∣Nh(λ+τh) - Nh(λ)∣ ≤ ∣Nh(λ + τh) + Nh(λ)∣ ≤ C'h~π((λ + τh)n+λn). Since h~^1 ≤ ∣T, h ≤ 1, and λ ∈ (λθ— Εθ,λθ+εθ), there is a constant C3 > 0 such that ∣Nh(λ+τh) - Nh(λ)∣ ≤ C3h1 “"(I + ∣τt)n + 1 Case 3: Suppose 5θ ≤ hτ ≤ Choose the integer k so that k5θ ≤ hτ < (k -∣-1)5q ∙ Then ∣Nh(λ + τh) - Nh(λ)∣ ≤ h(λ+τh) - Nh(λ+kSθ)∣ k-1 + Σ ∣N∣j(λ-∣-(f+l)5θ) — Nj1(λ+£άθ)|. £■=0 For λ∈ (λθ — y,λθ +y), each term above may be bound by Case 1, because λ -f- ££q ∈ (Xq—εθ,λθ+εθ) for each £ = 0,l,∙ ∙∙,k. So in this case, ∣Nh(λ + τh) - Nh(λ)∣ ≤ kC2h1~π ≤ ⅛ C2h1^π for λ∈ (λθ-^,λθ+^). Putting the three cases together then gives that there is a Cθ >0 such -42- that (A.19) is true for τ ≥ 0. The proof is similar for τ ≤ 0. Proof of Theorem 6: If λθ is not critical for V, then there is an eθ > 0 so that no value in [λθ-£θι^θ+€θ] is critical for V. Lemma 6.1 : The point {0} is isolated in . 0+e , 0.. '.^0*t0,Λ J Given Lemma 6.1, we may find a p ∈ Cg°(R) satisfying (3.8) such that supp <, ∩ ⅛cι.t,jιχ0+ts, - {0} . Then Theorems 3 and 4 and Proposition 5 prove Theorem 6. Proof a£ Lemma 6.1 : We must show, for classics periodic trajectories of 0 and energies in (^q~^qAq+€q)< that there is a tθ >0 such period t that ∣t∣ > t∩U . This would prove that {0} is isolated in L,lΛx o-Cc,,ΛQTtoJ . , „ -,. The trajectories obey (A.2) and (A.3). Define, for fixed (y,∏) ∈ R 2n with energy |7?|2 + V(y) ∈ (∖θ — eθ ,λθ -+-eθ) D(t) = (x-y)∙∈ - (∈ — 7?) ∙3xV(x). Then D(t) = 0 for a periodic trajectory starting at (y√7) and having period t. Since g ≡ D'(t) = 2∣∈∣2 + ∣3xV∣2 - (x—y) ∙3xV+2(ξ-7?) ∙3xxV ∙ζ we see that D'(0) = 2∣77∣2 + ∣3χV(y)∣2 ≥ d' > 0, for some d' > 0 depending only on λθ and eθ (otherwise V has a critical point in contradicting our hypotheses). Furthermore, from (A.3) -e0 ’ and ÷e0 ’ (3.2) the trajectory is bounded, and from (3.1) the quantities [3xV∣ and j)3xxV∣J are bounded on the trajectory. Using these facts and (A.2), we obtain the lower bound Dz(t) ≥ D'(0) - d∣t -43- for some d > 0 depending only on λθ and eθ . Since D(t) = β D'(t) dt ≥ D'(0)∣t∣ - ⅜ d∣t∣2, we see that the zeroes of D(t) for which t ≠ 0 obey ∣t∣ ≥ tθ This completes the proof of Lemma 6.1, and hence Theorem 6. 2D'(0) >0, -44- APPENDIX B The Method of Stationary Phase In this appendix we review several results on the method of stationary phase which we need in order to evaluate the small h asymptotics of integrals of the form f ∖ i≠(×)∕h d.τn x, Jr a(x)e ,r, ,X (B.l) where a ∈ Cq0(Fπi), φ ∈ Co°(Fm) and φ is real valued. Suppose for the moment that ∣7≠∣ > 0 on supp a. If supp a is sufficiently small that φ may be used as a coordinate in a new coordinate system {yu!xp∙∙∙4∏,J}^j over supp a (take y∣(x) = ≠(x)), then we have f .(x)ei*ω'h dmx _ ∫ f(y1)eiy∙zh dy1 - f(⅛), where f(y1) = ∫ a(x(y)) 3(x) dy2 3(y) that the integral (1) is 0(h +oo ) as h dyγn. Since f ∈ Cθ (1R), we conclude 0. We can always reduce to this case by use of some partition of unity over supp a . A result of Hörmander [9], which is proven by clever integrations by parts, gives uniform bounds. Theorem 8 ([9, Theorem 7.7,1]): Let K C Rm be compact, X an open neighborhood of K, and k a nonnegative integer. If a ∈ C∩o(K), φ ∈ Coo(X) and φ real valued, then .λ eiφ(x)∕h ∫ a (x) **,χ^" ij cjx for h ≤chk 23 s*p∣d αa∣ ∣V<⅛llαl 2k ∣a∣<k > 0. Here C (which depends on φ, but not a) is bounded when -45- 23 ∣Dα≠∣ stays bounded, ∣α∣≤k + l This theorem tells us that (B.l) vanishes rapidly as h — 0 when the phase φ has no stationary points in supp a . By a stationary point of φ, we mean that V0 = 0 at that point. Now consider a simple case ' wherein the phase φ has a stationary point. Let the dimension m = 1, put ≠(x) = iχ2, and let a ∈ Cq°(F). The phase φ is stationary at x =0. Since the τ ∕,^j ihf2∕2 (2πih)ιz2 e , we may compute (B.l) Fourier transform of (B.2) ∫a(x)eiχ2z2hdx = ^∫a(∈)(27rih)iz2e-lh^2/2 df . Using 23 (it)j∕d! ≤ ∣t∣k∕k! for t ∈ F in (B.2) gives e*x z2^1 is j-0 (B.3) ∫a (x)eiχ2z2hdx - (2τrih)iz2 jj c-^⅛^ a(2j) (0) 1=0 ι~ll^2iΞl- dc ≤ i ∫ I â (se)(2τrih)ιz21 vx> ≤Chk+* 2k+1 « ≤ ∙ ,t lDJa|L2) j=0 In the right-hand side of (B.3) the following bound was used ∫∣S(z)∣ ∣∈∣2kd∈ = ∫∣S(z) I I∈i2k( 1 ÷i∈∣) (l + ∣e∣)'1 dÇ ≤ (∫∣so 1 + ∣e∣)2k+1 ∣2d∈)ιz2 (∫( 1 ÷∣ei)^2d∈)ιz2 ≤ σ 2k+1 I« H 2k÷1 I. ]Γ ∣∣a(cej ∣l∑ = cz ∑ j=0 j=0 ∙ „ lbJa|lL2. So we see from (B.3) that (B.2) is not 0(h-r°°) as h — 0. This is due to the presence of the point of stationary phase in the integral at x = 0. As (B.3) -46- shows, we can compute the small h asymptotic behavior to any order we desire, and the result depends on the value of a and its derivatives at the location of the point of stationary phase. This last example can extended be to higher dimensions in a straightforward manner, yielding: Let A be an m X m real symmetric non­ ([9, Theorem 7.7.3]): Theorem 9 degenerate matrix. Then for every integer k > 0 and integer s > m∕2, |ja(x)ei<X,Ax)/2h dx - Sk(h) V ≤ ckdA-1Ih)m72+k lalâk+S IdM∙, L for h > 0 and where Sk(h) = (2τh)mz2 idet ΑΓ1/2 e17rs*nAZ4 ' Σ j=o ^d>a^1d^ a(0)· j' Finally, we consider the more general case where ≠(x) is stationary, but not degenerate, at a point x = xθ . Since φ(x) = ≠(x0) + i<(x-x0),≠"(x0)(x-x0)> + gχ0(x), where Bχ0(×) = 0(×) — √>(×q) — ⅛<(×~×θ),≠"(x0)(x-xθ)>. (B.4) We then write for (1), e iφ(x0)∕h rj . . ig×0(x)∕hγ i<(x-x0),≠"(χ0)(χ-x,-,)>∕h , J < a(x)e je dx and apply Theorem 9 (with the quantity in braces replacing a there). One obtains Theorem 10 ([9, Theorem 7.7.5]): Let K C Rπι be compact, X an open -47- neighborhood of K and k a positive integer. If a ∈ Cq0(K), φ ∈ Coo(X), ≠'(×0) = 0, det φ"(x∩) l0 ≠ 0 and ≠z(x) ≠ 0 for x ∈ K ∖ {x∩}, 0' then (B.5) ∫a(x) i≠(x)∕h dx - (2τrh)mz2∣detφz'(x0)∣∙ ∣-1∕2 iz2e iπsgn≠zz(x0)∕4 k-1 Σ lj, j=o ≤ Ch m72+k ∑ sup ∣D aa∣. sup∣D ∣α∣≤2k Here C stays bounded when ∑ sup∣Dαφ∣ and i≠zz(x∩)* 1fl stay bounded ∣α∣≤3k + l π u 11 in X. The Lja are given by Lja = Y" (ij 2υ χz ! υ ! )^1 < D, φ" (xθf1 D)υ (g‰ a) (xθ). υ-μ=j 2υ≥3∕x There is no problem in extending this to the case where φ and hence x 0 may depend on some parameters y [9, Theorem 7.7.6], as long as ∣α∣≤3k+l sup∣Dχ φ(x,y)∣ and ∣Φχχ(x0(y),y )^M stay bounded for x ’ π ∈ X and all y of interest. For reference, we list Lθa and L∣a explicitly; (B.6) Lθa = a(xθ) L∣a = — i(i <D,BD)a(xθ) + I<Da, BD <D, BD> φ >(xθ) + ∣(<D,BD>2≠)(xθ) a(xθ) + ^<D<D,BD>φ,BD<D,BD>φ>(x0) a(xθ)], where the matrix B = φz'(x0)^^. In the proofs of Theorems 2, 3 and 4 we use the results of (B.5) and (B.6). -48- REFERENCES [1] M. Bander, Corrections to the Thomas-Fermi model of the atom, Annals of Physics 144 (1982) 1-14. [2] R. Beals, A general calculus of pseudodifferential operators, Duke Math. Ji 42 (1975) 1-42. [3] M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rep. Prog. Phys. 35 (1972) 315-397. [4] J. Cħazarain, Spectre d’un Hamiltonien quantique et mécanique classique, Comm. P.D.E. 5 (1980) 595-644. [5] P.A.M. Dirac, Note on exchange phenomena in the Thomas atom, Proc. Camb. Phil. Soc. 26 (1930) 376-385. [6] L.L. Foldy, A note on atomic binding energies, Phys. Rev. 83 (1951) 397-399. [7] B. Helffer and D. Robert, Comportment semi-classique du spectre des Hamiltoniens quantiques elliptiques, Ann. Inst. Fourier (Grenoble) 31 (1981) 169-223. [8] B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Func. Anal. 53 (1983) 246-268. [9] L. Hörmander, The Analysis of Linear Partial Differential Operators 1, Springer-Verlag, New York, 1983, Chapter VII. [10] L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Springer-Verlag, New York, 1985, Chapter XXIX [11] V. Ya. Ivrii, On quasiclassical spectral asymptotics for the -49- Schrödinger operator on manifolds with boundary for h-pseudodifferential operators acting in bundles, Soviet Math. Dokl. 26 (1982) 285-289. [12] E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Physics 53 (1981) 603-641. [13] E.H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules, and solids, Adv. Math. 23 (1977) 22-116. [14] E.A. Milne, The total energy of binding of a heavy atom, Proc. Camb. Phil. Soc. 23 (1927) 794-799. [15] V. Petkov and D. Robert, Asymptotique semi-classique du spectre d’hamiltoniens quantiques et trajectoires classiques périodiques, Comm. P.D.E. 10 (1985) 365-390. [16] M. Reed and B. Simon, Methods of Modem Mathematical Physics 11: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975. [17] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978. [18] J. Schwinger, Thomas-Fermi model: The leading correction, Phys. Rev. A 22 (1980) 1827-1832. [19] J.M.C. Scott, The binding energy of the Thomas-Fermi atom, Phil. Mag. 43 (1952) 859-867. [20] R. Shakeshaft, L. Spruch and J.B. Mann, Truncated expansion of the ground-state energy of a neutral atom in powers Z'1/3: coefficients of the leading terms, J. Phys. B 14 (1981) L121L125. [21] R. Shakeshaft and L. Spruch, Remarks on the existence and accuracy -50- of the Z^ιz3 expansion of the nonrelativistιc ground-state energy of a neutral atom, Phys. Rev. A 23 (1981) 2118-2126. [22J H. Siedentop and R. Weikard, On the leading energy correction for the statistical model of the atom: Interacting case, (to appear). [23] W. Thirring, A lower bound with best possible constant for Coulomb Hamiltonians, Ccmm. Math. Phys. 79 (1981) 1-7. [24] L.H. Thomas, The calculation of atomic fields, Proc. Camb. Phil. Soc. 23 (1926) 542-548. [25] L.A. Young, Binding energy of light atoms, Phys. Rev. 34 (1929) 12261227.