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This IMA Volume in Mathematics and its Applications QUASICLASSICAL METHODS is based on the proceedings of a very successful one-week workshop with the same title, which was an integral part of the 1994-1995 IMA program on "Waves and Scattering." We would like to thank Jeffrey Rauch and Barry Simon for their excellent work as organizers of the meeting. We also take this opportunity to thank the National Science Foun­ dation (NSF), the Army Research Office (ARO) and the Office of Naval Research (ONR), whose financial support made the workshop possible. A vner Friedman Robert Gulliver v PREFACE There are a large number of problems where qualitative features of a partial differential equation in an appropriate regime are determined by the behavior of an associated ordinary differential equation. The example which gives the area its name is the limit of quantum mechanical Hamil­ tonians (Schrodinger operators) as Planck's constant h goes to zero, which is determined by the corresponding classical mechanical system. A sec­ ond example is linear wave equations with highly oscillatory initial data. The solutions are described by geometric optics whose centerpiece are rays which are solutions of ordinary differential equations analogous to the clas­ sical mechanics equations in the example above. Much recent work has concerned with understanding terms beyond the leading term determined by the quasi classical limit. Two examples of this involve Weyl asymptotics and the large-Z limit of atomic Hamiltonians, both areas of current research.



Tutorial Overviews

h-Pseudodifferential Operators and Applications: An Introduction

The aim of these lectures1 is to present the basic techniques in semiclassical analysis based on the use of h-pseudodifferential operators. One of our aims is to prepare to the comprehension of the recent results on the Scott’s conjecture. In particular, we hope it can help for the understanding of the recent proofs of the Scott’s conjecture for molecules by Ivrii and Sigal [107]. We present also some results turning around the Gutzwiller’s formula [69] and the Schnirelman’s theorem [141].
According to the short time we had, the following notes are sometimes sketchy and some arguments are oversimplified. This has the advantage to outline the basic ideas. We refer mainly to the book [135] or to [98] for precise and rigorous statements. We hope simply that this short presentation permits to see what can be done with this theory by entering rapidly in the applications.
Of course, a great part of this theory was developed initially in another context (see [98]) but it seems more efficient to work directly in the semiclassical context which has in particular the interest to remain quite near the physical intuition. We have also tried to give a rather extensive list of references but we are sure that we still forget important references.
Let us mention also that the most complete treatise on this subject is a project of book being written now by V. Ivrii [106] whose reading is probably rather difficult but which presents the most sophisticated developments of the theory. We heard also about different projects of books by Y. Colin de Verdi ere [32] and by Dimassi-Sjöstrand.
Bernard Helffer

Semi-Classical Methods with Emphasis on Coherent States

These notes cover and extend a review of four lectures whose aim was to present some results on semi-classical methods for the Schrödinger equation obtained by using coherent states. The goal was not to be by any mean exhaustive but rather to focus on applications susceptible to be related to some new -both numerical and experimental-recent results in quantum mechanics, with emphasis on the situations where the underlying classical system is chaotic.
T. Paul

Workshop Research Papers

Approximative Theories for Large Coulomb Systems

We review several approximations of mean field type to the quantum mechanics of large atoms and molecules and discuss their advantages and disadvantages.
Volker Bach

Semiclassical Analysis for the Schrödinger Operator with Magnetic Wells (After R. Montgomery, B. Helffer-A. Mohamed)

In this lecture1 we present some survey on the semiclassical analysis of the Schrödinger operator with magnetic fields with emphasis on the recent results by R.Montgomery [31] and extensions obtained in collaboration with A.Mohamed [14]. The main point is the analysis of the asymptotic behavior, in the semi-classical sense, of the ground state energy for the Schrödinger operator with a magnetic field. We consider the case when the locus of the minima of the intensity of the magnetic field is compact and our study is sharper when this locus is an hypersurface or a finite union of points.
Bernard Helffer

On the Asymptotic Distribution of Eigenvalues in Gaps

Virtually all results on eigenvalue asymptotics for differential operators have their roots in Weyl’s celebrated law for the distribution of the eigenvalues
$$ 0 < E_1 < E_2 \leqslant E_3 \leqslant \ldots ,E_k \to \infty {\text{ }}as{\text{ }}k \to \infty , $$
of the Dirichlet Laplacian -△ on an open, bounded domain Ω ⊂ R m : If N(λ) denotes the number of eigenvalues E k < λ, then
$$ N\left( \lambda \right) \sim c_d vol\left( \Omega \right)\lambda ^{m/2} ,\lambda \to \infty , $$
under mild regularity assumptions on the boundary δΩ here c m is a universal constant which depends only on the dimension m.
Rainer Hempel

Asymptotics of the Ground State Energy of Heavy Molecules in the Strong Magnetic Field

Multiparticle quantum theory is one of the main topics of modern mathematical physics, and one of the central questions in this theory is the problem of the high-density limit. There are different versions of this problem including the analysis of a heavy atom, and the analysis of a molecule consisting of heavy atoms. These two versions are the most popular and my project deals mainly with them.
The first step in the analysis is usually the Thomas-Fermi approximation, which leads to a non-linear partial differential system describing density and effective potential. This part of the theory is basically done.
However, justification of this approximation, error estimates and the obtaining of additional correction terms (scott and Dirac-schwinger) is a much more difficult matter requiring quite different techniques. Up to now the main tool has been variational methods of mathematical physics. After no less than 20 years of intensive investigations there remain major open problems, and even recently essential progress was obtained.
In some steps of the analysis there arise problems lying within the theory of semiclassical spectral asymptotics. This is a highly developed theory with the very strong machinery. However, problems specific for the multiparticle quantum theory have never been treated, and these problems have essential differences from standard problems of this theory. As a result these problems were treated either by variational methods as well (which led to non-accurate error estimate and the impossibility of recovering correction terms) or by separation of variables and investigation of ordinary differential equation by the WKB method (this approach has provided very precise error estimate but works only in the very special cases).
Only recently M.Sigal and me applied semiclassical spectral asymptotics methods to the multiparticle quantum theory problems and justified the scott correction term for the ground state energy for large molecules. Automatically this provided some progress in other problems as well.
Now I am starting a big project which I call “Multiparticle Quantum Theory and Semiclassical Spectral Asymptotics”. The objective of this project is to solve the class of problems in semiclassical spectral asymptotics that arise in multiparticle quantum theory, using the machinery already developed or developing new machinery as appropriate. In the process this will integrate this theory into the toolbox of multiparticle quantum theory. Moreover, it will provide an extension of the best results obtained by separation of variables and WKB to general problems. For example, it provides a justification of the Dirac-schwinger correction for molecules, and it should lead to much improved estimates of the minimal distance between nuclei, and of the maximal charge of negative ions. My main emphasis will be on semiclassical spectral asymptotics theory backed by semiclassical microlocal analysis which are domains where I came from to MQT. The first step will be to consider heavy atoms and molecules in a strong magnetic field, which seems to be the most challenging problem of this type.
Victor Ivrii

Local Trace Formulæ

Consider a Schrödinger operator H =-ħ 2 Δ+V(x) with V smooth, on ℝ n (in which case we assume V tends to infinity at infinity and therefore H has discrete spectrum) or on a compact Riemannian manifold, M.
T. Paul, A. Uribe

A Proof of the Strong Scott Conjecture

The strong Scott conjecture says that the electronic density of a big atom converges—after suitable rescaling—to the hydrogenic density
$$ \rho ^H \left( \tau \right): = q\sum\limits_{v,E_v \leqslant 0} {\left| {\psi _v \left( \tau \right)} \right|^2 } $$
$$\left( { - \Delta - \frac{1} {{\left| \tau \right|}}} \right)\psi _v = E_v \psi _v , $$
and q is the number of spin states per electron. This conjecture was recently proven by A. Iantchenko, E. H. Lieb, and the speaker. Here we give a partial result which is easy to present but caputeres already the essential idea of the full result. Finally, we discuss some related extensions.
Heinz Siedentop

Lieb-Thirring Inequalities for the Pauli Operator in Three Dimensions

Motion of a particle with spin in a magnetic field is described by the Pauli operator, that is by the operator
$$\begin{array}{*{20}{c}} {{\mathbb{P}_0} = {{\left( {\sum \cdot ( - i\nabla - a)} \right)}^2} = {{( - i\nabla - a)}^2}\mathbb{I}{\text{ - }}\sum \cdot {\text{B}},}&{\mathbb{I}{\text{ = }}\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)} \end{array}$$
acting in L 2(ℝ3) ⊕ L 2(ℝ3). Here a = (α1α2α3) is a vector-potential, B = (B 1, B 2, B 3) = rot a is the magnetic field and Σ is the vector of the 2 x 2 Pauli matrices σ1, σ2, σ3 (see[3]). As seen from (1.1), the operator \( \mathbb{D}_0 \) is non-negative. If one perturbs it by a real-valued function V (electric potential) decreasing at infinity, then the resulting operator may have some negative discrete spectrum. The main goal of the paper is to establish Lieb-Thirring type estimates for the momenta
$$ M_\gamma = \sum\limits_k {\left| {\Lambda _k } \right|} ^\gamma ,\gamma > 0, $$
of the negative eigenvalues Λ k of the operator \( \mathbb{P} = \mathbb{P}_0 + V\mathbb{I} \). Analogous question was studied in [16] for the Pauli operator acting on L 2(ℝ2) ⊕ L 2(ℝ2) and the present paper can be regarded as a continuation of [16]. It is well-known that without any magnetic field S γ, satisfies the following estimate 1:
$$ M_\gamma \leqslant C_\gamma \int {V - \left( x \right)^{\gamma + \frac{3} {2}} dx,} $$
which is usually referred to as the Lieb-Thirring inequality if γ > 0 and the Rosenblum-Lieb-Cwickel inequality if γ = 0. Using the diamagnetic inequality (see [1]) one can extend this estimate to the spinless operator (-i∇ - a)2 + V with a ≠ 0 as well.
Alexander V. Sobolev

Exact Anharmonic Quantization Condition (In One Dimension)

An exact version of the Bohr-Sommerfeld quantization scheme is constructed for any one-dimensional homogeneous anharmonic oscillator (having the potential q 2M ). It identifies the exact spectrum as the fixed point of a nonlinear transformation acting upon level sequences within a suitable domain. This mapping itself is explicitly given as a combination of a standard Bohr-Sommerfeld quantization step with a feedback operation from the resulting spectrum. An approximate linear theory suggests, and numerical tests confirm, that our mapping is contractive up to very high (possibly all) degrees of anharmonicity. The exact spectrum is then constructively specified as the attractor of semiclassically correct level sequences. (This type of approach ought to extend to general polynomial potentials.)
André Voros


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