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A semi-classical trace formula for Schrödinger operators. (English) Zbl 0729.35093
Consider the Schrödinger operator \(S(\hslash)=-\hslash^ 2\Delta +V\) on \({\mathbb{R}}^ n\), where \(V\in C^{\infty}({\mathbb{R}}^ n)\), \(V>0\). If \(\bar V_{\infty}=\lim_{| x| \to \infty} \inf V(x)\), the intersection Spec S(\(\hslash)\cap (-\infty,\underline V_{\infty})\) consists of eigenvalues \(\lambda_ 1(\hslash)\leq \lambda_ 2(\hslash)\leq...\). In this extensive paper, the authors analyse the distribution of the \(\lambda_ j(\hslash)\) around a fixed energy level E as \(\hslash \to 0\), generalizing and complementing the results of V. Guillemin and A. Uribe [Invent. Math. 96, No.2, 385-423 (1989; Zbl 0686.58040)] on compact manifolds, on circular symmetry and the trace formula. In particular, the authors relate this asymptotic distribution to the classical dynamics of a particle with energy E in a potential V.
The principal aim of the paper is to extend the results of Guillemin and Uribe to Schrödinger operators on \({\mathbb{R}}^ n\), without introducing extraneous assumptions on the behavior of V at infinity.
If \(S_{\hslash}=-\hslash \Delta +V\) on \({\mathbb{R}}^ n\), with V smooth, and \(0<E^ 2<\liminf_{| x| \to \infty} V(x)\) the spectrum of \(S_{\hslash}\) near \(E^ 2\) consists of finitely many eigenvalues \(\lambda_ j(\hslash)\). The authors study the asymptotic distribution of these eigenvalues about \(E^ 2\) as \(\hslash \to 0\) and make a detailed analysis of the case when the flow on \(B_ E=\{| \xi |^ 2+V(x)=E^ 2\}\) is periodic.
Reviewer: D.M.Bors (Iaşi)

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35J10 Schrödinger operator, Schrödinger equation
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