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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)

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
Full Text: DOI
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