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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
##### Keywords:
Schrödinger operator; trace formula
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##### References:
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