A semi-classical trace formula for Schrödinger operators in the case of a critical energy level
type: Item
instance of: scholarly article
title: A semi-classical trace formula for Schrödinger operators in the case of a critical energy level
zbMATH Open document ID: 0891.35130
author: David Khuat-Duy
published in: Journal of Functional Analysis
publication date: 1998-07-19T00:00:00Z
full work available at URL: https://basepub.dauphine.fr/handle/123456789/6195
review text: The author considers a semiclassical Schrödinger operator with smooth potential \(V\) and discrete spectrum, and studies the smoothed spectral density: \[ \gamma(E,h)= \sum_{E-\lambda\leq E_k\leq E+\lambda} \varphi\Biggl({E_k- E\over h}\Biggr), \] where \(\lambda>0\) is fixed, \(h\) is the semiclassical parameter, the \(E_k\)s are the eigenvalues of the operator, \(\varphi\) is a Schwartz function with compactly supported Fourier transform, and \(E\) is a critical energy level, that is the energy surface associated to \(E\) is assumed to contain the critical set \(\theta= \{(x,0); \nabla V(x)=0\}\). Under additional geometrical assumptions, the author shows the existence of a full asymptotic expansion of \(\gamma(E,h)\) as \(h\) goes to \(0\), in powers of \(\sqrt h\) and \(\ln h\). This asymptotics includes the contributions of the periodic orbits of the linearized flow on \(\theta\).
reviewed by: André Martinez
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 81Q20
zbMATH DE Number: 1077101
zbMATH Keywords: microlocal analysis
zbMATH Keywords: periodic orbits
zbMATH Keywords: semiclassical asymptotics
MaRDI profile type: Publication
OpenAlex ID: W2046392742
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DOI: 10.1006/JFAN.1996.3035
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description: scientific article; zbMATH DE number 1077101