Semiclassical spectral invariants for Schrödinger operators
type: Item
instance of: scholarly article
title: Semiclassical spectral invariants for Schrödinger operators
zbMATH Open document ID: 1273.35198
published in: Journal of Differential Geometry
publication date: 2012-09-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.0919
full work available at URL: https://projecteuclid.org/euclid.jdg/1343133702
review text: The paper is essentially devoted to the study of the semiclassical spectral measure of the semiclassical Schrödinger operator \(S_{h}= -(h^{2}/2)\Delta + V(x)\), where \(V\) is a smooth nonnegative potential and \(V^{-1}([0,a])\) is compact for some \(a>0\). A semiclassical spectral theorem for elliptic operators (Theorem 4.1 from [\textit{B. Helffer} and \textit{D. Robert}, J. Funct. Anal. 53, 246--268 (1983; Zbl 0524.35103)]) is used to compute the first two terms of the asymptotic expansion for \((2\pi h)^{n} \sum {f(\lambda_{i}(h))}\), \(\lambda_{i}(h)\), \(1\leq i\leq N(h)\), being the eigenvalues of \(S_{h}\) in \([0,a)\). The algorithm for computing higher order terms is also given. The method is next applied to prove inverse spectral results for single well potentials \(V\), for the perturbed Schrödinger operator \(P_{h} = -(h^{2}/2)\Delta + V(x) + h^{2}V_{1}(x)\) and for the magnetic Schrödinger operator in dimension 2 with radially symmetric electric potential and radially symmetric magnetic field. In the last two sections of the paper the relation between the spectral measure of \(S_{h}\) and the Birkhoff canonical form in one dimension is studied.
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35R20
Mathematics Subject Classification ID: 45Q05
Mathematics Subject Classification ID: 81Q20
zbMATH DE Number: 6081390
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: semiclassical spectral asymptotics
zbMATH Keywords: inverse spectral theorems
zbMATH Keywords: Birkhoff canonical form
author: Victor Guillemin
author: Zuoqin Wang
MaRDI profile type: Publication
Wikidata QID: Q115170155
OpenAlex ID: W2096156693
arXiv ID: 0905.0919
DOI: 10.4310/JDG/1343133702
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description: scientific article; zbMATH DE number 6081390