Semiclassical asymptotics for the spectral function of long-range Schrödinger operators
type: Item
instance of: scholarly article
title: Semiclassical asymptotics for the spectral function of long-range Schrödinger operators
zbMATH Open document ID: 0692.35069
DOI: 10.1016/0022-1236(89)90121-3
author: Christian Gérard
author: André Martinez
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: The authors study the spectral function for the semiclassical Schrödinger operator, \(P=-h^ 2\Delta +V(x)\) on \({\mathbb{R}}^ n\), when V is a long range potential, satisfying some general hypothesis so that complex scaling method may be applied. After fixing the general hypothesis (on the nature of V), the authors study the case of an energy level which does not contain trapped trajectories and obtain expansions, in h, of the spectral function of P, locally uniformly. Next, authors study a case with trapped trajectories, the ``potential well inside an island'', which gives rise to the well-known shape resonances. Finally, the authors obtain pointwise expansions which roughly speaking are the sum of a non-trapping term corresponding to the potential obtained by filling up the well, and a term of Breit-Wigner type corresponding to the resonances.
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 81Q15
zbMATH DE Number: 4132712
zbMATH Keywords: semiclassical asymptotics
zbMATH Keywords: espace function
zbMATH Keywords: spectral function
zbMATH Keywords: semiclassical Schrödinger operator
zbMATH Keywords: long range potential
zbMATH Keywords: complex scaling
zbMATH Keywords: trapped trajectories
zbMATH Keywords: shape resonances
zbMATH Keywords: Breit-Wigner
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90121-3
OpenAlex ID: W1977399553
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description: scientific article; zbMATH DE number 4132712