Precise estimates for tunneling and eigenvalues near a potential barrier
type: Item
instance of: scholarly article
title: Precise estimates for tunneling and eigenvalues near a potential barrier
zbMATH Open document ID: 0668.34022
DOI: 10.1016/0022-0396(88)90153-2
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: The authors study the semiclassical Schrödinger operator \(P=-h^ 2(d^ 2/dx^ 2)+V(x)\) on \(L^ 2({\mathbb{R}})\) with a real analytic potential V presenting a barrier between two potential wells \((V(0)=V'(0)=0\), \(V''(0)<0\), \(\liminf_{| x| \to \infty}V(x)>0)\). This operator has a discrete spectrum near 0 (the potential barrier). Asymptotic expansions as h tends to 0 of the eigenvalues near 0 and of the difference between two successive eigenvalues (splitting) are given. In addition to known results new estimates of the coefficients in these expansions are established. In order to prove the results exact solutions of the eigenvalue equation are constructed along the lines of \textit{A. Voros} [Ann. Inst. H. Poincaré 29, 211-338 (1983; Zbl 0526.34046)] and J. Ecalle, and their WKB expansion (the ``exact'' WKB method) is used.
reviewed by: Joachim Weidmann
Mathematics Subject Classification ID: 34L99
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34E20
zbMATH DE Number: 4093926
zbMATH Keywords: Asymptotic expansions
zbMATH Keywords: eigenvalues
zbMATH Keywords: WKB expansion
zbMATH Keywords: WKB method
author: Christian Gérard
author: Alain Grigis
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90153-2
OpenAlex ID: W2074907694
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description: scientific article; zbMATH DE number 4093926