Estimations de l'effet tunnel pour le double puits. (Estmates of tunneling for the double wells)
type: Item
instance of: scholarly article
title: Estimations de l'effet tunnel pour le double puits. (Estmates of tunneling for the double wells)
zbMATH Open document ID: 0564.35028
author: André Martinez
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1987-01-01T00:00:00Z
review text: This paper is devoted to the Schrödinger operator \(P=-h^ 2\Delta +V\), (h small), in the case where V is a \(C^{\infty}\) function, symmetrical about a hyperplane of \({\mathbb{R}}^ n\), and admitting two non-degenerate minima (called wells). Using the WKB method and introducing a pseudo- differential calculus adapted to the WKB developments, we find estimates of the splitting for the energy levels which are O(h). The lower bound we obtain is of order \(h^{1/2}e^{-S_ 0/h}\) (where \(S_ 0\) is the Agmon distance between the wells), while the upper bound is of order \(h^{-a}e^{-S_ 0/h}\) where \(a\geq 0\) depends on the energy level we consider. We also find an upper bound involving the modified Agmon metric associated to \(V-E_ 0h\), where \(E_ 0h\) is the first term in the WKB development of the eigenvalue.
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 3899388
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: WKB method
zbMATH Keywords: pseudo-differential calculus
zbMATH Keywords: splitting for the energy levels
zbMATH Keywords: Agmon metric
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3899388