Krein’s formula and one-dimensional multiple-well. (English) Zbl 0562.47002

Combes, J. M.; Duclos, P.; Seiler, R.

J. Funct. Anal. 52, 257-301 (1983).

The article is the first of a series concerning two problems for one- dimensional Schrödinger operators and their classical limits (as Planck’s constant \(\hslash \to 0)\). The first is the behaviour of discrete eigenvalues in the classical limit for multiple-well potentials; the second is the so-called shape resonance problem. This paper is mainly concerned with the first problem. It is shown that all discrete eigenvalues possess asymptotic expansions in powers of \(\hslash\) when \(\hslash \to 0\), and a formula for the coefficients is given. The multiple-well problem is considered by using Dirichlet (Neumann) boundary conditions to separate the wells and then removing the boundary conditions by a finite rank perturbation (Krein’s formula). Also use is made of the exponential decay of eigenvectors. These methods are developed in this article.

Reviewer: J.R.L.Webb

MSC:

47A10	Spectrum, resolvent
35J10	Schrödinger operator, Schrödinger equation
81Q15	Perturbation theories for operators and differential equations in quantum theory

Keywords:

Dirichlet-Neumann boundary conditions; one-dimensional Schrödinger operators; Planck’s constant; behaviour of discrete eigenvalues in the classical limit for multiple-well potentials; shape resonance problem; finite rank perturbation; Krein’s formula; decay of eigenvectors

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