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