On the number of negative eigenvalues of a Schrödinger operator with point interactions
type: Item
instance of: scholarly article
title: On the number of negative eigenvalues of a Schrödinger operator with point interactions
zbMATH Open document ID: 1173.34359
author: Osamu Ogurisu
published in: Letters in Mathematical Physics
publication date: 2009-04-20T00:00:00Z
review text: Let \(X=(x_1,\dots,x_n)\in \mathbb R^n\) and \(\alpha=(\alpha_1,\dots,\alpha_n)\in \mathbb R^n\). The one-dimensional Schrödinger operator \(L_{X,\alpha}\) is defined on \(\mathbb R\) by the expression \(-d^2/dx^2+\sum_{i=1}^n\alpha_i\delta(x-x_i)\), where \(\delta\) is the Dirac \(\delta\)-function. The state function \(\psi\) belongs to the Sobolev space \(W_2^2(R\setminus X)\), satisfying in the points of \(X\) (`points of interaction') conjugation conditions \(\psi(x_i+0)=\psi(x_i-0)\), \(\psi'(x_i+0)-\psi'(x_i-0)=\alpha_i\psi(x_i)\). It is known that the operator \(L_{X,\alpha}\) is selfadjoint and that its spectrum consists of the positive semiaxis and not more than \(n\) simple negative eigenvalues. \textit{S. Albeverio} and \textit{L. Nizhnik} proved in [Lett. Math. Phys. 65, No.~1, 27--35 (2003; Zbl 1056.34093)] that in the case \(n=1\) a negative eigenvalue exists only if \(\alpha_1<0\). The present authors consider the case when \(n>1\) and the number \(m\) of negative intensities \(\alpha_i\) is also greater than~1. In this situation, they obtain sufficient conditions for the operator \(L_{X,\alpha}\) to have at least \(m\) negative eigenvalues.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 5544980
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: point interactions
zbMATH Keywords: negative eigenvalues
MaRDI profile type: Publication
full work available at URL: s11005-008-0258-3
OpenAlex ID: W2043230918
cites work: Q4455605
cites work: On the number of negative eigenvalues of a one-dimensional Schrödinger operator with point interactions
cites work: Q3994511
cites work: Q4692748
cites work: Q5526485
cites work: Q3955947
DOI: 10.1007/S11005-008-0258-3
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description: scientific article; zbMATH DE number 5544980