On the spectrum of a class of one-dimensional Schrödinger type operators with generalized potential
type: Item
instance of: scholarly article
title: On the spectrum of a class of one-dimensional Schrödinger type operators with generalized potential
zbMATH Open document ID: 0926.47027
DOI: 10.1007/BF02358984
author: G. A. Ajgunov
published in: Mathematical Notes
publication date: 1999-11-16T00:00:00Z
review text: Let \(Q(x)=p(x)\) for \(x\leq 0\) and \(q(x)\) for \(x>0\), where \(p\) is a continuous real valued function such that \(\int_{-\infty}^0| p(x)| dx<\infty\), and \(q\) is a continuous real valued periodic function of period \(1\) on the half-line. Consider the set of \(L_2(-\infty,\infty)\) functions \(D(H)=\{g: g\in W_2^2({\mathbb{R}}| {\mathbb{N}})\cap W_2^1({\mathbb{R}}), g'(n+0)-g'(n-0)=\varepsilon g(n)\}\) for \(n\in{\mathbb{N}}\). Here \(\varepsilon\) is a real number. The author characterizes the eigenvalues and the continuous spectrum of the Schrödinger operator \(Hf=-f''(x)+Q(x)f(x)\) for \(f\in D(H)\). The continuous spectrum is bounded from below. The potential \(Q(x,\varepsilon)=p(x)\) for \(x\leq 0\) and \(q(x)+\varepsilon\sum_{n=1}^\infty\delta(x-n)\) for \(x>0\) is used. Here \(\delta(x)\) is the Dirac messure.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1257145
zbMATH Keywords: singular potential
zbMATH Keywords: point interaction
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: boundary value problem
zbMATH Keywords: spectrum structure
zbMATH Keywords: Dirac mass
reviewed by: Rémi Vaillancourt
MaRDI profile type: Publication
cites work: Q3810956
full work available at URL: https://doi.org/10.1007/bf02358984
OpenAlex ID: W2074276445
description: scientific article; zbMATH DE number 1257145