On the essential spectrum of Schrödinger operators with spherically symmetric potentials
type: Item
instance of: scholarly article
title: On the essential spectrum of Schrödinger operators with spherically symmetric potentials
zbMATH Open document ID: 0629.35028
DOI: 10.1007/BF01457359
author: Rainer Hempel
author: Andreas M. Hinz
author: Hubert Kalf
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164245
review text: This paper discusses how the essential spectrum, denoted by \(\sigma_ e\), of a higher-dimensional Schrödinger operator \(-\Delta +V(x)\) in \(L_ 2({\mathbb{R}}^{\nu})\), \(\nu\geq 2\), is connected with that of a one- dimensional Schrödinger operator \(-d^ 2/dr^ 2+q(r)\) in \(L_ 2({\mathbb{R}})\), when V behaves like q on a strip in rectangular coordinates or in the radial direction in spherical coordinates. Theorem 1: If V is a real-valued function in \(L_{2,loc}({\mathbb{R}}^{\nu})\) and q in \(L_{\infty,loc}({\mathbb{R}})\) such that \(-\Delta +V\) is essentially selfadjoint on \(C_ 0^{\infty}({\mathbb{R}}^{\nu})\) and \(-d^ 2/dr^ 2+q\) on \(C_ 0^{\infty}({\mathbb{R}})\) with closures T and t, respectively, and if V(x), \(x=(x_ 1,..,x_{\nu})\), behaves like \(q(x_ 1)\) for large \(| x_ 1|\) but does not change too much in \((x_ 2,...,x_{\nu})\), then \(\sigma_ e(T)\) includes the interval \([\lambda_ 0,\infty)\) whenever \(\lambda_ 0\in \sigma_ e(t).\) Theorem 2: If \(T=-\Delta +q(| \cdot |)\) and \(t=-d^ 2/dr^ 2+q\) are selfadjoint operators defined in the sense of quadratic forms, where q is symmetric, i.e. \(q(-r)=q(r)\), and \(q=q_+-q_-\) with \(q_{\pm}\) nonnegative, \(q_+\in L_{1,loc}({\mathbb{R}})\) and \(q_-\in L_{1,loc,unif}({\mathbb{R}})\), then \(\sigma_ e(T)=[\inf \sigma_ e(t),\infty).\) [See also the following review (Zbl 0629.35029).]
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 4023827
zbMATH Keywords: essential spectrum
zbMATH Keywords: higher-dimensional Schrödinger operator
zbMATH Keywords: essentially selfadjoint
reviewed by: Takashi Ichinose
MaRDI profile type: Publication
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OpenAlex ID: W2051172867
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description: scientific article; zbMATH DE number 4023827