Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-12T19:09:21.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral properties of singular Sturm—Liouville operators with indefinite weight sgn x

Published online by Cambridge University Press:  26 May 2009

Illya Karabash  and
Carsten Trunk
Illya Karabash
Affiliation:
Department of Partial Differential Equations, Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg str. 74, Donetsk 83114, Ukraine ([email protected]; [email protected])
Carsten Trunk
Affiliation:
Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a singular Sturm—Liouville expression with the indefinite weight sgn x. There is a self-adjoint operator in some Krein space associated naturally with this expression. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d2/dx2 + q) with non-real spectrum accumulating to a real point. The results obtained are applied to several classes of Sturm—Liouville operators.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 3 , June 2009 , pp. 483 - 503
Copyright
Copyright © Royal Society of Edinburgh 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)