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Article
Discreteness of the Spectrum of Second-Order Differential Operators and Associated Embedding Theorems
Journal of Differential Equations (2002)
  • Branko Ćurgus, Western Washington University
  • Thomas T. Read, Western Washington University
Abstract

Necessary and sufficient conditions and also simple sufficient conditions are given for the self-adjoint operators associated with the second-order linear differential expression τ(y) 1/w(-(py')'+qy) on [a,b) to have discrete spectrum. Here the coefficients of τ are non-negative and satisfy minimal smoothness conditions. These results follow from compact embedding theorems from a weighted one-dimensional Sobolev space with norm∫ab(p∣f′∣r+q∣f′∣r))1/r into a weighted Banach space with norm(∫abw∣f′∣s)1/s .

Keywords
  • Sturm–Liouville operator,
  • Discrete spectrum,
  • Compact embedding
Disciplines
Publication Date
2002
Publisher Statement
Copyright © 2002 Elsevier B.V. DOI: 10.1006/jdeq.2001.4152
Citation Information
Branko Ćurgus and Thomas T. Read. "Discreteness of the Spectrum of Second-Order Differential Operators and Associated Embedding Theorems" Journal of Differential Equations Vol. 184 Iss. 2 (2002)
Available at: http://works.bepress.com/branko_curgus/36/