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Singular Sturm‐Liouville Problems: The Friedrichs Extension and Comparison of Eigenvalues
Author(s) -
Niessen H.D.,
Zettl A.
Publication year - 1992
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-64.3.545
Subject(s) - sturm–liouville theory , mathematics , eigenvalues and eigenvectors , extension (predicate logic) , mathematical analysis , boundary value problem , transformation (genetics) , boundary (topology) , limit (mathematics) , principal (computer security) , singular solution , pure mathematics , biochemistry , chemistry , physics , quantum mechanics , computer science , programming language , gene , operating system
A new characterization of singular self‐adjoint boundary conditions for Sturm‐Liouville problems is given. These are an exact parallel of the regular case. They are given explicitly in terms of principal and non‐principal solutions. The special nature of the Friedrichs extension is clearly apparent and highlighted. Inequalities among the eigenvalues of different boundary conditions, separated and coupled, are obtained. Most of all we want to stress the method of proof. It is based on a very elementary transformation which transforms any singular non‐oscillatory limit‐circle endpoint into a regular one.