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A Sturm–Liouville problem depending rationally on the eigenvalue parameter
Author(s) -
Jonas Peter,
Trunk Carsten
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510573
Subject(s) - sturm–liouville theory , eigenvalues and eigenvectors , mathematics , operator (biology) , space (punctuation) , spectrum (functional analysis) , mathematical analysis , linear map , pure mathematics , mathematical physics , boundary value problem , physics , quantum mechanics , chemistry , computer science , biochemistry , repressor , transcription factor , gene , operating system
We consider the Sturm–Liouville problem (1.1) and (1.2) with a potential depending rationally on the eigenvalue parameter. With these equations a λ ‐linear eigenvalue problem is associated in such a way that L 2 ‐solutions of (1.1), (1.2) correspond to eigenvectors of a linear operator. If the functions q and u are real and satisfy some additional conditions, the corresponding linear operator is a definitizable self‐adjoint operator in some Krein space. Moreover we consider the problem (1.1) and (1.3) on the positive half‐axis. Here we use results on the absense of positive eigenvalues for Sturm–Liouville operators to exclude critical points of the associated definitizable operator. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)