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A nonself‐adjoint singular Sturm–Liouville problem with a spectral parameter in the boundary condition
Author(s) -
Allahverdiev Bilender P.
Publication year - 2005
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.200310269
Subject(s) - sturm–liouville theory , mathematics , boundary value problem , dissipative operator , dissipative system , eigenvalues and eigenvectors , operator (biology) , mathematical analysis , self adjoint operator , dilation (metric space) , spectral theory , hilbert space , combinatorics , physics , quantum mechanics , biochemistry , chemistry , repressor , transcription factor , gene
We consider nonself‐adjoint singular Sturm–Liouville boundary‐value problems in the limit‐circle case with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary‐value problem. We construct a self‐adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations that make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and specify its characteristic function in terms of solutions of the corresponding Sturm–Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and the Sturm–Liouville boundary‐value problem. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)