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Determinants of Regular Singular Sturm ‐ Liouville Operators
Author(s) -
Lesch Matthias
Publication year - 1998
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.19981940110
Subject(s) - mathematics , operator (biology) , bounded function , image (mathematics) , regular singular point , meromorphic function , mathematical analysis , pure mathematics , function (biology) , boundary (topology) , ordinary differential equation , differential equation , biochemistry , chemistry , repressor , artificial intelligence , evolutionary biology , biology , computer science , transcription factor , gene
Abstract We consider a regular singular Sturm‐Liouville operatoron the line segment (0,1]. We impose certain boundary conditions such that we obtain a semi‐bounded self‐adjoint operator. It is known (cf. Theorem 1.1 below) that the ζ‐function of this operatorhas a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to [RS] the ζ ‐ regularized determinant of L is defined byIn this paper we are going to express this determinant in terms of the solutions of the homogeneous differential equation L y = 0 generalizing earlier work of S. Levit and U. Smilansky [LS], T. Dreyfus and H. Dym [DD], and D. Burghelea, L. Friedlander and T. Kappeler [BFK1, BFK2). More precisely we prove the formulaHere ϕ ψ is a certain fundamental system of solutions for the homogeneous equation L y = 0, W (ϕ ψ), denotes their Wronski determinant, and v 0 , v 1 are numbers related to the characteristic roots of the regular singular points 0, 1.