Effective computation of traces, determinants, and $\zeta$-functions for Sturm-Liouville operators

DOI10.1016/J.JFA.2018.02.009arXiv1712.00928MaRDI QIDQ6294766

Klaus Kirsten, Fritz Gesztesy

Publication date: 4 December 2017

Abstract: The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, $zeta$-functions, and $zeta$-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and $zeta$-function regularized determinants. Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm-Liouville operators on compact intervals with various (separated and coupled) boundary conditions, and Schr"odinger operators on a half-line, are provided and further illustrated with an array of examples.

Mathematics Subject Classification ID

Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) (34L40) Linear operators belonging to operator ideals (nuclear, (p)-summing, in the Schatten-von Neumann classes, etc.) (47B10) Spectrum, resolvent (47A10) Green's functions for ordinary differential equations (34B27) Integral operators (47G10)

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