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Effective computation of traces, determinants, and \(\zeta\)-functions for Sturm-Liouville operators. (English) Zbl 06988448

Summary: 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 bounded intervals with various (separated and coupled) boundary conditions, and Schrödinger operators on a half-line, are provided and further illustrated with an array of examples.

MSC:

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