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

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, $ζ$-functions, and $ζ$-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and $ζ$-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 Schrödinger operators on a half-line, are provided and further illustrated with an array of examples.

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