Effective computation of traces, determinants, and $\zeta$-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, $\zeta$-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.