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

TL;DR

This paper introduces a unified method for efficiently computing traces, determinants, and $ta$-functions for Sturm-Liouville and Schrf6dinger operators, with practical applications and examples.

Contribution

It develops a comprehensive approach connecting Fredholm and $ta$-function determinants, applicable to various Sturm-Liouville and Schrf6dinger operators.

Findings

Unified formalism for trace and determinant computation

Application to general Sturm-Liouville operators with boundary conditions

Illustrative examples demonstrating effectiveness

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.