Functional determinants for general Sturm-Liouville problems

TL;DR

This paper derives explicit formulas for functional determinants of Sturm-Liouville operators with general boundary conditions, including cases with zero modes, using zeta-function techniques.

Contribution

It extends existing formulas to more general Sturm-Liouville problems with arbitrary boundary conditions and zero modes, broadening their applicability.

Findings

Derived explicit formulas for functional determinants with general boundary conditions

Included cases with zero modes after mode extraction

Applicable to operators with negative eigenvalues

Abstract

Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm-Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses the properties of generalised zeta-functions. The general form of the final results are the same for the entire range of problems considered.