Functional determinants for general Sturm-Liouville problems

TL;DR

The paper addresses functional determinants for general Sturm-Liouville problems with arbitrary linear boundary conditions, including operators with negative eigenvalues and zero modes. It develops a generalized zeta-function framework that reduces second-order problems to first-order systems and expresses determinant relations entirely in terms of boundary data and homogeneous solutions evaluated at the boundary. Two main contributions are provided: a nonzero-mode determinant-quotient formula and a zero-mode-extracted version, both applicable to single-component and multi-component (systems) of differential operators. These results hold across broad classes of weights $P(x)$ and boundary conditions, with potential applications in quantum fluctuations and transition-rate problems in physics.

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.

Figures (2)

• Figure 1: Contour $\gamma$ in the complex plane.
• Figure 2: Contour $\gamma$ in the complex plane.