Functional determinants by contour integration methods

TL;DR

This paper develops a simple, contour-integral framework for calculating ratios of functional determinants $rac{ m det L_1}{ m det L_2}$ of second-order differential operators, starting from Dirichlet boundary conditions and extending to general boundary conditions via a first-order reformulation. The method expresses the ratio in terms of solutions to the associated homogeneous equations and the chosen boundary data, yielding the canonical Dirichlet result $rac{ m det L_1}{ m det L_2}=rac{y_1(1)}{y_2(1)}$ and its generalizations, including cases with zero modes extracted. For zero modes, the authors derive a simple closed form, $rac{ m det' L_1}{ m det L_2}=-rac{raket{y_1|y_1}}{y'_1(1)^*y_2(1)}$ under appropriate normalization, by modifying the contour to omit the pole at $k=0$. The framework is then extended to general boundary conditions and to systems of differential operators, with a unified expression $rac{ m det L_1}{ m det L_2}=rac{ ext{det}(M+N Y_1(1))}{ ext{det}(M+N Y_2(1))}$ and its zero-mode counterpart, enabling practical evaluation in applications such as path integrals and transition-rate problems.

Abstract

We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible, the general idea is first illustrated on the simplest case: a second order differential operator with Dirichlet boundary conditions. The method is applicable to more general situations, and we discuss the way in which the formalism has to be developed to cover these cases. In particular, we also show that simple and elegant formulae exist for the physically important case of determinants where zero modes exist, but have been excluded.