Gelfand-Yaglom formula for functional determinants in higher dimensions

TL;DR

This paper generalizes the Gelfand-Yaglom formula to higher-dimensional differential operators, providing a new method to compute functional determinants in multiple dimensions, with applications to Laplace operators.

Contribution

The work extends the Gelfand-Yaglom formula from one dimension to higher dimensions, offering a novel approach for calculating determinants of multidimensional differential operators.

Findings

Derived a generalized Gelfand-Yaglom formula for higher dimensions

Applied the formula to 2D discrete Laplace operators

Obtained asymptotic expressions for determinants

Abstract

The Gelfand-Yaglom formula relates functional determinants of the one-dimensional second order differential operators to the solutions of the corresponding initial value problem. In this work we generalise the Gelfand-Yaglom method by considering discrete and continuum partial second order differential operators in higher dimensions. To illustrate our main result we apply the generalised formula to the two-dimensional massive and massless discrete Laplace operators and calculate asymptotic expressions for their determinants.