Discrete Determinants and the Gel'fand-Yaglom formula

TL;DR

This paper discusses the Gel'fand-Yaglom formula for discrete second-order difference operators, developing a matrix approach, applying it to boundary conditions, and exploring eigenvalues, potentials, and continuum limits with novel insights.

Contribution

It introduces a two-by-two matrix method for the Gel'fand-Yaglom formula in discrete settings and extends it to various boundary conditions and potentials, offering new analytical tools.

Findings

Developed a matrix approach to the Gel'fand-Yaglom formula.

Computed Euler-Rayleigh sums of eigenvalues.

Extended analysis to delta potentials and continuum limits.

Abstract

I present a partly pedagogic discussion of the Gel'fand-Yaglom formula for the functional determinant of a one-dimensional, second order difference operator, in the simplest settings. The formula is a textbook one in discrete Sturm-Liouville theory and orthogonal polynomials. A two by two matrix approach is developed and applied to Robin boundary conditions. Euler-Rayleigh sums of eigenvalues are computed. A delta potential is introduced as a simple, non-trivial example and extended, in an appendix, to the general case. The continuum limit is considered in a non--rigorous way and a rough comparison with zeta regularised values is made. Vacuum energies are also considered in the free case. Chebyshev polynomials act as free propagators and their properties are developed using the two-matrix formulation, which has some advantages and appears to be novel. A trace formula, rather than a determinant one, is derived for the Gel'fand-Yaglom function.