Heat Trace and Functional Determinant in One Dimension

TL;DR

This paper investigates spectral properties of Laplace-type operators on the circle, deriving approximations for heat traces and determinants, and exploring their relation to integrable hierarchies, with exact results in the infinite radius limit.

Contribution

It introduces new recursive formulas and differential equations for heat kernel coefficients, enabling precise approximations of heat traces and determinants on the circle.

Findings

Derived a differential equation for the heat kernel diagonal

Developed recursive formulas for heat kernel coefficients

Obtained exact formulas in the infinite radius limit

Abstract

We study the spectral properties of the Laplace type operator on the circle. We discuss various approximations for the heat trace, the zeta function and the zeta-regularized determinant. We obtain a differential equation for the heat kernel diagonal and a recursive system for the diagonal heat kernel coefficients, which enables us to find closed approximate formulas for the heat trace and the functional determinant which become exact in the limit of infinite radius. The relation to the generalized KdV hierarchy is discussed as well.