Green function diagonal for a class of heat equations

TL;DR

This paper constructs the heat kernel diagonal as a generalized Zeta function, linking it to operator determinants and providing explicit expressions for finite-gap potentials, advancing regularization techniques in heat equations.

Contribution

It introduces a novel approach to express the heat kernel diagonal via generalized Zeta functions and derives explicit formulas for finite-gap potentials.

Findings

Heat kernel diagonal expressed as a generalized Zeta function.

Gradient at zero defines the determinant of a differential operator.

Explicit expressions obtained for finite-gap potential coefficients.

Abstract

A construction of the heat kernel diagonal is considered as element of generalized Zeta function, that, being meromorfic function, its gradient at the origin defines determinant of a differential operator in a technique for regularizing quadratic path integral. Some classes of explicit expression in the case of finite-gap potential coefficient of the heat equation are constructed.