Smeared heat-kernel coefficients on the ball and generalized cone

TL;DR

This paper investigates smeared heat-kernel coefficients on generalized cones and balls, providing a method to compute these coefficients and explicitly calculating the $A_{5/2}$ term, which aids understanding of spectral geometry on manifolds with boundary.

Contribution

It introduces a scheme for calculating heat-kernel coefficients on manifolds with boundary, including explicit computation of the $A_{5/2}$ coefficient, using smeared zeta functions and conformal transformations.

Findings

Derived the form of heat-kernel coefficients on the ball and generalized cone.

Provided a method to compute the $A_{5/2}$ coefficient explicitly.

Restricted the form of heat-kernel coefficients on smooth manifolds with boundary.

Abstract

We consider smeared zeta functions and heat-kernel coefficients on the bounded, generalized cone in arbitrary dimensions. The specific case of a ball is analysed in detail and used to restrict the form of the heat-kernel coefficients $A_n$ on smooth manifolds with boundary. Supplemented by conformal transformation techniques, it is used to provide an effective scheme for the calculation of the $A_n$. As an application, the complete $A_{5/2}$ coefficient is given.