The exotic heat-trace asymptotics of a regular-singular operator revisited

TL;DR

This paper revisits the complex heat-trace asymptotics of regular-singular operators, clarifying their properties and connecting previous results through the general heat kernel construction, highlighting the non-polyhomogeneous nature of the heat kernel.

Contribution

It demonstrates how existing results on heat-trace asymptotics for regular-singular operators follow from Mooers' heat kernel construction, addressing a gap in understanding of the singular structure.

Findings

Heat-trace asymptotics exhibit exotic properties.

Results can be derived from Mooers' heat kernel construction.

The heat kernel is non-polyhomogeneous in this context.

Abstract

We discuss the exotic properties of the heat-trace asymptotics for a regular-singular operator with general boundary conditions at the singular end, as observed by Falomir, Muschietti, Pisani and Seeley as well as by Kirsten, Loya and Park. We explain how their results alternatively follow from the general heat kernel construction by Mooers, a natural question that has not been addressed yet, as the latter work did not elaborate explicitly on the singular structure of the heat trace expansion beyond the statement of non-polyhomogeneity of the heat kernel.