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

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 [“Unusual poles of the ζ-functions for some regular singular differential operators,” J. Phys. A 36(39), 9991–10010 (2003)]10.1088/0305-4470/36/39/302 as well as by Kirsten, Loya, and Park [“The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2/dr2 − 1./(4r2),” J. Math. Phys. 47(4), 043506 (2006)]10.1063/1.2189194. We explain how their results alternatively follow from the general heat kernel construction by Mooers [“Heat kernel asymptotics on manifolds with conic singularities,” J. Anal. Math. 78, 1–36 (1999)]10.1007/BF02791127, 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.