The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^2/dr^2 - 1/(4r^2)$

TL;DR

This paper investigates the spectral properties of a singular differential operator, revealing that different self-adjoint extensions lead to nonstandard behaviors in the resolvent, heat kernel, and zeta function, especially for the Friedrichs realization.

Contribution

It demonstrates how the spectral functions of the operator vary with self-adjoint extensions, highlighting unique asymptotic behaviors and branch points in the zeta function.

Findings

Standard properties are recovered only for the Friedrichs realization.

The heat kernel's small-time expansion includes terms like (log t)^{-k}.

The zeta function exhibits a logarithmic branch point at s=0.

Abstract

In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator $-d^2/dr^2 - 1/(4r^2)$ over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for $k\in \N$ we find terms like $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel. Furthermore, the zeta function has $s=0$ as a logarithmic branch point.