Krein's Formula And Heat-Kernel Expansion For Some Differential Operators With A Regular Singularity

TL;DR

This paper generalizes Krein's formula to differential operators with regular singularities, revealing a unique heat-kernel expansion with non-integer powers related to boundary conditions breaking scale invariance.

Contribution

It extends Krein's formula to operators with regular singularities and analyzes the resulting non-standard heat-kernel asymptotics.

Findings

Heat-kernel trace admits a non-standard small-t expansion with fractional powers.

Boundary conditions breaking scale invariance lead to integer powers of t^ν.

The generalized Krein's formula connects resolvents of different selfadjoint extensions.

Abstract

We get a generalization of Krein's formula -which relates the resolvents of different selfadjoint extensions of a differential operator with regular coefficients- to the non-regular case $A=-\partial_x^2+(\nu^2-1/4)/x^2+V(x)$, where $0<\nu<1$ and $V(x)$ is an analytic function of $x\in\mathbb{R}^+$ bounded from below. We show that the trace of the heat-kernel $e^{-tA}$ admits a non-standard small-t asymptotic expansion which contains, in general, integer powers of $t^\nu$. In particular, these powers are present for those selfadjoint extensions of $A$ which are characterized by boundary conditions that break the local formal scale invariance at the singularity.