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

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.

Submitted to Journal of Physics A, special issue corresponding to QFEXT'05, The Seventh Workshop on Quantum Field Theory under the Influence of External Conditions; IEEC, CSIC and University of Barcelona. Barcelona, Spain, 5-9 September 2005 (9 pages.)