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

H. Falomir·P. A. G. Pisani
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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 $A=-\partial_x^2+(ν^2-1/4)/x^2+V(x)$, where $0<ν<1$ 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 $t^ν$. 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.

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H. Falomir, P. A. G. Pisani. 2005-12-16. Krein's Formula And Heat-Kernel Expansion For Some Differential Operators With A Regular Singularity. https://doi.org/10.1088/0305-4470%2F39%2F21%2Fs25

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