Kreins Formula And Heat-Kernel Expansion For Some Differential Operators With A Regular Singularity

We get a generalization of Kreins formula -which relates the resolvents of different selfadjoint extensions of a differential operator with regular coefficients- to the non-regular case $A=-partial_x^2+( u^2-1/4)/x^2+V(x)$, where $0< u<1$ and $V(x)$ is an analytic function of $xinmathbb{R}^+$ 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 $t^ u$. 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.