  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.
