On the resolvent and spectral functions of a second order differential operator with a regular singularity

We consider the resolvent of a second order differential operator with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents unusual powers of $lambda$ which depend on the singularity. The consequences for the pole structure of the $zeta$-function, and the small-$t$ asymptotic expansion of the heat-kernel, are also discussed.