  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.
