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 for the small-t asymptotic expansion of the heat kernel, are also discussed. (C) 2004 American Institute of Physics.