  To motivate our discussion, we consider a 1+1 dimensional scalar field
interacting with a static Coulomb-type background, so that the spectrum of
quantum fluctuations is given by a second-order differential operator on a
single coordinate r with a singular coefficient proportional to 1/r. We find
that the spectral functions of this operator present an interesting behavior:
the zeta function has multiple poles in the complex plane; accordingly,
logarithms of the proper time appear in the heat-trace expansion. As a
consequence, the zeta function does not provide a finite regularization of the
effective action. This work extends similar results previously derived in the
context of conical singularities.
