  In this article we analyze the resolvent, the heat kernel and the spectral
zeta function of the operator $-d^2/dr^2 - 1/(4r^2)$ over the finite interval.
The structural properties of these spectral functions depend strongly on the
chosen self-adjoint realization of the operator, a choice being made necessary
because of the singular potential present. Only for the Friedrichs realization
standard properties are reproduced, for all other realizations highly
nonstandard properties are observed. In particular, for $k\in \N$ we find terms
like $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel.
Furthermore, the zeta function has $s=0$ as a logarithmic branch point.
