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In this article we analyze the resolvent, the heat kernel and the spectralzeta 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 thechosen self-adjoint realization of the operator, a choice being made necessarybecause of the singular potential present. Only for the Friedrichs realizationstandard properties are reproduced, for all other realizations highlynonstandard properties are observed. In particular, for $k\in \N$ we find termslike $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel.Furthermore, the zeta function has $s=0$ as a logarithmic branch point.Comment: 28 pages, 4 figures, LaTe

The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2∕dr2−1∕(4r2)