  One of the many problems to which J.S. Dowker devoted his attention is the
effect of a conical singularity in the base manifold on the behavior of the
quantum fields. In particular, he studied the small-$t$ asymptotic expansion of
the heat-kernel trace on a cone and its effects on physical quantities, as the
Casimir energy. In this article we review some peculiar results found in the
last decade, regarding the appearance of non-standard powers of $t$, and even
negative integer powers of $\log{t}$, in this asymptotic expansion for the
selfadjoint extensions of some symmetric operators with singular coefficients.
Similarly, we show that the $\zeta$-function associated to these selfadjoint
extensions presents an unusual analytic structure.
