Spectral functions of non-essentially self-adjoint operators

One of the many problems to which 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 paper, 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 self-adjoint extensions of some symmetric operators with singular coefficients. Similarly, we show that the ζ-function associated with these self-adjoint extensions presents an unusual analytic structure. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. (paper)