Spectral functions of non essentially selfadjoint operators

Details Spectral functions of non essentially selfadjoint operators Spectral functions of non essentially selfadjoint operators

2012-02-28

arxiv.org/abs/1202.6332v2

Physics

Mathematical Physics

57 pages, 1 fugure. Submitted to the Journal of Physics A as a contribution to the special volume in honor of Stuart Dowker's

Scientific paper

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.

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