Exotic expansions and pathological properties of $ζ$-functions on conic manifolds

Details Exotic expansions and pathological properties of $ζ$-functions on conic manifolds Exotic expansions and pathological properties of $ζ$-functions on conic manifolds

2005-11-07

arxiv.org/abs/math/0511185v2

Mathematics

Spectral Theory

44 pages, 5 figures, AMS-LaTeX

Scientific paper

We give a complete classification and present new exotic phenomena of the meromorphic structure of $\zeta$-functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these $\zeta$-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

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