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Exotic Expansions and Pathological Properties of ζ-Functions on Conic Manifolds

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Abstract

We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ-functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these ζ-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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Authors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, TX, 76798, USA

    Klaus Kirsten

  2. Department of Mathematics, Binghamton University, Binghamton, NY, 13902, USA

    Paul Loya

  3. School of Mathematics, Korea Institute for Advanced Study, 207-43, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722, South Korea

    Jinsung Park

Authors
  1. Klaus Kirsten
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  2. Paul Loya
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  3. Jinsung Park
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Corresponding author

Correspondence to Klaus Kirsten.

Additional information

Appendix by Boris Vertman.

Universität Bonn, Mathematiches Institut, Beringstr. 6, 53115 Bonn, Germany

e-mail: vertman@math.uni-bonn.de

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Kirsten, K., Loya, P. & Park, J. Exotic Expansions and Pathological Properties of ζ-Functions on Conic Manifolds. J Geom Anal 18, 835–888 (2008). https://doi.org/10.1007/s12220-008-9028-9

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