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Zeta Functions of Elliptic Cone Operators

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Partial Differential Equations and Spectral Theory

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 211))

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Abstract

This paper is an overview of aspects of the singularities of the zeta function, equivalently, of the small time asymptotics of the trace of the heat semigroup, of elliptic cone operators. It begins with a brief description of classical results for regular differential operators on smooth manifolds, and includes a concise introduction to the theory of cone differential operators. The later sections describe recent joint work of the author with J. Gil and T. Krainer on the existence of the resolvent of elliptic cone operators and the structure of its asymptotic behavior as the modulus of the spectral parameter tends to infinity within a sector in C on which natural ray conditions on the symbol of the operator are assumed. These ideas are illustrated with examples.

Mathematics Subject Classification (2000). Primary: 58J50, 35P05, Secondary: 47A10, 58J35.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Gerardo A. Mendoza

Authors
  1. Gerardo A. Mendoza
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Correspondence to Gerardo A. Mendoza .

Editor information

Editors and Affiliations

  1. FB Mathematik u. Informatik, Inst. Mathematik, TU Clausthal, Erzstr. 1, Clausthal-Zellerfeld, 38678, Germany

    Michael Demuth

  2. Inst. Mathematik, Abt. Analysis, Universität Potsdam, Potsdam, Brandenburg, Germany

    Bert-Wolfgang Schulze

  3. FB Mathematik, Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, Göttingen, 37073, Germany

    Ingo Witt

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Mendoza, G.A. (2011). Zeta Functions of Elliptic Cone Operators. In: Demuth, M., Schulze, BW., Witt, I. (eds) Partial Differential Equations and Spectral Theory. Operator Theory: Advances and Applications(), vol 211. Springer, Basel. https://doi.org/10.1007/978-3-0348-0024-2_5

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