Asymptotic and Exact Expansions of Heat Traces

Open Access


We study heat traces associated with positive unbounded operators with compact inverses. With the help of the inverse Mellin transform we derive necessary conditions for the existence of a short time asymptotic expansion. The conditions are formulated in terms of the meromorphic extension of the associated spectral zeta-functions and proven to be verified for a large class of operators. We also address the problem of convergence of the obtained asymptotic expansions. General results are illustrated with a number of explicit examples.


Heat traces Asymptotic expansions Spectral theory Zeta-functions General Dirichlet series 

Mathematics Subject Classification (2010)

Primary: 11M36 Secondary: 30B50 35K08 41A60 47B15 81Q10 


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Authors and Affiliations

  1. 1.Faculty of Physics, Astronomy and Applied Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland

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