Abstract
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.
Highlights
Given a positive, possibly unbounded, operator P with a compact resolvent, acting on a separable infinite-dimensional Hilbert space H one can define the associated heat operator e−t P for t > 0
The existence of an asymptotic expansion of Tr e−t P was proven for P being a classical positive elliptic pseudodifferential operator of order m ∈ N
The Mellin transform has a direct application to the study of the asymptotic expansions of heat traces
Summary
Possibly unbounded, operator P with a compact resolvent, acting on a separable infinite-dimensional Hilbert space H one can define the associated heat operator e−t P for t > 0. The existence of an asymptotic expansion of Tr e−t P was proven for P being a classical positive elliptic pseudodifferential operator of order m ∈ N (see [39] and references therein). We discuss the limitations of the method and compare its usefulness with the Tauberian theorems commonly used in this domain
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
