Survival probability (heat content) and the lowest eigenvalue of Dirichlet Laplacian

Details Survival probability (heat content) and the lowest eigenvalue of Dirichlet Laplacian Survival probability (heat content) and the lowest eigenvalue of Dirichlet Laplacian

2011-05-13

arxiv.org/abs/1105.2626v1

Int. J. Mod. Phys. B 25, 1993 (2011)

Mathematics

Spectral Theory

Accepted in IJMPB

Scientific paper

10.1142/S0217979211101004

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Pad\'e interpolation between the short-time and the long-time behavior of the survival probability, i.e. between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.

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