An inverse problem for a general annular-bounded domain in R-2 with mixed boundary conditions and its physical applications

This paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R-2. The asymptotic expansion of the trace of the heat kernel theta(t) = Sigma(D=1)(infinity) exp (-tmu(v)), where \{mu(v)\}(v=1)(infinity) are the eigenvalues of the negative Laplacian -Delta2 = -Sigma(k=1)(2) (partial derivative/partial derivativex(k))(2) in the (x(1), x(2))-plane, is studied for short-time t of a general annular-bounded domain Omega in R-2 together with its smooth inner boundary partial derivativeOmega(1) and its smooth outer boundary partial derivativeOmega(2), where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Gamma(i)(i=1,..., m) of partial derivativeOmega(1) and on the piecewise smooth components Gamma(i)(i=m+1,..., n) of partial derivativeOmega(2) such that partial derivativeOmega(1) = boolean ORi=1m Gamma(i) and partial derivativeOmega(2) = boolean ORi=m+1n Gamma(i), are considered. In this paper, one may extract information on the geometry of Omega by analyzing the asymptotic expansions of theta(t) for short-time t. Some applications of theta(t) for an ideal gas enclosed in Omega are given. Thermodynamic quantities of an ideal gas enclosed in Omega are examined. We use an asymptotic expansion for high temperatures to obtain the partition function of an ideal gas showing the leading corrections to the internal energy due to a finite container. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it. (C) 2002 Elsevier Science Inc. All rights reserved.