The spectral function (μ) over cap (t) = Sigma(J=1)(infinity) exp(-itmu(J)(1/2)) where {muJ}(J=1)(infinity) are the eigenvalues of the negative Laplacian -Delta(2) = -Sigma(v=1)(2) (partial derivative\partial derivativex(v))(2) in R-2 is studied for small \t\ for a variety of domains, where -infinity < t < infinity and i = root-1. The dependencies of (μ) over cap (t) on the connectivity of domains and the boundary conditions are analyzed. Particular attention is given to a general multi-connected bounded domain in R-2 together with Dirichlet, Neumann and Robin boundary conditions on the boundaries partial derivativeOmega (J = 1,..., m) of the domain Omega. Some geometrical properties of Omega (e.g., the area of Omega, the total lengths of the boundaries partial derivativeOmega(J), the curvatures of partial derivativeOmega(J), the number of holes of Omega, etc.) are determined from the asymptotic expansions of (μ) over cap (t) for small \t\. (C) 2002 Elsevier Inc. All rights reserved.
