Higher dimensional inverse problem of the wave equation for a general multi-connected bounded domain with a finite number of smooth mixed boundary conditions

The spectral function (\&mu;) over cap (t) = Sigma(J=1)(infinity) exp(-itmu(J)(1/2)) where \{mu(J)\}(J=1)(infinity) are the eigenvalues of the negative Laplacian -Delta(3) = -Sigma(v=1)(3) (partial derivative/partial derivativex(v))(2) in the (x(1),x(2),x(3))-space. is studied for small \textbackslash{}t\textbackslash{} for a variety of bounded domains, where -infinity < t < infinity and i = root-1. The dependences of (t) on the connectivity of bounded domains and the boundary conditions are analysed. Particular attention is given to a general multi-connected bounded domain 0 in R-3 together with a finite number of smooth Dirichlet, Neumann and Robin boundary conditions on the smooth boundaries partial derivativeOmega(J) (J = 1,.... m) of the domain Omega. Some geometrical properties of Omega (e.g.. the volume, the surface area, the mean curvature and the Gaussian curvature of Omega) are determined from the, asymptotic expansions of (\&mu;) over cap (t) for small \textbackslash{}t\textbackslash{}. (C) 2002 Elsevier Science Inc. All rights reserved.