The wave equation approach for solving inverse eigenvalue problems of a multi-connected region in R-3 with Robin conditions

The asymptotic expansion for \textbackslash{}t\textbackslash{} of the trace of the wave kernel (\&mu;) over cap (t) = Sigma(nu=1)(infinity) exp(-itmu(upsilon)(1/2)), where \{mu(upsilon)\}(infinity) (upsilon=1) are the eigenvalues of the negative Laplacian -del(2) = - Sigma(beta = 1)(3),(partial derivative / partial derivativex(beta))(2) in the (x(1), x(2), x(3))-space where i = root-1 and -infinity < t < infinity, is studied for a general multiply-connected bounded domain Omega in R-3 surrounded by simply connected bounded domains Q(j) with smooth bounding surfaces S-j (j = 1,. . . , n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components S-i{*} (i = 1 + k(j-1), . . . k(j)) of the bounding surfaces S-j is considered, such that S-j = U-i=1+kj-1(kj) S-i({*}), where k(0) = 0. The basic problem is to extract information on the geometry Omega by using the wave equation approach from complete knowledge of its eigenvalues. Some geometric quantities of Omega (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion of (\&mu;) over cap (t) for small \textbackslash{}t\textbackslash{}. (C) 2003 Elsevier Inc. All rights reserved.