The wave equation approach for solving inverse eigenvalue problems of a multi-connected region in R3 with Robin conditions

Abstract

The asymptotic expansion for | t| of the trace of the wave kernel μ ̂ (t)=∑ ∞ υ=1 exp(− itμ 1/2 υ) , where { μ υ } υ=1 ∞ are the eigenvalues of the negative Laplacian −∇ 2=−∑ 3 β=1 (∂/∂ x β ) 2 in the ( x 1, x 2, x 3)-space where i= −1 and −∞< t<∞, is studied for a general multiply-connected bounded domain Ω in R 3 surrounded by simply connected bounded domains Ω 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 =∪ k j i=1+ k j−1 S i *, where k 0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from complete knowledge of its eigenvalues. Some geometric quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion of μ ̂ (t) for small | t|.