Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

Abstract

The spectral function Θ(t)=∑i=1∞exp(−tλj), where {λj}j=1∞ are the eigenvalues of the negative Laplace-Beltrami operator −Δ, is studied for a compact Riemannian manifold Ω of dimension “k” with a smooth boundary ∂Ω, where a finite number of piecewise impedance boundary conditions (∂∂ni+γi)u=0 on the parts ∂Ωi(i=1,…,m) of the boundary ∂Ω can be considered, such that ∂Ω=∪i=1m∂Ωi, and γi(i=1,…,m) are assumed to be smooth functions which are not strictly positive.