Can one hear the density of a drum? Weyl's law for inhomogeneous media

Abstract

We generalize Weyl's law to inhomogeneous bodies in $d$ dimensions. Using a perturbation scheme recently obtained by us in Ref. \cite{Amore09}, we have derived an explicit formula, which describes the asymptotic behavior of the eigenvalues of the negative laplacian on a closed $d$-dimensional cubic domain, either with Dirichlet or Neumann boundary conditions. For homogeneous bodies, the leading term in our formula reduces to the standard expression for Weyl's law. We have also used Weyl's conjecture to obtain a non-perturbative extension of our formula and we have compared our analytical results with the precise numerical results obtained using the Conformal Collocation Method of Refs. \cite{Amore08,Amore09}.