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

TL;DR

This paper extends Weyl's law to inhomogeneous media in multiple dimensions, providing explicit asymptotic formulas for eigenvalues of the Laplacian, validated by numerical comparisons.

Contribution

It introduces a perturbative and non-perturbative extension of Weyl's law for inhomogeneous bodies, generalizing classical results to more complex media.

Findings

Derived explicit asymptotic formulas for eigenvalues in inhomogeneous media.

Validated analytical formulas with high-precision numerical results.

Extended Weyl's law using perturbation and Weyl's conjecture.

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}.