Exact sum rules for inhomogeneous drums

TL;DR

This paper derives general sum rules for eigenvalues of inhomogeneous drums of arbitrary shape and density, extending previous one-dimensional results to higher dimensions and specific geometries.

Contribution

It provides a unified framework for calculating eigenvalue sum rules for inhomogeneous drums in any shape and boundary condition, generalizing earlier one-dimensional formulas.

Findings

Derived explicit sum rules for eigenvalues of inhomogeneous drums.

Extended formulas to higher dimensions and specific geometries.

Applied results to circular annuli and sectors with exact sum rules.

Abstract

We derive general expressions for the sum rules of the eigenvalues of drums of arbitrary shape and arbitrary density, obeying different boundary conditions. The formulas that we present are a generalization of the analogous formulas for one dimensional inhomogeneous systems that we have obtained in a previous paper. We also discuss the extension of these formulas to higher dimensions. We show that in the special case of a density depending only on one variable the sum rules of any integer order can be expressed in terms of a single series. As an application of our result we derive exact sum rules for the a homogeneous circular annulus with different boundary conditions, for a homogeneous circular sector and for a radially inhomogeneous circular annulus with Dirichlet boundary conditions.