Heterogeneous systems in $d$ dimensions: lower spectrum

TL;DR

This paper introduces iterative methods to systematically approximate the lower spectrum of the Helmholtz equation in heterogeneous systems across various boundary conditions, applicable in multiple dimensions.

Contribution

It presents a novel iterative approach that converges to the lower spectrum of the Helmholtz equation in heterogeneous media, independent of inhomogeneity strength.

Findings

Methods work for different boundary conditions.

Applications demonstrated in 1D and 2D cases.

Convergence is guaranteed with a suitable ansatz.

Abstract

We show that the properties of the lower part of the spectrum of the Helmholtz equation for an heterogeneous system in a finite region in $d$ dimensions, where the solutions to the homogeneous problems are known, can be systematically approximated by means of iterative methods. These methods only require the specification of an arbitrary ansatz and necessarily converge to the desired solution, regardless of the strength of the inhomogeneities, provided that the ansatz has a finite overlap with it. Different boundary conditions at the borders of the domain can be assumed. Applications in one and two dimensions are used to illustrate the methods.