Boundary perturbations and the Helmholtz equation in three dimensions

TL;DR

This paper introduces a novel three-dimensional perturbative method to compute eigenvalues of the Helmholtz equation with arbitrary boundaries, expanding boundary shapes in spherical harmonics and validating the approach against numerical solutions.

Contribution

The paper presents the first formulation of boundary perturbation theory for the 3D Helmholtz equation with arbitrary boundaries, extending 2D methods to three dimensions.

Findings

Method accurately predicts eigenvalues for various boundary shapes.

Perturbation remains effective for large boundary deviations.

Works well for higher excited states.

Abstract

We propose an analytic perturbative scheme for determining the eigenvalues of the Helmholtz equation, $(\nabla^2 + k^2) \psi = 0$, in three dimensions with an arbitrary boundary where $\psi$ satisfies either the Dirichlet boundary condition ($\psi =0$ on the boundary) or the Neumann boundary condition (the normal gradient of $\psi$, $\frac{\partial \psi}{\partial n}$ is vanishing on the boundary). Although numerous works are available in the literature for arbitrary boundaries in two dimensions, to best of our knowledge the formulation in three dimensions is proposed for the first time. In this novel prescription, we have expanded the arbitrary boundary in terms of spherical harmonics about an equivalent sphere and obtained perturbative closed-form solutions at each order for the problem in terms of corrections to the equivalent spherical boundary for both the boundary conditions. This formulation is in parallel with the standard time-independent Rayleigh-Schr{\"o}dinger perturbation theory. The efficiency of the method is tested by comparing the perturbative values against the numerically calculated eigenvalues for spheroidal, superegg and superquadric shaped boundaries. It is shown that this perturbation works quite well even for wide departure from spherical shape and for higher excited states too. We believe this formulation would find applications in the field of quantum dots and acoustical cavities.