A semi-empirical formula for the eigenspectrum of the 2-dimensional Helmholtz equation with Dirichlet or Neumann condition on a supercircular boundary

TL;DR

This paper introduces a semi-empirical formula to accurately estimate the eigenvalues of the 2D Helmholtz equation with Dirichlet or Neumann boundary conditions on supercircular domains, simplifying spectral calculations.

Contribution

A single-parameter semi-empirical formula is proposed for the Helmholtz eigenvalues on supercircular boundaries, applicable to both Dirichlet and Neumann conditions, enhancing previous perturbative approaches.

Findings

The formula accurately predicts low-lying eigenvalues across various supercircular exponents.

It extends previous work by also covering Neumann boundary conditions.

Results show remarkable agreement with numerical estimates.

Abstract

In a recent paper \cite{chak} Chakraborty et al have put forward a perturbative formulation for solving the 2 dimensional homogeneous Helmholtz equation with the Dirichlet condition on a supercircular boundary. In this note a single parameter (supercircular exponent or exponent) semi-empirical formula, giving the eigenspectrum, is presented for the same problem. The same formula now is also applicable for the Neumann type boundary condition. The formula is put to test by comparing the obtained eigenvalues for several low lying states with their corresponding numerical estimates. It is seen that the formula gives results with a remarkable accuracy for a wide range of the supercircular exponent.