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

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.