Progress in Weyl's problem achieved by computational methods

Abstract

The smoothed eigenvalue distribution for the scalar, and the electromagnetic vector, wave equations are studied for large, but finite wavenumbers k by counting the first 10 6 eigenvalues for various shapes of the domain. The results have implications on the Fermi-gas model of nuclear matter, the electron gas as well as the long-wave acoustic vibration modes in small crystals, the laws of black-body radiation, the acoustics of complicated resonators, and the thermodynamics of perfect gases in a finite volume. The relevance of the computational procedure is compared to that of the analytical methods yielding asymptotic expansions for the eigenvalue distribution which are valid in the limit of infinite k. As an illustrative example for the computational procedure, we present the calculation of the electromagnetic mode density in a lossless cavity resonator with the shape of a circular cylinder. This calculation comprehends the computation of the first 10 4 zeros of the Bessel functions.