A closed form solution of the Helmholtz equation for a class of chaotic resonators

We study numerically the solution of the Helmholtz equation in a classically chaotic two-dimensional region in the shape of a bow-tie. The quantum ergodicity of classically chaotic systems has been studied extensively, both theoretically and experimentally, in mathematics and in physics. Despite this long tradition, we are able to present a new rigorous result using only elementary calculus. In particular, a closed form solution is derived by using multipole expansions. Our results have been validated by an integral equation method based on layer potential which is solved via the Nyström discretization method.