Asymptotic behaviour of quasi-eigenvalues of the Laplace operator in the case of the outside of a circular disc

An analisys of the exact solution is used to obtain the asymptotic behaviour as $|K| \to \infty$ of the quasi-eigenvalues $K$, closest to the $Im~K = 0$ axis, of the Laplace operator in the case of the outside of a circular disc (it is assumed that the Neumann boundary condition holds on the disc itself). A geometrical interpretation is given for the asymptotic expressions for the quasi-eigenfunctions of the Laplace operator, in terms of geometrical optics.