Splitting of the poles of a Helmholtz resonator

It is shown that in a neighborhood of a two-fold eigenvalue of the Neumann problem there are two poles of the Green's function of the Helmholtz resonator. Asymptotic expressions for them with respect to a small parameter $\varepsilon$ (the linear size of the hole) are constructed, and the principal terms of the asymptotics are written out for the corresponding scattering and radiation problems.