On scattering by cylinder with narrow slit and with shell of finite depth

The method of matching asymptotic expansions is applied to construction of asymptotics of poles for the Green function analytic continuation of a two-dimensional analog of Helmholtz resonator with a shell of a finite depth and with Neumann's boundary condition. The explicit formulae for the principle terms of asymptotics of poles, for corresponding generalized eigenfunctions and for solutions of scattering problem are obtained.