On scattering by a cylinder with a narrow slit and walls of finite thickness

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A two-dimensional analog of the Helmholtz resonator with walls of finite thickness and Neumann's boundary conditions is considered. The method of matching asymptotic expansions is used to find the asymptotics for the poles of the analytic continuation of the Green's function. For the corresponding eigendistributions and the solutions of the scattering problem, the leading terms in the asymptotics of the poles are obtained directly.