Indefinite metric and scattering by a domain with a small hole

For the problem of plane waves scattered by a domain with a small hole, we suggest a model based on the theory of self-adjoint extensions of symmetric operators in a space with indefinite metric. For two-dimensional problems of scattering on a line with a hole and on a semi-ellipse connected by a hole with a half-plane, we justify the choice of extension that guarantees the coincidence of the model solution with the solution of the “actual” problem in the far zone with a high degree of accuracy.