Example of point potential with inner structure

This paper is devoted to constructing a new example of a point potential with the specific form of its scattering amplitude. This problem was inspired by numerous works on zero-range potentials. Scattering amplitude obtained in the model below has the form $(-ik+a-bk^2)^{-1}$ for arbitrary constants $a$ and $b > 0$ while in classical works of Bethe and Pierls coefficient $b = 0$. Potentials with scattering amplitude of this kind may be obtained using the technique of self-adjoint extensions of Laplace operator, but the model of interest in this paper is the point potential represented as a limit of classical rectangular potential barrier with radius converging to zero. The scattering amplitude of the constructed system stabilizes as its support reduces into one point. The special form of the scattering amplitude in this case can be understood as an existence of an inner structure of the scatterer.