Zero-range Model of p-scattering by a Potential Well

Abstract

A well-known method of zero-range potentials consists of replacing a deep potential well of a small radius by a boundary condition at the point of the centre of the well. However, in passing to the limit from a deep and narrow potential well to the zero-range model, information, concerning pscattering and scatterings of higher orders, disappears. Traditional zero-range model describes only bound states and s-scattering. The principal mathematical difficulty, which arises in the mathematical construction of a zero-range model, describing p-scattering, is that p-scattered waves have a square nonintegrable singularity at the point, where the well should be located. It is not possible to construct the corresponding energy operator in $L_2(\mathbb{R}^3)$ . We construct the energy operator in some Hilbert space, which naturally arises from the problem and includes $L_2(\mathbb{R}^3)$. We explicitly construct the complete system of generalized eigenfunctions in this space.

Mathematics Subject Classification (2000). 35Q40, 81V45.