Centre for Mathematical Sciences
Lund Institute of Technology
Lund University
Box 118
SE-22100 Lund, Sweden

We are going to present an exactly solvable model of point interaction leading to nontrivial scattering amplitude in $ p$-channel. This model is a generalization of the celebrated Berezin-Faddeev model which is used to describe the $ s$-scattering. The operator under investigation is given by the formal expression

$\displaystyle - \Delta + \alpha_x \partial_x \delta + \alpha_y \partial_y
\delta + \alpha_z \partial_z \delta , $

where $ \alpha_{x,y,z} $ are arbitrary real constants and $ \delta $ is Dirac's delta function. To determine this operator rigorously the theory of finite rank singular perturbations of self-adjoint operators is used. It is proven that this operator can be determined in a certain finite-dimensional extension of the Hilbert space $ W_2^1
({\bf R}^3) .$ Spectral and scattering properties of the described model are investigated. In the case $ \alpha_x = \alpha _y =
\alpha_z $ one gets a spherically symmetric model (the operator commutes with the rotations) with nontrivial $ p$-component of the scattering amplitude. Possible generalizations to include higher order interactions are discussed.

EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts