SELF-ADJOINT POINT INTERACTION MODEL OF $p$-SCATTERING

P KURASOV

kurasov@maths.lth.se

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

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

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.