On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations

Abstract

For a singular perturbation $A = A_0 + \sum^n_{i, j=1}t_{ij} \langle \psi_j, \cdot \rangle \psi_i,\quad n \leq \infty$ of a positive self-adjoint operator $A_0$ with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations $A_T$ is carried out and the scattering matrix $\mathfrak{S}_{(A_T, A_0)}(\delta)$ is calculated in terms of parameters $t_{ij}$ under some additional restrictions on singular elements $\psi_{j}$ that provides the possibility of application of the Lax -Phillips approach in the scattering theory.