On the $S$-matrix of Schrödinger operator with nonlocal $\delta$-interaction

Abstract

Schrödinger operators with nonlocal $\delta$-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the $S$-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The $S$-matrix $S(z)$ is analytical in the lower half-plane ${\mathbb{C}}_{-}$ when the Schrödinger operator with nonlocal $\delta$-interaction is positive self-adjoint. Otherwise, $S(z)$ is a meromorphic matrix-valued function in ${\mathbb{C}}_{-}$ and its properties are closely related to the properties of the corresponding Schrödinger operator. Examples of $S$-matrices are given.