On the $S$-matrix of Schr\"odinger operator with nonlocal $\delta$-interaction

TL;DR

This paper investigates the $S$-matrix of Schrödinger operators with nonlocal delta interactions using Lax-Phillips scattering theory, deriving formulas and analyzing its analytical properties in relation to the operator's positivity.

Contribution

It introduces two formulas for the $S$-matrix and establishes conditions for the applicability of the Lax-Phillips approach in this context.

Findings

Derived formulas for the $S$-matrix using different methods.

Characterized the analytical properties of the $S$-matrix based on operator positivity.

Provided examples illustrating the $S$-matrix behavior.

Abstract

Schr\"{o}dinger 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 Schr\"{o}dinger 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 Schr\"{o}dinger operator. Examples of $S$-matrices are given.