Open AccessOn the S-matrix of Schrödinger operator with nonlocal δ-interaction
Author(s) -
Anna Główczyk,
Sergiusz Kużel
Publication year - 2021
Publication title -
rocznik akademii górniczo-hutniczej im. stanisława staszica. opuscula mathematica/opuscula mathematica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 16
eISSN - 2300-6919
pISSN - 1232-9274
DOI - 10.7494/opmath.2021.41.3.413
Subject(s) - mathematics , resolvent , operator (biology) , matrix (chemical analysis) , complex plane , meromorphic function , mathematical physics , matrix function , schrödinger's cat , reflection (computer programming) , s matrix , mathematical analysis , pure mathematics , function (biology) , plane (geometry) , scattering , quantum mechanics , symmetric matrix , eigenvalues and eigenvectors , physics , geometry , materials science , repressor , chemistry , computer science , composite material , biology , biochemistry , evolutionary biology , transcription factor , programming language , gene
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.