Opuscula Math. 41, no. 3 (2021), 413-435

Opuscula Mathematica

On the S-matrix of Schrödinger operator with nonlocal δ-interaction

Anna Główczyk
Sergiusz Kużel

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.

Keywords: Lax-Phillips scattering scheme, scattering matrix, \(S\)-matrix, nonlocal \(\delta\)-interaction, non-cyclic function.

Mathematics Subject Classification: 47B25, 47A40.

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  • Anna Główczyk (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-5205-827X
  • AGH University of Science and Technology, Faculty of Applied Mathematics AGH, Al. Mickiewicza 30, 30-059 Kraków
  • Communicated by Alexander Gomilko.
  • Received: 2020-09-01.
  • Accepted: 2021-02-01.
  • Published online: 2021-04-19.
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Cite this article as:
Anna Główczyk, Sergiusz Kużel, On the S-matrix of Schrödinger operator with nonlocal δ-interaction, Opuscula Math. 41, no. 3 (2021), 413-435, https://doi.org/10.7494/OpMath.2021.41.3.413

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