Opuscula Math. 41, no. 3 (2021), 413-435
https://doi.org/10.7494/OpMath.2021.41.3.413

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)
• 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.
• Accepted: 2021-02-01.
• Published online: 2021-04-19.