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.

Full text (pdf)

  1. V.M. Adamyan, Nondegenerate unitary couplings of semiunitary operators, Funct. Anal. Appl. 7 (1973), no. 4, 255-267.
  2. V.M. Adamyan, B.S. Pavlov, Null-range potentials and M.G. Krein's formula for generalized resolvents, J. Sov. Math. 42 (1988), 1537-1550.
  3. N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Spaces, Dover Publication Inc, New York, 1993.
  4. S. Albeverio, L. Nizhnik, Schrödinger operators with nonlocal point interactions, J. Math. Anal. Appl. 332 (2007), 884-895.
  5. S. Albeverio, L. Nizhnik, Schrödinger operators with nonlocal potentials, Methods Funct. Anal. Topology 19 (2013), no. 3, 199-210.
  6. S. Albeverio, R. Hryniv, L. Nizhnik, Inverse spectral problems for nonlocal Sturm-Liouville operators, Inverse Problems 23 (2007), 523-536.
  7. F. Bagarello, J.-P. Gazeau, F.H. Szafraniec, M. Znojil (eds.), Non-Selfadjoint Operators in Quantum Physics. Mathematical Aspects , J. Wiley & Sons, Hoboken, New Jersey, 2015.
  8. J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering matrices and Dirichlet-to-Neumann maps, J. Funct. Anal. 273 (2017), 1970-2025.
  9. C.M. Bender, P.E. Dorey, T.C. Dunning, A. Fring, D.W. Hook, H.F. Jones, S. Kuzhel, G. Levai, R. Tateo, PT-Symmetry in Quantum and Classical Physics, World Scientific, Singapore, 2019
  10. J. Brasche, L.P. Nizhnik, One-dimensional Schrödinger operators with general point interactions, Methods Funct. Anal. Topology 19 (2013), no. 1, 4-15.
  11. K.D. Cherednichenko, A.V. Kiselev, L.O. Silva, Functional model for extensions of symmetric operators and applications to scattering theory, Networks and Heterogeneous Media 13 (2018), no. 2, 191-215.
  12. P.A. Cojuhari, S. Kuzhel, Lax-Phillips scattering theory for PT-symmetric \(\rho\)-perturbed operators, J. Math. Phys. 53 (2012), 073514.
  13. R.G. Douglas, H.S. Shapiro, A.L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier 20 (1970), 37-76.
  14. M. Gawlik, A. Główczyk, S. Kuzhel, On the Lax-Phillips scattering matrix of the abstract wave equation, Banach J. Math. Anal. 13 (2019), no. 2, 449-467.
  15. M.L. Gorbachuk, V.I. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Kluwer, Dordrecht, 1991.
  16. S. Kuzhel, On the determination of free evolution in the Lax-Phillips scattering scheme for second-order operator-differential equations, Math. Notes 68 (2000), 724-729.
  17. S. Kuzhel, Nonlocal perturbations of the radial wave equation. Lax-Phillips approach, Methods Funct. Anal. Topology 8 (2002), no. 2, 59-68.
  18. S. Kuzhel, On the inverse problem in the Lax-Phillips scattering theory method for a class of operator-differential equations, St. Petersburg Math. J. 13 (2002), 41-56.
  19. S. Kuzhel, On conditions of applicability of the Lax-Phillips scattering scheme to investigation of abstract wave equation, Ukrainian Math. J. 55 (2003), 621-630.
  20. A. Kuzhel, S. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht, 1998.
  21. S. Kuzhel, M. Znojil, Non-self-adjoint Schrödinger operators with nonlocal one-point interactions, Banach J. Math. Anal. 11 (2017), no. 4, 923-944.
  22. P. Lax, Translation invariant spaces, Acta Math. 101 (1959), 163-178.
  23. P. Lax, R. Phillips, Scattering Theory, Revised Edition, Academic Press, London, 1989.
  24. N.K. Nikolski, Operators, Functions, and Systems: an Easy Reading, Volume I, AMS, USA, 2002.
  25. M. Reed, B. Simon, Methods of Modern Mathematical Physics, Volume III: Scattering Theory, Academic Press, New York, 1979.
  26. K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, Berlin, 2012.
  • 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.
Opuscula Mathematica - cover

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

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.