Opuscula Math. 40, no. 5 (2020), 569-597
https://doi.org/10.7494/OpMath.2020.40.5.569

 
Opuscula Mathematica

On some extensions of the A-model

Rytis Juršėnas

Abstract. The A-model for finite rank singular perturbations of class \(\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}\), \(m \in \mathbb{N}\), is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces \((\mathfrak{H}_n)_{n\in\mathbb{Z}}\) admit an orthogonal decomposition \(\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n\), with the corresponding projections satisfying \(P^{\pm}_{n+1}\subseteq P^{\pm}_n\), nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.

Keywords: finite rank higher order singular perturbation, cascade (A) model, peak model, Hilbert space, scale of Hilbert spaces, Pontryagin space, ordinary boundary triple, Krein \(Q\)-function, Weyl function, gamma field, symmetric operator, proper extension, resolvent.

Mathematics Subject Classification: 47A56, 47B25, 47B50, 35P05.

Full text (pdf)

  1. S. Albeverio, S.-M. Fei, P. Kurasov, Many body problems with "spin"-related contact interactions, Rep. Math. Phys. 47 (2001) 2, 157-166.
  2. S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, London Mathematical Society Lecture Note Series 271, Cambridge University Press, UK, 2000.
  3. Y. Arlinskii, S. Belyi, V. Derkach, E. Tsekanovskii, On realization of the Krein-Langer class \(N_{\kappa}\) of matrix-valued functions in Pontryagin spaces, Math. Nachr. 281 (2008) 10, 1380-1399.
  4. T. Azizov, I. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, Inc., 1989.
  5. J. Behrndt, V.A. Derkach, S. Hassi, H. de Snoo, A realization theorem for generalized Nevanlinna families, Operators and Matrices 5 (2011) 4, 679-706.
  6. J. Behrndt, S. Hassi, H. de Snoo, Functional models for Nevanlinna families, Opuscula Math. 28 (2008) 3, 233-245.
  7. J. Behrndt, A. Luger, C. Trunk, On the negative squares of a class of self-adjoint extensions in Krein spaces, Math. Nachr. 286 (2013) 2-3, 118-148.
  8. V. Derkach, On Weyl function and generalized resolvents of a Hermitian operator in a Krein space, Integr. Equ. Oper. Theory 23 (1995) 4, 387-415.
  9. V. Derkach, Boundary triplets, Weyl functions, and the Krein formula, volume 1-2 of Operator Theory, Springer, Basel, 2015, Chapter 10, 183-218.
  10. V.A. Derkach, S. Hassi, A reproducing kernel space model for \(N_{\kappa}\)-functions, Proc. Amer. Math. Soc. 131 (2003) 12, 3795-3806.
  11. V. Derkach, S. Hassi, M. Malamud, Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions, Math. Nachr. 293 (2020) 7, 1278-1327.
  12. V. Derkach, S. Hassi, M. Malamud, H. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358 (2006) 12, 5351-5400.
  13. V. Derkach, S. Hassi, M. Malamud, H. de Snoo, Boundary relations and generalized resolvents of symmetric operators, Russ. J. Math. Phys. 16 (2009) 1, 17-60.
  14. V. Derkach, S. Hassi, M. Malamud, H. de Snoo, Boundary triplets and Weyl functions. Recent developments, [in:] S. Hassi, H.S.V. de Snoo, F.H. Szafraniec (eds), Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Series, vol. 404, Cambridge University Press, UK, 2012, Chapter 7, 161-220.
  15. A. Dijksma, P. Kurasov, Yu. Shondin, High order singular rank one perturbations of a positive operator, Integr. Equ. Oper. Theory 53 (2005), 209-245.
  16. A. Dijksma, H. Langer, Y. Shondin, Rank one perturbations at infinite coupling in Pontryagin spaces, J. Func. Anal. 209 (2004) 1, 206-246.
  17. A. Dijksma, Y. Shondin, Singular point-like perturbations of the Bessel operator in a Pontryagin space, J. Diff. Equ. 164 (2000) 1, 49-91.
  18. S. Hassi, H.S.V. de Snoo, F.H. Szafraniec, Componentwise and Cartesian decompositions of linear relations, Dissertationes Math. 465 (2009), 1-59.
  19. S. Hassi, H.S.V. de Snoo, F.H. Szafraniec, Infinite-dimensional perturbations, maximally nondensely defined symmetric operators, and some matrix representations, Indag. Math. 23 (2012) 4, 1087-1117.
  20. S. Hassi, S. Kuzhel, On symmetries in the theory of finite rank singular perturbations, J. Func. Anal. 256 (2009), 777-809.
  21. S. Hassi, M. Malamud, V. Mogilevskii, Unitary equivalence of proper extensions of a symmetric operator and the Weyl function, Integr. Equ. Oper. Theory 77 (2013) 4, 449-487.
  22. S. Hassi, Z. Sebestyén, H.S.V. de Snoo, F.H. Szafraniec, A canonical decomposition for linear operators and linear relations, Acta Math. Hungar. 115 (2007) 4, 281-307.
  23. R. Juršėnas, Computation of the unitary group for the Rashba spin-orbit coupled operator, with application to point-interactions, J. Phys. A: Math. Theor. 51 (2018) 1, 015 203.
  24. R. Juršėnas, The peak model for the triplet extensions and their transformations to the reference Hilbert space in the case of finite defect numbers, arXiv:1810.07 416, 2020.
  25. P. Kurasov, \(\mathcal{H}_{-n}\)-perturbations of self-adjoint operators and Krein's resolvent formula, Integr. Equ. Oper. Theory 45 (2003) 4, 437-460.
  26. P. Kurasov, Triplet extensions I: Semibounded operators in the scale of Hilbert spaces, Journal d'Analyse Mathematique 107 (2009) 1, 252-286.
  27. P. Kurasov, Yu.V. Pavlov, On field theory methods in singular perturbation theory, Lett. Math. Phys. 64 (2003) 2, 171-184.
  28. K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, Dordrecht, Heidelberg, New York, London, 2012.
  • Communicated by Aurelian Gheondea.
  • Received: 2019-09-24.
  • Revised: 2020-05-12.
  • Accepted: 2020-07-30.
  • Published online: 2020-10-10.
Opuscula Mathematica - cover

Cite this article as:
Rytis Juršėnas, On some extensions of the A-model, Opuscula Math. 40, no. 5 (2020), 569-597, https://doi.org/10.7494/OpMath.2020.40.5.569

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.