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Funktsional. Anal. i Prilozhen., 2010, Volume 44, Issue 2, Pages 87–91 (Mi faa2982)  

This article is cited in 18 scientific papers (total in 18 papers)

Brief communications

One-dimensional Schrödinger operator with $\delta$-interactions

A. S. Kostenkoa, M. M. Malamudb

a School of Mathematical Sciences, Dublin Institute of Technology
b Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences, Donetsk

Abstract: The one-dimensional Schrödinger operator $\mathrm{H}_{X,\alpha}$ with $\delta$-interactions on a discrete set is studied in the framework of the extension theory. Applying the technique of boundary triplets and the corresponding Weyl functions, we establish a connection of these operators with a certain class of Jacobi matrices. The discovered connection enables us to obtain conditions for the self-adjointness, lower semiboundedness, discreteness of the spectrum, and discreteness of the negative part of the spectrum of the operator $\mathrm{H}_{X,\alpha}$.

Keywords: Schrödinger operator, point interactions, self-adjointness, lower semiboundedness, discreteness


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English version:
Functional Analysis and Its Applications, 2010, 44:2

Bibliographic databases:

Document Type: Article
UDC: 517.984
Received: 08.07.2009

Citation: A. S. Kostenko, M. M. Malamud, “One-dimensional Schrödinger operator with $\delta$-interactions”, Funktsional. Anal. i Prilozhen., 44:2 (2010), 87–91

Citation in format AMSBIB
\by A.~S.~Kostenko, M.~M.~Malamud
\paper One-dimensional Schr\"odinger operator with $\delta$-interactions
\jour Funktsional. Anal. i Prilozhen.
\yr 2010
\vol 44
\issue 2
\pages 87--91

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    This publication is cited in the following articles:
    1. Albeverio S., Kostenko A., Malamud M., “Spectral theory of semibounded Sturm–Liouville operators with local interactions on a discrete set”, J. Math. Phys., 51:10 (2010), 102102, 24 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. K. A. Mirzoev, T. A. Safonova, “Singulyarnye operatory Shturma–Liuvillya s negladkimi potentsialami v prostranstve vektor-funktsii”, Ufimsk. matem. zhurn., 3:3 (2011), 105–119  mathnet  zmath
    3. Mirzoev K.A., Safonova T.A., “Singular Sturm-Liouville operators with distribution potential on spaces of vector functions”, Dokl. Math., 84:3 (2011), 791–794  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. Markov V.G., “Nekotorye svoistva neznakoopredelennykh operatorov Shturma-Liuvillya”, Matematicheskie zametki YaGU, 19:1 (2012), 44–59  zmath  elib
    5. Albeverio S. Kostenko A. Malamud M. Neidhardt H., “Spherical Schrodinger Operators with Delta-Type Interactions”, J. Math. Phys., 54:5 (2013), 052103  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. K. A. Mirzoev, “Sturm–Liouville operators”, Trans. Moscow Math. Soc., 75 (2014), 281–299  mathnet  crossref  elib
    7. J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl, “Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials”, J. Spectr. Theory, 4:4 (2014), 715–768  crossref  mathscinet  zmath  isi  scopus
    8. S.G. Pyatkov, “Existence of maximal semidefinite invariant subspaces and semigroup properties of some classes of ordinary differential operators”, Oper. Matrices, 8:1 (2014), 237–254  crossref  mathscinet  zmath  isi  elib  scopus
    9. B. P. Osilenker, “On linear summability methods of fourier series in polynomials orthogonal in a discrete Sobolev space”, Siberian Math. J., 56:2 (2015), 339–351  mathnet  crossref  mathscinet  isi  elib  elib
    10. K. A. Mirzoev, T. A. Safonova, “On the Deficiency Index of the Vector-Valued Sturm–Liouville Operator”, Math. Notes, 99:2 (2016), 290–303  mathnet  crossref  crossref  mathscinet  isi  elib
    11. A. S. Kostenko, M. M. Malamud, D. D. Natyagajlo, “Matrix Schrödinger Operator with $\delta$-Interactions”, Math. Notes, 100:1 (2016), 49–65  mathnet  crossref  crossref  mathscinet  isi  elib
    12. Medet Nursultanov, “Spectral Properties of the Schrödinger Operator with $\delta$-Distribution”, Math. Notes, 100:2 (2016), 263–275  mathnet  crossref  crossref  mathscinet  isi  elib
    13. Koshmanenko V. Dudkin M., “Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators”, Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators, Operator Theory Advances and Applications, 253, Springer Int Publishing Ag, 2016, 1–237  crossref  mathscinet  isi
    14. Z. S. Aliyev, A. G. Geidarov, “Spectral Properties of the Sturm–Liouville Operator with $\delta$-Potential and with Spectral Parameter in the Boundary Condition”, Math. Notes, 101:5 (2017), 913–918  mathnet  crossref  crossref  mathscinet  isi  elib
    15. T. R. Gadyl'shin, F. Kh. Mukminov, “Perturbation of second order nonlinear equation by delta-like potential”, Ufa Math. J., 10:2 (2018), 31–43  mathnet  crossref  isi
    16. I. N. Broitigam, K. A. Mirzoev, “O defektnykh chislakh operatorov, porozhdennykh yakobievymi matritsami s operatornymi elementami”, Algebra i analiz, 30:4 (2018), 1–26  mathnet
    17. N. N. Konechnaja, K. A. Mirzoev, A. A. Shkalikov, “On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients”, Math. Notes, 104:2 (2018), 244–252  mathnet  crossref  crossref  isi  elib
    18. Exner P., Kostenko A., Malamud M., Neidhardt H., “Spectral Theory of Infinite Quantum Graphs”, Ann. Henri Poincare, 19:11 (2018), 3457–3510  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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