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TMF, 2010, Volume 163, Number 1, Pages 34–44 (Mi tmf6485)  

This article is cited in 15 scientific papers (total in 15 papers)

Asymptotics of the discrete spectrum of a model operator associated with a system of three particles on a lattice

T. H. Rasulov

Bukhara State University, Bukhara, Uzbekistan

Abstract: We consider a model Schrödinger operator $H_\mu$ associated with a system of three particles on the three-dimensional lattice $\mathbb Z^3$ with a functional parameter of special form. We prove that if the corresponding Friedrichs model has a zero-energy resonance, then the operator $H_\mu$ has infinitely many negative eigenvalues accumulating at zero (the Efimov effect). We obtain the asymptotic expression for the number of eigenvalues of $H_\mu$ below $z$ as $z\to-0$.

Keywords: model operator, Friedrichs model, Birman–Schwinger principle, Efimov effect, Hilbert–Schmidt operator, zero-energy resonance, discrete spectrum

DOI: https://doi.org/10.4213/tmf6485

Full text: PDF file (436 kB)
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English version:
Theoretical and Mathematical Physics, 2010, 163:1, 429–437

Bibliographic databases:

Received: 02.06.2009
Revised: 09.10.2009

Citation: T. H. Rasulov, “Asymptotics of the discrete spectrum of a model operator associated with a system of three particles on a lattice”, TMF, 163:1 (2010), 34–44; Theoret. and Math. Phys., 163:1 (2010), 429–437

Citation in format AMSBIB
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\jour TMF
\yr 2010
\vol 163
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\pages 34--44
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\jour Theoret. and Math. Phys.
\yr 2010
\vol 163
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\pages 429--437
\crossref{https://doi.org/10.1007/s11232-010-0033-3}
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  • http://mi.mathnet.ru/eng/tmf/v163/i1/p34

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. T. H. Rasulov, “Essential spectrum of a model operator associated with a three-particle system on a lattice”, Theoret. and Math. Phys., 166:1 (2011), 81–93  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    2. T. Kh. Rasulov, A. A. Rakhmonov, “Uravnenie Faddeeva i mestopolozhenie suschestvennogo spektra odnogo trekhchastichnogo modelnogo operatora”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(23) (2011), 170–180  mathnet  crossref
    3. T. Kh. Rasulov, Kh. Kh. Turdiev, “Nekotorye spektralnye svoistva obobschennoi modeli Fridrikhsa”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(23) (2011), 181–188  mathnet  crossref
    4. T. Kh. Rasulov, “O suschestvennom spektre odnogo modelnogo operatora, assotsiirovannogo s sistemoi trekh chastits na reshetke”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 3(24) (2011), 42–51  mathnet  crossref
    5. Dell'Antonio G.F., Muminov Z.I., Shermatova Y.M., “On the number of eigenvalues of a model operator related to a system of three particles on lattices”, J. Phys. A: Math. Theor., 44:31 (2011), 315302  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Yu. Kh. Eshkabilov, R. R. Kucharov, “Essential and discrete spectra of the three-particle Schrödinger operator on a lattice”, Theoret. and Math. Phys., 170:3 (2012), 341–353  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. T. Kh. Rasulov, “Struktura suschestvennogo spektra modelnogo operatora, assotsiirovannogo s sistemoi trekh chastits na reshetke”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(27) (2012), 34–43  mathnet  crossref  zmath
    8. Yu. Kh. Eshkabilov, “On the discrete spectrum of partial integral operators”, Siberian Adv. Math., 23:4 (2013), 227–233  mathnet  crossref  mathscinet  elib
    9. T. Kh. Rasulov, R. T. Mukhitdinov, “The finiteness of the discrete spectrum of a model operator associated with a system of three particles on a lattice”, Russian Math. (Iz. VUZ), 58:1 (2014), 52–59  mathnet  crossref
    10. R. R. Kucharov, Yu. Kh. Eshkabilov, “On the number of negative eigenvalues of a partial integral operator”, Siberian Adv. Math., 25:3 (2015), 179–190  mathnet  crossref  mathscinet
    11. M. I. Muminov, T. H. Rasulov, “Infiniteness of the number of eigenvalues embedded in the essential spectrum of a $2\times2$ operator matrix”, Eurasian Math. J., 5:2 (2014), 60–77  mathnet
    12. G. P. Arzikulov, Yu. Kh. Eshkabilov, “On the essential and the discrete spectra of a Fredholm type partial integral operator”, Siberian Adv. Math., 25:4 (2015), 231–242  mathnet  crossref  mathscinet
    13. T. Kh. Rasulov, Z. D. Rasulova, “Cpektr odnogo trekhchastichnogo modelnogo operatora na reshetke s nelokalnymi potentsialami”, Sib. elektron. matem. izv., 12 (2015), 168–184  mathnet  crossref
    14. Yu. Kh. Èshkabilov, “Spectrum of a model three-particle Schrödinger operator”, Theoret. and Math. Phys., 186:2 (2016), 268–279  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. G. P. Arzikulov, Yu. Kh. Eshkabilov, “O spektralnykh svoistvakh odnogo trekhchastichnogo modelnogo operatora”, Izv. vuzov. Matem., 2020, no. 5, 3–10  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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