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

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Teoreticheskaya i Matematicheskaya Fizika, 2010, Volume 163, Number 1, Pages 34–44
DOI: https://doi.org/10.4213/tmf6485 (Mi tmf6485)

This article is cited in 23 scientific papers (total in 23 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

Full-text PDF (436 kB) Citations (23)

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

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.

Received: 02.06.2009
Revised: 09.10.2009

English version:
Theoretical and Mathematical Physics, 2010, Volume 163, Issue 1, Pages 429–437
DOI: https://doi.org/10.1007/s11232-010-0033-3

Bibliographic databases:

Document Type: Article

Language: Russian

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

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https://doi.org/10.4213/tmf6485

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This publication is cited in the following 23 articles:

D. Zh. Kulturaev, Yu. Kh. Eshkabilov, “On the Spectral Properties of Selfadjoint Partial Integral Operators with a Nondegenerate Kernel”, Sib Math J, 65:2 (2024), 475
M. I. Muminov, J. A. Pardaev, “The Spectrum of Discrete Schrödinger Operator on a Three Dimensional Triangular Lattice with a Finite-range Potential”, Lobachevskii J Math, 45:4 (2024), 1722
J. I. Abdullaev, Sh. H. Ergashova, “Eigenvalues of the Schrödinger Operator Corresponding to a System of Three Fermions on a One Dimensional Lattice”, Lobachevskii J Math, 45:8 (2024), 3821
J. I. Abdullaev, A. M. Khalkhuzhaev, Kh. Sh. Makhmudov, “The Infiniteness of the Number of Eigenvalues of the Schrödinger Operator of a System of Two Particles on a Lattice”, Lobachevskii J Math, 45:10 (2024), 4828
Bekzod I. Bahronov, Tulkin H. Rasulov, PHYSICAL MESOMECHANICS OF CONDENSED MATTER: Physical Principles of Multiscale Structure Formation and the Mechanisms of Nonlinear Behavior: MESO2022, 2899, PHYSICAL MESOMECHANICS OF CONDENSED MATTER: Physical Principles of Multiscale Structure Formation and the Mechanisms of Nonlinear Behavior: MESO2022, 2023, 030007
Tulkin H. Rasulov, Elyor B. Dilmurodov, Khilola G. Khayitova, PHYSICAL MESOMECHANICS OF CONDENSED MATTER: Physical Principles of Multiscale Structure Formation and the Mechanisms of Nonlinear Behavior: MESO2022, 2899, PHYSICAL MESOMECHANICS OF CONDENSED MATTER: Physical Principles of Multiscale Structure Formation and the Mechanisms of Nonlinear Behavior: MESO2022, 2023, 030005
D. Zh. Kulturaev, Yu. Kh. Eshkabilov, “O spektralnykh svoistvakh samosopryazhennykh chastichno integralnykh operatorov s nevyrozhdennymi yadrami”, Vladikavk. matem. zhurn., 24:4 (2022), 91–104
Kucharov R.R. Khamraeva R.R., “Non-Compact Perturbations of the Spectrum of Multipliers Given With Functions”, Nanosyst.-Phys. Chem. Math., 12:2 (2021), 135–141
G. P. Arzikulov, Yu. Kh. Eshkabilov, “About the spectral properties of one three-partial model operator”, Russian Math. (Iz. VUZ), 64:5 (2020), 1–7
Yu. Kh. Èshkabilov, “Spectrum of a model three-particle Schrödinger operator”, Theoret. and Math. Phys., 186:2 (2016), 268–279
T. Kh. Rasulov, Z. D. Rasulova, “Cpektr odnogo trekhchastichnogo modelnogo operatora na reshetke s nelokalnymi potentsialami”, Sib. elektron. matem. izv., 12 (2015), 168–184
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
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
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
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
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
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
Yu. Kh. Eshkabilov, “On the discrete spectrum of partial integral operators”, Siberian Adv. Math., 23:4 (2013), 227–233
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
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

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

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