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Izvestiya: Mathematics, 2012, Volume 76, Issue 5, Pages 946–966
DOI: https://doi.org/10.1070/IM2012v076n05ABEH002611
(Mi im7330)
 

This article is cited in 11 scientific papers (total in 11 papers)

Asymptotics of the eigenvalues of a discrete Schrödinger operator with zero-range potential

S. N. Lakaev, Sh. Yu. Kholmatov

A. Navoi Samarkand State University
References:
Abstract: We consider a family of discrete Schrödinger operators $H_{\mu}(k)$, $k\in\mathfrak{G}\subset\mathbb{T}^d$. These operators are associated with the Hamiltonian ${H}_{\mu}$ of a system of two identical quantum particles (bosons) moving on the $d$-dimensional lattice $\mathbb{Z}^d$, $d\geqslant 3$, and interacting by means of a pairwise zero-range (contact) attractive potential $\mu>0$. It is proved that for any $k\in\mathfrak{G}$ there is a number $\mu(k)>0$ which is a threshold value of the coupling constant; for $\mu>\mu(k)$ the operator $H_{\mu}(k)$, $k\in\mathfrak{G}\subset\mathbb{T}^d$, has a unique eigenvalue $z(\mu, k)$ placed to the left of the essential spectrum. The asymptotic behaviour of $z(\mu, k)$ is found as $\mu\to\mu(k)$ and as $\mu\to+\infty$ and also as $k\to k^*$ for every value of the quasi-momentum $k^*=k^*(\mu)$ belonging to the manifold $\{k\in\mathfrak{G}\colon\mu(k)=\mu\}$, where $\mu\in\bigl(\inf_{k\in\mathfrak{G}}\mu(k),\sup_{k\in\mathfrak{G}}\mu(k)\bigr)$.
Keywords: discrete Schrödinger operator, Hamiltonian system of two particles, zero-range (contact) potential, eigenvalue, asymptotic behaviour.
Received: 01.03.2011
Revised: 24.10.2011
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2012, Volume 76, Issue 5, Pages 99–118
DOI: https://doi.org/10.4213/im7330
Bibliographic databases:
Document Type: Article
UDC: 517.984.46
MSC: Primary 81Q10; Secondary 81U05
Language: English
Original paper language: Russian
Citation: S. N. Lakaev, Sh. Yu. Kholmatov, “Asymptotics of the eigenvalues of a discrete Schrödinger operator with zero-range potential”, Izv. RAN. Ser. Mat., 76:5 (2012), 99–118; Izv. Math., 76:5 (2012), 946–966
Citation in format AMSBIB
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\by S.~N.~Lakaev, Sh.~Yu.~Kholmatov
\paper Asymptotics of the eigenvalues of a~discrete Schr\"odinger operator with zero-range potential
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\pages 99--118
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  • https://doi.org/10.1070/IM2012v076n05ABEH002611
  • https://www.mathnet.ru/eng/im/v76/i5/p99
  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Russian version PDF:231
    English version PDF:23
    References:86
    First page:37
     
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