Skip to main content
Log in

On the Number and Location of Eigenvalues of the Two Particle Schrödinger Operator on a Lattice

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the discrete spectrum of the two-particle Schrödinger operator \(\widehat{H}_{\lambda\mu}(K),\) \(K\in\mathbb{T}^{2},\) associated to the Bose-Hubbard Hamiltonian \(\widehat{\mathbb{H}}_{\lambda\mu}\) of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice \(\mathbb{Z}^{2}\) with interaction magnitudes \(\lambda\in\mathbb{R}\) and \(\mu\in\mathbb{R},\) respectively. Under certain conditions on \(\lambda,\,\mu\in\mathbb{R}\) we prove that the discrete Schrödinger operator \(\widehat{H}_{\lambda\mu}(0)\) can have zero, one, two or three eigenvalues below the bottom or above the top of the essential spectrum. Moreover, we show the conditions for existence of three eigenvalues, where two of them are situated below the bottom of the essential spectrum, and other one above its top.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

REFERENCES

  1. S. N. Lakaev, Sh. Yu. Kholmatov, and Sh. I. Khamidov, ‘‘Bose–Hubbard models with on-site and nearest-neighbor interactions: Exactly solvable case,’’ Phys. A (Amsterdam, Neth.) 54, 245201 (2021).

  2. S. N. Lakaev and I. N. Bozorov, ‘‘The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice,’’ Theor. Math. Phys. 158, 360–376 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Hiroshima, Z. Muminov, and U. Kuljanov, ‘‘Threshold of discrete Schrödinger operators with delta-potentials on \(N\)-dimensional lattice,’’ Lin. Multilin. Algebra 70, 919–954 (2022).

    Article  MATH  Google Scholar 

  4. I. Bloch, ‘‘Ultracold quantum gases in optical lattices,’’ Nat. Phys. 1 (5), 23–30 (2005).

    Article  Google Scholar 

  5. D. Jaksch, C. Bruder, J. Cirac, C. W. Gardiner, and P. Zoller, ‘‘Cold bosonic atoms in optical lattices,’’ Phys. Rev. Lett. 81, 3108–3111 (1998).

    Article  Google Scholar 

  6. D. Jaksch and P. Zoller, ‘‘The cold atom Hubbard toolbox,’’ Ann. Phys. 315, 52–79 (2005).

    Article  MATH  Google Scholar 

  7. M. Lewenstein, A. Sanpera, and V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems (Oxford Univ. Press, Oxford, 2012).

    Book  MATH  Google Scholar 

  8. M.-S. Heo, T. T. Wang, C. A. Christensen, T. M. Rvachov, D. A. Cotta, J.-H. Choi, Y.-R. Lee, and W. Ketterle, ‘‘Formation of ultracold fermionic NaLi Feshbach molecules,’’ Phys. Rev. A 86, 021602 (2012).

  9. Z. E. Muminov, N. Aliev, and Sh. S. Lakaev, ‘‘On the essential spectrum of three-particle discrete Schrödinger operators with short-range potentials,’’ Lobachevskii J. Math. 43, 1303–1315 (2021).

    MATH  Google Scholar 

  10. C. Ospelkaus, S. Ospelkaus, L. Humbert, P. Ernst, K. Sengstock, and K. Bongs, ‘‘Ultracold heteronuclear molecules in a 3D optical lattice,’’ Phys. Rev. Lett. 97, 120402 (2006).

  11. K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P. Büchler, and P. Zoller, ‘‘Repulsively bound atom pairs in an optical lattice,’’ Nature (London, U.K.) 441, 853–856 (2006).

    Article  Google Scholar 

  12. J. J. Zirbel, K.-K. Ni, S. Ospelkaus, T. L. Nicholson, M. L. Olsen, P. S. Julienne, C. E. Wieman, J. Ye, and D. S. Jin, ‘‘Heteronuclear molecules in an optical dipole trap,’’ Phys. Rev. A 78, 013416 (2008).

  13. Yu. Kondratiev, O. Kutoviy, and S. Pirogov, ‘‘Correlation functions and invariant measures in continuous contact model,’’ Infin. Dim. Anal. Quantum Probab. Relat. Top. 11, 231–258 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  14. Yu. Kondratiev, S. Molchanov, S. Pirogov, and E. Zhizhina, ‘‘On ground state of some non local Schrödinger operators,’’ Applic. Anal. 96, 1390–1400 (2016).

    Article  MATH  Google Scholar 

  15. Yu. Kondratiev and A. Skorokhod, ‘‘On contact processes in continuum,’’ Infin. Dim. Anal. Quantum Probab. Relat. Top. 9, 187–198 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Albeverio, S. N. Lakaev, K. A. Makarov, and Z. I. Muminov, ‘‘The threshold effects for the two-particle Hamiltonians,’’ Commun. Math. Phys. 262, 91–115 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  17. P. A. Faria Da Veiga, L. Ioriatti, and M. O’Carroll, ‘‘Energy-momentum spectrum of some two-particle lattice Schrödinger Hamiltonians,’’ Phys. Rev. E 66, 016130 (2002).

  18. S. N. Lakaev, A. M. Khalkhuzhaev, and Sh. S. Lakaev, ‘‘Asymptotic behavior of an eigenvalue of the two-particle discrete Schrödinger operator,’’ Theor. Math. Phys. 171, 800–811 (2012).

    Article  MATH  Google Scholar 

  19. S. N. Lakaev and Sh. Yu. Kholmatov, ‘‘Asymptotics of the eigenvalues of a discrete Schrödinger operator with zero-range potential,’’ Izv. Math. 76, 946–966 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  20. S. N. Lakaev and Sh. Yu. Kholmatov, ‘‘Asymptotics of eigenvalues of two-particle Schrödinger operators on lattices with zero-range interaction,’’ J. Phys. A: Math. Theor. 44 (13) (2011).

  21. I. N. Bozorov and A. M. Khurramov, ‘‘On the number of eigenvalues of the lattice model operator in one-dimensional case,’’ Lobachevskii J. Math. 43, 353–365 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. N. Lakaev and E. Özdemir, ‘‘The existence and location of eigenvalues of the one particle Hamiltonians on lattices,’’ Hacettepe J. Math. Stat. 45, 1693–1703 (2016).

    MathSciNet  MATH  Google Scholar 

  23. S. N. Lakaev and I. U. Alladustova, ‘‘The exact number of eigenvalues of the discrete Schrödinger operators in one-dimensional case,’’ Lobachevskii J. Math. 42, 1294–1303 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  24. S. N. Lakaev, A. T. Boltaev, and F. M. Almuratov, ‘‘On the discrete spectra of Schrödinger-type operators on one dimensional lattices,’’ Lobachevskii J. Math. 43, 1523–1536 (2022).

    Article  MATH  Google Scholar 

  25. Sh. Kholmatov and M. Pardabaev, ‘‘On spectrum of the discrete bilaplacian with zero-range perturbation,’’ Lobachevskii J. Math. 42, 1286–1293 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. N. Lakaev and A. T. Boltaev, ‘‘Threshold phenomena in the spectrum of the two-particle Schrödinger operator on a lattice,’’ Theor. Math. Phys. 198, 363–375 (2019).

    Article  MATH  Google Scholar 

  27. Z. E. Muminov, Sh. U. Alladustov, and Sh. S. Lakaev, ‘‘Threshold analysis of the three dimensional lattice Schrödinger operator with non-local potential,’’ Lobachevskii J. Math. 41, 1094–1102 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  28. Z. E. Muminov, Sh. U. Alladustov, and Sh. S. Lakaev, ‘‘Spectral and threshold analysis of a small rank perturbation of the discrete Laplacian,’’ J. Math. Anal. Appl. 496, 124827 (2021).

  29. Z. E. Muminov and Sh. S. Lakaev, ‘‘On spectrum and threshold analysis for discrete Schrödinger operator,’’ AIP Conf. Proc. 2365, 050011 (2021).

  30. J. I. Abdullayev and A. M. Toshturdiyev, ‘‘Invariant subspaces of the Shrödinger operator with a finite support potential,’’ Lobachevskii J. Math. 43, 728–737 (2022).

    Article  MathSciNet  Google Scholar 

  31. Z. Muminov, U. Kulzhanov, and Sh. Lakaev, ‘‘On the spectrum of the two-particle Shrödinger operator with point interaction,’’ Lobachevskii J. Math. 42, 598–605 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  32. Z. Muminov, U. Kulzhanov, and G. Ismoilov, ‘‘Two-particle Shrödinger operator with point interaction,’’ Lobachevskii J. Math. 43, 1537–1545 (2022).

    Article  MATH  Google Scholar 

  33. B. Simon, ‘‘Notes on infinite determinants of Hilbert space operators,’’ Adv. Math. 24, 244–273 (1977).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The authors acknowledge support from the Foundation for Basic Research of the Republic of Uzbekistan (Grant no. FZ-20200929224).

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to S. N. Lakaev or Sh. I. Khamidov.

Additional information

(Submitted by T. K. Yuldashev)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lakaev, S.N., Khamidov, S.I. On the Number and Location of Eigenvalues of the Two Particle Schrödinger Operator on a Lattice. Lobachevskii J Math 43, 3541–3551 (2022). https://doi.org/10.1134/S1995080222150173

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080222150173

Keywords:

Navigation