Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


We consider a family of discrete Schrö dinger operators hd(k), where k is the two-particle quasi-momentum varying in 𝕋d=(−π,π]d , associated to a system of two-particles on the d - dimensional lattice ℤd, d>1. The CwikelLieb-Rozenblum (CLR)-type estimates are written for hd(k) when the Fermi surface Ek-1(𝔢m(k)) of the associated dispersion relation is a one point set at em(k), the bottom of the essential spectrum. Moreover, when the Fermi surface Ek-1(𝔢m(k)) is of dimension d−1 or d−2, we obtain the necessary and sufficient conditions for the existence of infinite discrete spectrum of hd(k) , while in the case dimEk-1(𝔢m(k)) ≤ d-3, the discrete spectrum of hd(k) is finite.

First Page


Last Page



1. Abdullaev J.I., Lakaev S.N. Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice, Theoretical and Mathematical Physics, 136, 2 1096–1109 (2003). 136, No.2, 231–245. (2003).

2. Abdullaev J.I., Lakaev S.N. On the Spectral Properties of the Matrix-Valued Friedrichs Model. Advances in soviet Mathematics. American Mathematical Society. Vol. 5, 1–37 (1991).

3. Albeverio S., Lakaev S.N. and Abdullaev J.I. On the structure of the essential spectrum for the three-particle Schrödinger operators on lattices. Mathematische Nachrichten, Vol. 280, No. 7, 1–18 (2007).

4. Albeverio S., Lakaev S.N., Makarov K.A., Muminov Z.I. The Threshold Effects for the Two-particle Hamiltonians on Lattices, Comm.Math.Phys. Vol. 262, 91–115 (2006).

5. Albeverio S., Lakaev S.N., Muminov Z.I. Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics. Ann. Henri Poincar\'e. Vol. 5, 743–772 (2004).

6. Damanik D., Hundertmark D., Killip R., Simon B. Variational estimates for eiscrete Schrödinger operators with potentials of indefinite sign. Comm.Math.Phys. Vol. 238, 545–562 (2003).

7. Faria da Viega P.A., Ioriatti L. and O'Carrol M. Energy-momentum spectrum of some two-particle lattice Schrödinger Hamiltonian. Phys. Rev. E(3) Vol. 66, 016130, 9 (2002).

8. I. Gohberg, S. Goldberg, M. A. Kaashoek. Basic Classes of Linear Operators, Birkhäuser Verlag, Basel, 2003.

9. Gruber P.M., Lekkerkerker C.G. Geometry of Numbers. Elseviar science publishers B.V., 1987.

10. Hofmann K.H., Morris S.A.: The Structure of Compact Groups A Primer for Students - A Handbook for the Expert. Walter De Gruyter Inc; 2 Revised edition (August 22, 2006). 858 p.

11. Klaus M. On the bound state of Schrödinger operators in one dimension. Ann. Phys. Vol. 108, 288–300 (1977).

12. Landau L.D., Lifshitz E.M. Quantum Mechanics: Non-relativistic Theory. Course of Theoretical Physics, Vol. 3. Reading, Mass: Addison-Wesley, 1958.

13. Mattis D.C. The few-body problem on a lattice. Rev. Modern Phys. Vol. 58, 361–379 (1986).

14. MolVain12 Molchanov S., Vainberg B. Bargmann type estimates of the counting function for general Schrödinger operators. Journal of Mathematical Sciences, Vol. 184, Issue 4, 457–508 (July 2012).

15. Mogilner A. Hamiltonians in solid state physics as multi-particle discrete Schrödinger operators: Problems and results. Advances in Soviet Mathematics Vol. 5, 139–194 (1991).

16. Pankrashkin K. Variational principle for hamiltonians with degenerate bottom. In the book I. Beltita, G. Nenciu, R. Purice (Eds.). Mathematical Results in Quantum Mechanics. Proceedings of the QMath10 Conference (World Scientific, 2008) 231–240 Preprint arXiv:0710.4790.

17. Reed M. and Simon B. Methods of modern mathematical physics. VI: Analysis of Operators, Academic Press, New York, 1979.

18. Rozenblum G.V. and Solomyak M.Z. On the spectral estimates for the Schrödinger operator on $\mathbbZ^d$, $d \geq 3$. [in Russian], Probl. Mat. Anal. Vol. 41, 107–126 (2009); English transl.: J. Math. Sci., New York, Vol. 159, No. 2, 241–263 (2009).

19. The bound state of weakly coupled Schrödinger operators in one and two-dimentionas, Ann. Phys. Vol. 97, 279–288 (1976).

20. Yafaev D.R. The discrete spectrum in the singular Friedrichs model. Amer.Math.Soc.Trans. Vol. 2, Issue 189, 255–274 (1999).

21. Yafaev D.R. Scattering theory: Some old and new problems, Lecture Notes in Mathematics, 1735. Springer-Verlag, Berlin, 2000, 169 pp.



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.