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Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice

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Abstract

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of the family of the operators h(k) depends on the particle interaction energy and the total quasimomentum \(k \in \mathbb{T}^3\) (where \(\mathbb{T}^3\) is a three-dimensional torus). Depending on the particle interaction energy, we obtain conditions under which the left edge of the continuous spectrum is simultaneously a multiple virtual level and an eigenvalue of the operator h(0).

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References

  1. L. D. Faddeev, Trudy Mat. Inst. Steklov., 69, 3–122 (1963).

    MathSciNet  Google Scholar 

  2. D. C. Mattis, Rev. Modern Phys., 58, 361–379 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  3. S. Albeverio, S. N. Lakaev, K. A. Makarov, and Z. I. Muminov, Commun. Math. Phys., 262, 91–115 (2006).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. D. R. Yafaev, Math. USSR-Sb., 23, 535–559 (1974).

    Article  Google Scholar 

  5. A. V. Sobolev, Commun. Math. Phys., 156, 101–126 (1993).

    Article  MATH  ADS  Google Scholar 

  6. D. R. Yafaev, Theor. Math. Phys., 25, 1065–1072 (1975).

    Article  Google Scholar 

  7. S. A. Vugal’ter and G. M. Zhislin, Trans. Moscow Math. Soc., 49, 87–114 (1987).

    Google Scholar 

  8. G. M. Zhislin, Theor. Math. Phys., 68, 815–823 (1986).

    Article  MathSciNet  Google Scholar 

  9. S. N. Lakaev, Theor. Math. Phys., 89, 1079–1086 (1991).

    Article  MathSciNet  Google Scholar 

  10. S. N. Lakaev and M. I. Muminov, Theor. Math. Phys., 135, 849–871 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  11. E. L. Lakshtanov and R. A. Minlos, Funct. Anal. Appl., 39, 31–45 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. A. Faria da Veiga, L. Ioriatti, and M. O’Carroll, Phys. Rev. E, 66, 016130 (2002).

    Article  MathSciNet  ADS  Google Scholar 

  13. M. I. Muminov, Theor. Math. Phys., 153, 1671–1676 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. I. Muminov and A. M. Hurramov, Theor. Math. Phys., 177, 1693–1705 (2013).

    Article  Google Scholar 

  15. V. P. Maslov, Theor. Math. Phys., 159, 684–685 (2009).

    Article  Google Scholar 

  16. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Acad. Press, New York (1978).

    Google Scholar 

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Authors and Affiliations

  1. Samarkand State University, Samarkand, Uzbekistan

    M. E. Muminov & A. M. Khurramov

  2. Universiti Technologi, Johor, Malaysia

    M. E. Muminov

Authors
  1. M. E. Muminov
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  2. A. M. Khurramov
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Correspondence to M. E. Muminov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 180, No. 3, pp. 329–341, July, 2014.

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Muminov, M.E., Khurramov, A.M. Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice. Theor Math Phys 180, 1040–1050 (2014). https://doi.org/10.1007/s11232-014-0198-2

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  • DOI: https://doi.org/10.1007/s11232-014-0198-2

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