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Existence and analyticity of bound states of a two-particle Schrödinger operator on a lattice

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Abstract

We consider the two-particle discrete Schrödinger operator Hμ(K) corresponding to a system of two arbitrary particles on a d-dimensional lattice ℤ d, d ≥ 3, interacting via a pair contact repulsive potential with a coupling constant μ > 0 (\(K \in \mathbb{T}^d\) is the quasimomentum of two particles). We find that the upper (right) edge of the essential spectrum can be either a virtual level (for d = 3, 4) or an eigenvalue (for d ≥ 5) of Hμ (K). We show that there exists a unique eigenvalue located to the right of the essential spectrum, depending on the coupling constant μ and the two-particle quasimomentum K. We prove the analyticity of the corresponding eigenstate and the analyticity of the eigenvalue and the eigenstate as functions of the quasimomentum \(K \in \mathbb{T}^d\) in the domain of their existence.

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References

  1. A. I. Mogilner, “Hamiltonians in solid-state physics as multiparticle discrete Schrödinger operators: Problems and results,” in: Many Particle Hamiltonians: Spectra and Scattering (Advances Sov. Math., Vol. 5, R. A. Minlos, ed.), Amer. Math. Soc., Providence, R. I. (1991), pp. 139–194.

    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); arXiv:math-ph/0501013v1 (2005).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. S. Albeverio, F. Gesztesy, and R. Høgh-Krohn, Ann. Inst. H. Poincaré A, n. s., 37, 1–28 (1982).

    MATH  Google Scholar 

  5. S. N. Lakaev, Funct. Anal. Appl., 27, No. 3, 166–175 (1993).

    Article  MathSciNet  Google Scholar 

  6. J. Rauch, J. Funct. Anal., 35, 304–315 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Tamura, Nagoya Math. J., 130, 55–83 (1993).

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. S. Albeverio, R. Høgh-Krohn, and T. T. Wu, Phys. Lett. A, 83, 105–109 (1971).

    Article  ADS  Google Scholar 

  10. Yu. N. Ovchinnikov and I. M. Sigal, Ann. Physics, 123, 274–295 (1989).

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. S. Albeverio, S. N. Lakaev, and Z. I. Muminov, Ann. Henri Poincaré, 5, 743–772 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hesker Denschlag, K. J. Daley, A. Kantian, H. P. Büchler, and P. Zoller, Nature, 441, 853–856 (2006); arXiv:cond-mat/0605196v1 (2006).

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  15. V. Bach, P. W. De Siqueira, and S. N. Lakaev, “Bounds on the discrete spectrum of lattice Schrödinger operators,” Preprint mp-arc 10-143 (2010).

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Authors and Affiliations

  1. Samarkand State University, Samarkand, Uzbekistan

    S. N. Lakaev & S. S. Ulashov

Authors
  1. S. N. Lakaev
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  2. S. S. Ulashov
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Correspondence to S. N. Lakaev.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 170, No. 3, pp. 393–408, March, 2012.

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Lakaev, S.N., Ulashov, S.S. Existence and analyticity of bound states of a two-particle Schrödinger operator on a lattice. Theor Math Phys 170, 326–340 (2012). https://doi.org/10.1007/s11232-012-0033-6

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  • DOI: https://doi.org/10.1007/s11232-012-0033-6

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