Skip to main content

On the spectrum of the three-particle Hamiltonian on a unidimensional lattice

Abstract

On a unidimensional lattice, the Hamiltonian of a system of three arbitrary particles is considered (with dispersion relations), where the particles interact pairwise via zero-range (contact) attractive potentials.We prove that the discrete spectrum of the corresponding Schrödinger operator is finite for all values of the total quasimomentum if the masses of two particles are finite. We also prove that the discrete spectrum of the Schrödinger operator is infinite if the masses of two particles in a three-particle system are infinite.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    S. Albeverio, S. N. Lakaev, and Z. I. Muminov, “Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics,” Ann. Henri Poincaré 5, 743 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  2. 2.

    R. D. Amado and J. V. Noble, “Efimov’s effect: a new pathology of three-particle systems. II,” Phys. Rev. D. 5, 1992 (1972).

    Article  Google Scholar 

  3. 3.

    M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Kluwer Academic Publishers,Dordrecht, 1987) [Spectral Theory of Self-AdjointOperators in Hilbert Space (Izd. Leningrad. Univ., Leningrad, 1980)].

    Google Scholar 

  4. 4.

    V. N. Efimov, “Weakly-bound states of three resonantly-interacting particles,” Sov. J. Nuclear Phys. 12, 589 (1971) [Yad. Fiz. 12, 1080 (1970)].

    Google Scholar 

  5. 5.

    L. D. Faddeev and S. P. Merkur’ev, Quantum Scattering Theory for Several Particle Systems (Kluwer Academic Publishers, Dordrecht, 1993) [Quantum Scattering Theory for Several Particle Systems (Nauka, Moscow, 1985)].

    Google Scholar 

  6. 6.

    S. N. Lakaev, “On the infinite number of three-particle bound states of a system of three quantum lattice particles,” Theoret.Math. Phys. 89, 1079 (1991) [Teor.Mat. Fiz. 89, 94 (1991)].

    Article  MathSciNet  Google Scholar 

  7. 7.

    S. N. Lakaev and M. E.Muminov, “Essential and discrete spectra of the three-particle Schrödinger operator on a lattice,” Theoret.Math. Phys. 135, 849 (2003) [Teor.Mat. Fiz. 135, 478 (2003)].

    Article  MATH  Google Scholar 

  8. 8.

    M. E. Muminov, “The infiniteness of the number of eigenvalues in the gap in the essential spectrum for the three-particle Schrödinger operator on a lattice,” Theoret. Math. Phys. 159, 667 (2009) [Teor.Mat. Fiz. 159, 299 (2009)].

    Article  MathSciNet  MATH  Google Scholar 

  9. 9.

    M. E.Muminov and N.M. Aliev, “Spectrumof the three-particle Schrödinger operator on a one-dimensional lattice,” Theoret.Math. Phys. 171, 754 (2012) [Teor.Mat. Fiz. 171, 387 (2012)].

    Article  Google Scholar 

  10. 10.

    Yu. N. Ovchinnikov and I. M. Sigal, “Number of bound states of three-body systems and Efimov’s effect,” Ann. Physics 123, 274 (1979).

    Article  MathSciNet  Google Scholar 

  11. 11.

    M. Reed and B. Simon, Methods of Modern Mathematical Physics. Vol. IV. Analysis of Operators (Academic Press, New York–San Francisco–London, 1978).

    MATH  Google Scholar 

  12. 12.

    H. Tamura, “The Efimov effect of three-body Schrödinger operators,” J. Funct. Anal. 95, 433 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  13. 13.

    D. R. Yafaev, “On the theory of the discrete spectrum of the three-particle Schrödinger operator,” Math. USSR, Sb. 23, 535 (1976) [Mat. Sb. 94, 567 (1974)].

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to N. M. Aliev.

Additional information

Original Russian Text © N.M. Aliev and M.E. Muminov, 2014, published in Matematicheskie Trudy, 2014, Vol. 17, No. 2, pp. 3–22.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aliev, N.M., Muminov, M.E. On the spectrum of the three-particle Hamiltonian on a unidimensional lattice. Sib. Adv. Math. 25, 155–168 (2015). https://doi.org/10.3103/S1055134415030013

Download citation

Keywords

  • three-particle system on a lattice
  • Schrödinger operator
  • essential spectrum
  • discrete spectrum
  • compact operator