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, log in via an institution to check access.
Similar content being viewed by others
References
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).
R. D. Amado and J. V. Noble, “Efimov’s effect: a new pathology of three-particle systems. II,” Phys. Rev. D. 5, 1992 (1972).
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)].
V. N. Efimov, “Weakly-bound states of three resonantly-interacting particles,” Sov. J. Nuclear Phys. 12, 589 (1971) [Yad. Fiz. 12, 1080 (1970)].
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)].
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)].
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)].
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)].
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)].
Yu. N. Ovchinnikov and I. M. Sigal, “Number of bound states of three-body systems and Efimov’s effect,” Ann. Physics 123, 274 (1979).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Vol. IV. Analysis of Operators (Academic Press, New York–San Francisco–London, 1978).
H. Tamura, “The Efimov effect of three-body Schrödinger operators,” J. Funct. Anal. 95, 433 (1991).
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)].
Author information
Authors and Affiliations
Corresponding author
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
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1055134415030013