Abstract
For a broad class of short-range pairwise attraction potentials, we study threshold phenomena in the spectrum of the two-particle Schrödinger operator associated with the energy operator of the s–d exchange model. We prove that the bound state (eigenvalue) either exists or does not exist depending on the exchange interaction parameter, the system quasimomentum, and dimension of the lattice.
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References
A. I. Baz’, Ya. B. Zel’dovich, and A. M. Perelomov, Scattering, Reactions, and Decay in Nonrelativistic Quantum Mechanics [in Russian], Nauka, Moscow (1971); English transl. prev. ed., Israel Program for Scientific Translations, Jerusalem (1969).
A. I. Mogil’ner, “Hamiltonians in solid state physics as multiparticle discrete Schroedinger operators: Problems and results,” in: Many-Particle Hamiltonians: Spectra and Scattering (Adv. Sov. Math., Vol. 5, R. A. Minlos, ed.), Amer. Math. Soc., Providence, R. I. (1991), pp. 139–194.
E. L. Nagaev, Physics of Magnetic Semiconductors [in Russian], Nauka, Moscow (1979); English transl., Mir, Moscow (1983).
A. K. Motovilov, W. Sandhas, and V. B. Belyaev, “Perturbation of a lattice spectral band by a nearby resonance,” J. Math. Phys., 42, 2490–2506 (2001); arXiv:cond-mat/9909414v2 (1999).
S. N. Lakaev and Sh. M. Latipov, “Existence and analyticity of eigenvalues of a two-channel molecular resonance model,” Theor. Math. Phys., 169, 1658–1667 (2011).
M. Klaus and B. Simon, “Coupling constant thresholds in nonrelativistic quantum mechanics: I. Short-range two-body case,” Ann. Phys., 130, 251–281 (1980).
M. Klaus, “On the bound state of Schrödinger operators in one dimension,” Ann. Phys., 108, 288–300 (1977).
Zh. I. Abdullaev and S. N. Lakaev, “Finiteness of discrete spectrum of three particle Schrödinger operator on the lattice,” Theor. Math. Phys., 111, 467–479 (1997).
S. Albeverio, S. N. Lakaev, K. A. Makarov, and Z. I. Muminov, “The threshold effects for the two-particle Hamiltonians on lattices,” Commun. Math. Phys., 262, 91–115 (2006); arXiv:math-ph/0501013v1 (2005).
B. Simon, “The bound states of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys., 97, 279–288 (1976).
S. N. Lakaev and I. N. Bozorov, “The number of bound states of a one-particle Hamiltonian on a threedimensional lattice,” Theor. Math. Phys., 158, 360–376 (2009).
S. Albeverio, S. N. Lakaev, and Z. I. Muminov, “Schrödinger operators on lattices: The Efimov effect and discrete spectrum asymptotics,” Ann. Henri Poincaré, 5, 743–772 (2004).
S. N. Lakaev, “On Efimov’s effect in a system of three identical quantum particles,” Funct. Anal. Appl., 27, 166–175 (1993).
S. N. Lakaev and Sh. U. Alladustov, “Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice,” Theor. Math. Phys., 178, 336–346 (2014).
A. V. Sobolev, “The Efimov effect: Discrete spectrum asymptotics,” Commun. Math. Phys., 156, 101–126 (1993).
Yu. N. Ovchinnikov and I. M. Sigal, “Number of bound states of three-body systems and Efimov’s effect,” Ann. Phys., 123, 274–295 (1989).
D. R. Yafaev, “On the theory of the discrete spectrum of the three-particle Schrödinger operator,” Math. USSRSb., 23, 535–559 (1974).
H. Tamura, “Asymptotics for the number of negative eigenvalues of three-body Schrödinger operators with Efimov effect,” in: Spectral and Scattering Theory and Applications (Adv. Stud. Pure Math., Vol. 23, K. Yajima, ed.), Math. Soc. Japan, Tokyo (1994), pp. 311–322.
S. Albeverio, S. N. Lakaev, and A. M. Khalkhujaev, “Number of eigenvalues of the three-particle Schrödinger operators on lattices,” Markov Proc. Relat. Fields, 18, 387–420 (2012).
V. Bach, W. de Siqueira Pedra, and S. N. Lakaev, “Bounds on the discrete spectrum of lattice Schrödinger operators,” J. Math. Phys., 59, 022109 (2017); arXiv:1709.02966v2 [math-ph] (2017).
M. Reed and E. Simon, Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Acad. Press, New York (1979).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 198, No. 3, pp. 418–432, March, 2019. Received April 24, 2018. Revised July 2, 2018.
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Lakaev, S.N., Boltaev, A.T. Threshold Phenomena in the Spectrum of the Two-Particle Schrödinger Operator on a Lattice. Theor Math Phys 198, 363–375 (2019). https://doi.org/10.1134/S0040577919030036
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DOI: https://doi.org/10.1134/S0040577919030036