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 Funktsional. Anal. i Prilozhen., 2005, Volume 39, Issue 1, Pages 39–55 (Mi faa30)

Two-Particle Bound State Spectrum of Transfer Matrices for Gibbs Fields (Fields on the Two-Dimensional Lattice. Adjacent Levels)

E. L. Lakshtanova, R. A. Minlosb

a M. V. Lomonosov Moscow State University
b Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: This paper is a continuation of the authors' paper published in no. 3 of this journal in the previous year, where a detailed statement of the problem on the two-particle bound state spectrum of transfer matrices was given for a wide class of Gibbs fields on the lattice $\mathbb Z^{\nu+1}$ in the high-temperature region $(T \gg 1)$. In the present paper, it is shown that for $\nu=1$ the so-called “adjacent” bound state levels (i.e., those lying at distances of the order of $T^{-\alpha}$, $\alpha>2$, from the continuous spectrum) can appear only for values of the total quasimomentum $\Lambda$ of the system that satisfy the condition $|\Lambda-\Lambda_j^{\textup{mult}}|< c/T^2$ (here $c$ is a constant), where $\Lambda_j^{\textup{mult}}$ are the quasimomentum values for which the symbol $\{\omega_\Lambda(k), k \in \mathbb T^1\}$ has two coincident extrema. Conditions under which such levels actually appear are also presented.

Keywords: transfer matrices, bound state, Fredholm operator, total quasimomentum, adjacent level

DOI: https://doi.org/10.4213/faa30

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English version:
Functional Analysis and Its Applications, 2005, 39:1, 31–45

Bibliographic databases:

UDC: 517.9

Citation: E. L. Lakshtanov, R. A. Minlos, “Two-Particle Bound State Spectrum of Transfer Matrices for Gibbs Fields (Fields on the Two-Dimensional Lattice. Adjacent Levels)”, Funktsional. Anal. i Prilozhen., 39:1 (2005), 39–55; Funct. Anal. Appl., 39:1 (2005), 31–45

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa30
• https://doi.org/10.4213/faa30
• http://mi.mathnet.ru/eng/faa/v39/i1/p39

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This publication is cited in the following articles:
1. M. I. Muminov, “Positivity of the two-particle Hamiltonian on a lattice”, Theoret. and Math. Phys., 153:3 (2007), 1671–1676
2. Albeverio S., Lakaev S.N., Muminov Z.I., “The threshold effects for a family of Friedrichs models under rank one perturbations”, J. Math. Anal. Appl., 330:2 (2007), 1152–1168
3. S. N. Lakaev, I. N. Bozorov, “The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice”, Theoret. and Math. Phys., 158:3 (2009), 360–376
4. S. N. Lakaev, Sh. Yu. Kholmatov, “Asymptotics of the eigenvalues of a discrete Schrödinger operator with zero-range potential”, Izv. Math., 76:5 (2012), 946–966
5. M. I. Muminov, A. M. Hurramov, “Spectral properties of a two-particle Hamiltonian on a lattice”, Theoret. and Math. Phys., 177:3 (2013), 1693–1705
6. M. I. Muminov, A. M. Hurramov, “Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice”, Theoret. and Math. Phys., 180:3 (2014), 1040–1050
7. M. E. Muminov, A. M. Khurramov, “Spectral properties of two particle Hamiltonian on one-dimensional lattice”, Ufa Math. J., 6:4 (2014), 99–107
8. Muminov M.I., Khurramov A.M., “Spectral properties of a two-particle Hamiltonian on a d-dimensional lattice”, Nanosyst.-Phys. Chem. Math., 7:5 (2016), 880–887
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