Geodesic
Two-Particle Bound State Spectrum of Transfer Matrices for Gibbs Fields (Fields on the Two-Dimensional Lattice. Adjacent Levels)
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 39-55.

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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 $|\Lambda-\Lambda_j^{\text{\textup{mult}}}| c/T^2$ (here $c$ is a constant), where $\Lambda_j^{\text{\textup{mult}}}$ 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.
Mots-clés : transfer matrices
Keywords: bound state, Fredholm operator, total quasimomentum, adjacent level. 
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     title = {Two-Particle {Bound} {State} {Spectrum} of {Transfer} {Matrices} for {Gibbs} {Fields} {(Fields} on the {Two-Dimensional} {Lattice.} {Adjacent} {Levels)}},
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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). Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 39-55. http://geodesic.mathdoc.fr/item/FAA_2005_39_1_a3/

[1] Lakshtanov E. L., Minlos R. A., “Spektr dvukhchastichnykh svyazannykh sostoyanii transfer-matrits gibbsovskikh polei (uedinennoe svyazannoe sostoyanie)”, Funkts. analiz i ego pril., 38:3 (2004), 52–69 | DOI | MR | Zbl

[2] Kudryavtseva E. A., Lakshtanov E., “Classification of singularities and bifurcations of critical points of even functions”, Topological Methods in Theory of Integrable Systems, eds. A. V. Bolsinov, A. T. Fomenko, A. A. Oshemkov, Cambridge Scientific Publishers, 2005 | MR

[3] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii, Nauka, M. | MR

[4] Abdullaev J., Minlos R. A., “An extension of the Ising Model”, Probability Contributions to Statistical Mechanics, Adv. Sov. Math., 20, Amer. Math. Soc., Providence, RI, 1994, 1–20 | MR | Zbl

[5] Lakshtanov E. L., “Starshie vetvi spektra transfer-matritsy dlya obschikh spinovykh modelei s vzaimodeistviem na odin shag”, Vestnik MGU, ser. I, matem., mekh., 2004, no. 6, 3–7 | MR | Zbl

[6] Markushevich A. I., Kratkii kurs teorii analiticheskikh funktsii, Gostekhizdat, M., 1957 | MR