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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

The spectrum of two-particle bound states of transfer matrices of Gibbs fields. Part 3: Fields on the three-dimensional lattice


Authors: E. L. Lakshtanov and R. A. Minlos
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 69 (2008).
Journal: Trans. Moscow Math. Soc. 2008, 255-288
MSC (2000): Primary 82C10; Secondary 47N55, 60G60, 82C20
Published electronically: November 19, 2008
MathSciNet review: 2549449
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Abstract: In this paper, which continues our earlier papers, we investigate so-called ``adjacent'' bound states (that is, bound states which only appear in a neighbourhood of special values of the total quasimomentum of the system) of the transfer matrix of the general spin model on the 3-dimensional lattice in its two-particle subspace for high temperatures $ T=1/ \beta$. The case of double non-degenerate extrema of the ``symbol'' $ \omega_\Lambda(k)$, $ \Lambda \in \mathbb{T}^2$, $ k \in \mathbb{T}^2$, is studied. The corresponding points $ \Lambda$ are situated on certain ``double'' curves on the torus $ \mathbb{T}^2$. We also study the case of degenerate extrema $ \omega_\Lambda(k)$ situated on caustic curves on the torus. In the first case, conditions under which adjacent levels appear are indicated and the size of a neighbourhood of ``double'' curves where these levels ``live'' is estimated. In the second case, it is shown that for a degenerate extremum of $ \omega_\Lambda(k)$ ``with general position'' there are no adjacent levels in a neighbourhood of caustics.


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Additional Information

E. L. Lakshtanov
Affiliation: Aveiro University, Portugal
Email: lakshtanov@rambler.ru

R. A. Minlos
Affiliation: Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
Email: minl@iitp.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-08-00171-4
PII: S 0077-1554(08)00171-4
Keywords: Adjacent bound states, Gibbs fields, transfer matrix, lattice, spectrum, Fredholm determinant, total quasimomentum
Published electronically: November 19, 2008
Additional Notes: The first author thanks Frau Cristel Schröder for hospitality during his visit to Technische Universität München, where the main part of this paper was completed. This research was partially supported by the Centre for Research on Optimization and Control in Fundação para a Ciência e a Tecnologia and by the European Community Fund FEDER/POCTI
The second author thanks the Russian Foundation for Basic Research for financial support (grant no. 05-01-00449)
Article copyright: © Copyright 2008 American Mathematical Society