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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. E. L. Lakshtanov and R. A. Minlos, The spectrum of two-particle bound states of transfer matrices of Gibbs fields (an isolated bound state), Funktsional. Anal. i Prilozhen. 38 (2004), no. 3, 52–69 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 38 (2004), no. 3, 202–216. MR 2095134, 10.1023/B:FAIA.0000042805.04113.42
  • 2. E. L. Lakshtanov and R. A. Minlos, The spectrum of two-particle bound states of transfer matrices of Gibbs fields (fields on a two-dimensional lattice: adjacent levels), Funktsional. Anal. i Prilozhen. 39 (2005), no. 1, 39–55, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 39 (2005), no. 1, 31–45. MR 2132438, 10.1007/s10688-005-0015-7
  • 3. E.A. Kudryavtceva and E.L. Lakshtanov, Classification of singularities and bifurcation of critical points of even functions, Topological methods in Hamiltonian systems, Cambridge Sci. Publ., 2005, 151-174.
  • 4. V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR 966191
  • 5. E. L. Lakshtanov, Leading branches of the transfer matrix spectrum of a general spin model with nearest-neighbor interaction. The high-temperature regime, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (2004), 3–7, 69 (Russian, with Russian summary); English transl., Moscow Univ. Math. Bull. 59 (2004), no. 6, 1–5 (2005). MR 2157594
  • 6. D. R. Yafaev, Mathematical scattering theory, Translations of Mathematical Monographs, vol. 105, American Mathematical Society, Providence, RI, 1992. General theory; Translated from the Russian by J. R. Schulenberger. MR 1180965
  • 7. Sh. S. Mamatov and R. A. Minlos, Bound states of a two-particle cluster operator, Teoret. Mat. Fiz. 79 (1989), no. 2, 163–179 (Russian, with English summary); English transl., Theoret. and Math. Phys. 79 (1989), no. 2, 455–466. MR 1007792, 10.1007/BF01016525
  • 8. J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
  • 9. C. Boldrighini, R. A. Minlos, and A. Pellegrinotti, Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive, Probab. Theory Related Fields 129 (2004), no. 1, 133–156. MR 2052866, 10.1007/s00440-003-0331-x
  • 10. A. I. Markushevich, The theory of analytic functions: a brief course, “Mir”, Moscow, 1983. Translated from the Russian by Eugene Yankovsky. MR 708893

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2000): 82C10, 47N55, 60G60, 82C20

Retrieve articles in all journals with MSC (2000): 82C10, 47N55, 60G60, 82C20


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
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