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

E. L. Lakshtanov ; R. A. Minlos

Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 39-55

Résumé

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.

Détail

Mots-clés : transfer matrices
Keywords: bound state, Fredholm operator, total quasimomentum, adjacent level.