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

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/β. The case of double non-degenerate extrema of the “symbol” ωΛ(k), Λ ∈ T2, k ∈ T2, is studied. The corresponding points Λ are situated on certain “double” curves on the torus T2. We also study the case of degenerate extrema ωΛ(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 ωΛ(k) “with general position” there are no adjacent levels in a neighbourhood of caustics. 1. Brief reminders This paper is a continuation of our papers [1] and [2], where we explained in detail the problem concerning two-particle bound states of transfer matrices of Gibbs fields (with compact spin space consisting of more than 2 elements and with an arbitrary compactly supported interaction along “spatial” directions) in the high-temperature region β = 1/T 1. Recall that in [1] the original problem, after several reductions, was reduced to the following problem. There is a family of Hilbert spaces {HΛ,Λ ∈ T} labelled by the points of the ν-dimensional torus T : { HΛ = C⊕ L Λ 2 , Λ ∈ T } ,