The Spectrum of Two-Particle Bound States for the Transfer Matrices of Gibbs Fields (an Isolated Bound State)

This paper initiates a general study of the spectrum of two-particle bound states of transfer matrices for a fairly wide class of Gibbs fields at high temperature $T$. In the present first part of this study, a detailed statement of the problem is given and the existence of a so-called “isolated level” lying at a distance $\sim 1/{T^2}$ from the boundary of the continuous spectrum is established for all values of the total quasimomentum $\Lambda$ of the system. In the concluding part of the paper, we prove that there are no other bound states provided that $\Lambda$ is far from certain singular values. In the second part, we will consider bound states for $\Lambda$ close to the singular values. The distance from these states (adjacent levels, in the authors' terminology) to the continuous spectrum is at most of the order of $1/T^4$.