Bounds on the number of bound states in the transfer matrix spectrum for some weakly correlated lattice models

Abstract

We consider the interaction of particles in weakly correlated lattice quantum field theories. In the imaginary time functional integral formulation of these theories there is a relative coordinate lattice Schroedinger operator H which approximately describes the interaction of these particles. Scalar and vector spin, QCD and Gross-Neveu models are included in these theories. In the weakly correlated regime H = Ho + W where Ho = −γΔl, 0 < γ ≪ 1 and Δl is the d-dimensional lattice Laplacian: γ = β, the inverse temperature for spin systems and γ = κ3 where κ is the hopping parameter for QCD. W is a self-adjoint potential operator which may have non-local contributions but obeys the bound |W(x, y)| ⩽ cexp ( − a(|x| + |y|)), a large: $\exp \left( { - a} \right) = \left| {\beta /\beta _o } \right|^{\frac{1}{2}} (\left| {\kappa /\kappa _o } \right|)$exp−a=β/βo12(κ/κo) for spin (QCD) models. Ho, W, and H act in l2(Zd), d ⩾ 1. The spectrum of H below zero is known to be discrete and we obtain bounds on the number of states below zero. This number depends on the short range properties of W, i.e., the long range tail does not increase the number of states.