The Hvz Theorem for N -particle Schrödinger Operators on Lattice

The N -particle Schrödinger operator H(K), K ∈ (−π, π], K being the total quasi-momentum, with short-range pair potentials on lattice Z, d ≥ 1, is considered. For fixed total quasimomentum K , the structure of H(K) ’s essential spectrum is described and the analogue of the Hunziker – van Winter – Zhislin (HVZ) theorem is proved. INTRODUCTION In the sixties, fundamental results on the essential spectrum of the manyparticle continuous Schrödinger operators were obtained by Hunziker[1], van Winter [11], Zhislin[12]. The theorem describing the essential spectrum for a system of many particles was named the HVZ theorem in honor of these three authors. It affirms that the essential spectrum of an N particle Hamiltonian (after separating off the free center-of-mass motion) is bounded below by the lowest possible energy which two independent subsystems can have. Since then the result was generalized in many ways (see [2, 4, 6, 14, 17, 23] and for more extensive references [16]). For the multiparticle Chandrasekhar operators the HVZ theorem was proved in [18]. The HVZ theorem for atomic Brown-Ravenhall operators in the Born-Oppenheimer approximation was obtained in [19, 20, 21] in terms of two-cluster decompositions. In [22], the HVZ theorem is proved for a wide class of models which are obtained by projecting of multiparticle Dirac operators to subspaces dependent on the external electromagnetic field. In the continuous case one method of proving the HVZ theorem is the use of the diagrammatic techniques and the Weinberg-Van Winter equations through the theory of integral equations of Hilbert Schmidt kernel (see e.g. [3, 11, 1, 16]). Another set of equations for the resolvent are the FaddeevYakubovky equations [24, 25]. These equations became the base for the creation of new computing techniques in nuclear and atomic physics. Technically simple proofs of the HVZ theorem were given in [6]. The key idea