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In the space L2(T ν x T ν ), where T ν is a ν-dimensional torus, we study the spectral properties of the "threeparticle" discrete Schrödinger operator H = H0 + H1 + H2, where H0 is the operator of multiplication by a function and H1 and H2 are partial integral operators. We prove several theorems concerning the essential spectrum of H. We study the discrete and essential spectra of the Hamiltonians H t and h arising in the Hubbard model on the three-dimensional lattice.Keywords: discrete Schrödinger operator, Hubbard model, discrete spectrum of a discrete operator, essential spectrum of a discrete operator

Abstract. In this paper we construct several models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35 (secondary)

We study the existence of an infinite number of eigenvalues for a model "three-particle" Schrödinger operator H. We prove a theorem on the necessary and sufficient conditions for the existence of an infinite number of eigenvalues of the model operator H below the lower boundary of its essential spectrum.

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