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This monograph deals with the spectral properties of discrete Schrödinger operators (Hamiltonians) suitable for the system of (quantum) particles in the optical lattice. It should be noted that lattice systems, even when one-particle, have a variety of effects. One the main mechanisms in the theoretical justification of experimental results in various fields such as physics of solid bodies, physics of high-energy systems, and physics of ultracold optical systems. Currently, the spectral properties of operetors consisting of set of Laurent-Toeplitz-type convolution operator and multiplication operator into account only the zero-range and one-range interactions are important place in a wide class of physical models. The threshold effect be the basis in mathematical physics for the Efimov effect (in three-dimensional space) and the super-Efimov effect (in two-dimensional antisymmetric spaces). This monograph can be used by mathematicians, those working in the field of mathematical physics., i.e. in the spectral theory of self-adjoint operators, quantum mechanics, solid state physics, quantum field theory.