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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. These operators contain very important information about the properties of a system and analyzing the discrete and essential spectra of the operators are of particular significance in solid state physics, quantum mechanics and particle physics. Particularly, investigating the spectrum of families of energy operators corresponding to the Hamiltonians of systems of one, two and many quantum particles on lattices is very important. These studies include finding the location of the essential spectrum of the discrete Schrödinger operators, obtaining sufficient conditions for the existence of eigenvalues (bound state energies) outside the essential spectrum 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, in particular, solutions of problems related to the spectrum of Hamiltonians of systems of two and three particles on a lattice.