Research of the difference Schrödinger operator for some physical models

In this paper, the discrete Schrödinger operator on a perturbed by the decreasing potential graph with vertices at the two intersecting lines is considered. We investigate spectral properties of this operator and the scattering problem for the above operator in the case of a small potential and also in the case when both a potential and velocity of a quantum particle are small. Asymptotic formulas for the probabilities of the particle propagation in all possible directions are obtained. In addition, we investigate the spectral properties of the discrete Schrödinger operator for the infinite band with zero boundary conditions. The scattering pattern is described. Simple formulas for transmission and reflection coefficients near boundary points of the subbands (this corresponds to small velocities of quantum particles) for small potentials are obtained. We consider a one-particle discrete Schrödinger operator with a periodic potential perturbed by a function which is periodic in two variables and exponentially decreases in third variable. In the paper, we also investigate the scattering problem for this operator near the extreme point of the eigenvalue of the periodic Schrödinger operator in the cell with respect to the third component of the quasimomentum, i.e. for the small perpendicular component of the angle of incidence of a particle on the potential barrier. Simple formulas of the propagation and reflection probabilities are obtained.