On the spectrum of the discrete Laplacian under small perturbations

The aim of research work is to find a sufficient condition for the existence of eigenvalues of discrete Schrödinger operator with finite-range potential on a lattice, to determine the location of the essential spectrum, and to analyze the eigenvalues of the operators.
In this talk we determine the location of the essential spectrum of the discrete Schrödinger operator with a point potential on a $d$-dimensional lattice, and find a sufficient condition for existence of the operator’s eigenvalue; we derive an asymptotic formula for the eigenvalue of the discrete Schrödinger operators depending on the parameters of potential and kinetic energies of the operator; then we find a sufficient condition for the existence of eigenvalue of discrete Schrödinger operator with two delta potentials on a one-dimensional lattice and study the number of eigenvalues to the left of the essential spectrum of discrete Schrödinger operator, depending on two parameters, and discuss boundary eigenvalues and resonances.