Lp boundedness, r-nuclearity and approximation of pseudo-differential operators on $\mathrm{\hslash }{\mathbb{Z}}^n$

In this work sufficient conditions on the order of the symbol are developed to ensure boundedness, compactness and r-nuclearity of pseudo-differential operators in $\mathrm{\hslash }{\mathbb{Z}}^n$. In addition, these conditions allow us to obtain growth estimates for the eigenvalues of some elliptic operators, in particular perturbed discrete Schrödinger operator.