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

TL;DR

This paper establishes conditions on symbols of pseudo-differential operators on discrete lattices to ensure their boundedness, compactness, and nuclearity, and derives eigenvalue growth estimates for related elliptic operators.

Contribution

It provides new sufficient conditions linking symbol order to operator properties on $ abla ext{Z}^n$, including eigenvalue growth estimates for elliptic operators.

Findings

Conditions for boundedness, compactness, and r-nuclearity of pseudo-differential operators.

Growth estimates for eigenvalues of elliptic operators on discrete lattices.

Application to perturbed discrete Schrödinger operators.

Abstract

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 $\hbar\mathbb{Z}^n$. In addition, these conditions allow us to obtain growth estimates for the eigenvalues of some elliptic operators, in particular perturbed discrete Schr\"odinger operator.