On spectral estimates for the Schr\"odinger operators in global dimension 2

TL;DR

This paper reviews recent advances in eigenvalue estimates for Schr"odinger operators in two dimensions and extends these results to operators on combinatorial and metric graphs related to the lattice Z^2.

Contribution

It establishes new eigenvalue estimate counterparts for Schr"odinger operators on combinatorial and metric graphs based on the lattice Z^2.

Findings

Reviewed recent eigenvalue estimates for 2D Schr"odinger operators.

Extended these estimates to operators on lattice-based graphs.

Provided theoretical foundations for spectral analysis on graph structures.

Abstract

The problem of finding eigenvalue estimates for the Schr\"odinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. We discuss these results and establish their counterparts for the operators on the combinatorial and metric graphs corresponding to the lattice Z^2.