Landscape Theory for Schr\"odinger Operators with General Hopping Terms on a Finite Lattice

TL;DR

This paper introduces a power series expansion method to analyze discrete Schr"odinger operators, extending previous results on invertibility and eigenvector bounds to long-range and higher-dimensional lattice cases.

Contribution

It presents an alternative proof technique using power series expansion, generalizing landscape theory results to more complex and higher-dimensional lattice operators.

Findings

Extended landscape function bounds to long-range operators

Demonstrated applicability to higher-dimensional lattices

Provided a new proof method based on power series expansion

Abstract

Findings by M. L. Lyra, S. Mayboroda and M. Filoche relate invertibility and positivity of a class of discrete Schr\"odinger matrices with the existence of the "Landscape Function", which provides an upper bound on all eigenvectors simultaneously. Their argument is based on the variational principles. We consider an alternative method of proving these results, based on the power series expansion, and demonstrate that it naturally extends the original findings to the case of long range operators. The method of proof by power series expansion can also be employed in other scenarios, such as higher dimensional lattices.