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

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.