Magnetic sparseness and Schr\"odinger operators on graphs

TL;DR

This paper investigates magnetic Schr"odinger operators on graphs, introducing a magnetic sparseness concept linked to the frustration index, which characterizes the spectral properties and eigenvalue behavior of these operators.

Contribution

It extends graph sparseness to include magnetic effects via the frustration index, establishing equivalence with the form domain being an 2 space and deriving spectral criteria.

Findings

Magnetic sparseness is equivalent to the form domain being 2.

Criteria for spectrum discreteness are established.

Eigenvalue asymptotics are derived.

Abstract

We study magnetic Schr\"odinger operators on graphs. We extend the notion of sparseness of graphs by including a magnetic quantity called the frustration index. This notion of magnetic sparse turn out to be equivalent to the fact that the form domain is an $\ell^{2}$ space. As a consequence, we get criteria of discreteness for the spectrum and eigenvalue asymptotics.