Magnetic-Sparseness and Schrödinger Operators on Graphs

Abstract

We study magnetic Schrödinger 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 ℓ 2 space. As a consequence, we get criteria of discreteness for the spectrum and eigenvalue asymptotics.

Fractional order Orlicz-Sobolev spaces

“…where, for z ∈ C n , |z| p p := |ℜz| p + |ℑz| p , | · | is the euclidean norm in R n and ℜz, ℑz denotes the real an imaginary parts of z respectively. See for instance [13] for existence results for −∆ A p and [1] for this operator in the context of graphs.…”

Magnetic fractional order Orlicz–Sobolev spaces

“…Let A : Ω → R n be a measurable magnetic potential such that |A| < ∞ a.e. in Ω and let u ∈ W 1,1 loc (R n ; C). Then the following diamagnetic inequality holds: Theorem 5.2.…”

“…where, for z ∈ C n , |z| p p := | z| p + | z| p , | • | is the Euclidean norm in R n . See for instance [13] for existence results for −∆ A p and [1] for this operator in the context of graphs.…”

Three conjectures of Ostrander on digraph Laplacian eigenvectors

“…Trofimov [54] proved that every infinite locally finite vertex-transitive graph has a nonconstant harmonic function, which grows at most exponentially with respect to the distance to a base point. Tointon [8,10,17,20,43,51]. A Schrödinger operator with a nonnegative potential on a finite digraph is either a Laplacian matrix or a perturbed Laplacian matrix.…”