Optimal Hardy inequalities for Schrödinger operators on graphs

Abstract

For a given subcritical discrete Schrödinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H − λw is subcritical in X for all λ < 1, null-critical in X for λ = 1, and supercritical near any neighborhood of infinity in X for any λ > 1. Our results rely on a criticality theory for Schrödinger operators on general weighted graphs.

An Improved Discrete Hardy Inequality

“…Remark 0.1. In [1] we show that (0.3) cannot be improved and is optimal in a certain sense. In particular, there is no function w w such that (0.3) holds with w instead of w.…”

A note on the surjectivity of operators on vector bundles over discrete spaces

“…If q 0 satisfies some Hardy inequality, see e.g. the discussion in [10,11], or the graph has positive Cheeger constant, see e.g. [1], then also certain V without a fixed sign induce a nonnegative form q V .…”

Shnol-type theorem for the Agmon ground state

“…Additionally, the authors show the existence of a weak positive solution (Allegretto-Piepenbrink-type theorem) by approximating such a solution under suitable assumptions on the Dirichlet form. In the following, we focus on regular Dirichlet forms over a discrete set X since the existence of a ground state with corresponding null-sequence is known under reasonable assumption, see [KL12,KPP16]. The statements of Theorem 3.3 stay valid in this setting while some estimates need to be adjusted since the chain rule does not hold for discrete Laplacians, see [KPP16].…”

“…Furthermore, a sequence (ϕ n ) of nonnegative functions in C 0 (X) is called null-sequence if there is an o ∈ X and a constant C > 0 such that ϕ n (o) = C for each n ∈ N and Q(ϕ n ) → 0. Note that for a nonnegative quadratic form Q the existence of a positive harmonic function is guaranteed in the case of locally finite graphs [HK11], or in the critical case [KPP16]. In the following, the space of bounded real-valued functions on X equipped with the uniform norm is denoted by B(X, R).…”