Criticality theory for Schrödinger operators on graphs

Abstract

We study Schrödinger operators given by positive quadratic forms on infinite graphs. From there, we develop a criticality theory for Schrödinger operators on general weighted graphs.

Optimal Hardy inequalities for Schrödinger operators on graphs

“…Note that the null/positive-criticality of a critical form depends also on the weight w. By [15,Theorem 6.2], the form h − w is null-critical with respect to w if and only if the ground state ψ of the critical form h − w cannot be approximated by compactly supported functions, i.e., by convergence with respect to h − w and pointwise convergence.…”

“…The properness assumption (a) says that u −1 0 (I) is a finite set for any compact I ⊆ u 0 (X). Since u 0 (X) ⊆ (0, ∞) for positive superharmonic functions by the Harnack inequality, [15,Lemma 4.5], the assumption implies that 0 and ∞ are the only (possible) accumulation points of the range of u 0 (and without loss of generality we may assume sup u 0 = ∞, since otherwise, we can only replace u 0 with u 0 := 1/u 0 ). For example this can be achieved if the limit of u 0 in the one-point compactification…”

“…We next employ [15,Lemma 5.11] which states that since h v 1/2 is critical and (e n ) is a null-sequence for h v 1/2 it follows that for every function f ∈ C(X) we have lim n→∞ h (e n f ) = h (f ). Hence, for f ∈ C(X) with support in X \ W , by Fatou's lemma and the inequality in the assumption (noting that e n f ∈ C c (X \ W )) we obtain…”

On families of optimal Hardy-weights for linear second-order elliptic operators

“…Remark 2.8. Criticality theory has been extended to the case of half-linear operators of Schrödinger-type 25,26], and to the case of Schrödinger operators on discrete graphs [18].…”

Coverings and the heat equation on graphs: Stochastic incompleteness, the Feller property, and uniform transience

“…Uniform transience is a strengthening of transience (as well as of the Feller property) which was recently introduced for graphs in [56], following previous work in [3,52,53,92]. This is also related to the notion of uniform subcriticality which has been studied for elliptic operators defined on domains in Euclidean space in [82] and, more recently, for Schrödinger operators on weighted graphs in [59]. Intuitively, if transience means that heat (or a random walker) escapes to infinity eventually, uniform transience means that it does so in all directions.…”