Ian Zimmermann

Titel

An introduction to criticality theory

Abstract

A non−negative quadratic form E on an L² space (one may think, for example, of the Dirichlet energy on Euclidean space with a potential V, i.e. E(u) = ∫ |∇ u|² + V ∫ u² ) is said to be subcritical if it satisfies a so−called Hardy inequality, that is to say, if there exists a positive weight w such that E(u) ≥ ∫ u² w for all u. Otherwise, the form is called critical. In the case of graphs, criticality of the form turns out to be equivalent to recurrence of the graph with which the form is associated. The goal of this talk is to look at some connections between criticality of a form and the Green's function associated with the form, both in the quadratic case and for more general forms.