Cluster expansion of the resolvent for the Schrödinger operator on non-percolating graphs with applications to Simon–Spencer type theorems and localization

Abstract

The paper contains a generalization of the well-known 1D results on the absence of the a.c. spectrum ( in the spirit of the Simon–Spencer theorem) and localization to the wide class of “non-percolating” graphs, which include the Sierpiński lattice and quasi 1D trees. The main tools are cluster expansion of the resolvent and real analytic techniques (Kolmogorov’s lemma and similar estimates).