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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).
Cite this article
Stanislav Molchanov, Lukun Zheng, Cluster expansion of the resolvent for the Schrödinger operator on non-percolating graphs with applications to Simon–Spencer type theorems and localization. J. Spectr. Theory 7 (2017), no. 3, pp. 733–770