Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs

Michael Aizenman, Robert J Sims, Simone Warzel

Research output: Contribution to journalArticle

52 Citations (Scopus)

Abstract

The subject of this work is random Schrödinger operators on regular rooted tree graphs [script T sign] with stochastically homogeneous disorder. The operators are of the form H λ (ω)=T+U+λ V(ω) acting in l2(script T sign), with T the adjacency matrix, U a radially periodic potential, and V(ω) a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of H λ (ω) exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder in case U=0 and V(ω) given by iid random variables on script T sign. Our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder. The method yields the continuity of the ac spectral density of H λ (ω) at λ=0. The latter is shown to converge in the L 1-sense over closed Borel sets in which H 0 has no singular spectrum. The analysis extends to random potentials whose values at different sites need not be independent, assuming only that their joint distribution is weakly correlated across different tree branches.

Original languageEnglish (US)
Pages (from-to)363-394
Number of pages32
JournalProbability Theory and Related Fields
Volume136
Issue number3
DOIs
StatePublished - Nov 2006
Externally publishedYes

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Absolutely Continuous Spectrum
Random Operators
Disorder
Random Potential
Absolutely Continuous
Graph in graph theory
Borel Set
I.i.d. Random Variables
Periodic Potential
Rooted Trees
Spectral Density
Adjacency Matrix
Closed set
Joint Distribution
Persistence
Branch
Converge
Operator

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability
  • Analysis

Cite this

Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. / Aizenman, Michael; Sims, Robert J; Warzel, Simone.

In: Probability Theory and Related Fields, Vol. 136, No. 3, 11.2006, p. 363-394.

Research output: Contribution to journalArticle

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