### 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 l^{2}(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 language | English (US) |
---|---|

Pages (from-to) | 363-394 |

Number of pages | 32 |

Journal | Probability Theory and Related Fields |

Volume | 136 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2006 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Analysis

### Cite this

*Probability Theory and Related Fields*,

*136*(3), 363-394. https://doi.org/10.1007/s00440-005-0486-8

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

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 136, no. 3, pp. 363-394. https://doi.org/10.1007/s00440-005-0486-8

}

TY - JOUR

T1 - Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs

AU - Aizenman, Michael

AU - Sims, Robert J

AU - Warzel, Simone

PY - 2006/11

Y1 - 2006/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33747884265&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747884265&partnerID=8YFLogxK

U2 - 10.1007/s00440-005-0486-8

DO - 10.1007/s00440-005-0486-8

M3 - Article

AN - SCOPUS:33747884265

VL - 136

SP - 363

EP - 394

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3

ER -