## Abstract

We study self adjoint operators of the form H_{ω} = H_{0} + Σλ_{ω}(n) 〈δ_{n}, ·〉δ_{n}, where the δ_{n}'s are a family of orthonormal vectors and the λ_{ω}(n)'s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n, m), if the cyclic subspaces corresponding to the vectors δ_{n} and δ_{m} are not completely orthogonal, then the restrictions of H_{ω} to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that "well behaved" absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases.

Original language | American English |
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Pages (from-to) | 561-577 |

Number of pages | 17 |

Journal | Inventiones Mathematicae |

Volume | 141 |

Issue number | 3 |

DOIs | |

State | Published - 2000 |