July 2019 Rank-one perturbations and Anderson-type Hamiltonians
Constanze Liaw
Banach J. Math. Anal. 13(3): 507-523 (July 2019). DOI: 10.1215/17358787-2019-0001


Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators Hω=H+Vω on a separable Hilbert space H, where the perturbation is given by Vω=nωn(,φn)φn with a sequence {φn}H and independent identically distributed random variables ωn. We show that the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank-one perturbation. This result connects one of the least trackable perturbation problem (with almost surely noncompact perturbations) with one where the perturbation is “only” of rank-one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has nonzero Lebesgue measure.


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Constanze Liaw. "Rank-one perturbations and Anderson-type Hamiltonians." Banach J. Math. Anal. 13 (3) 507 - 523, July 2019. https://doi.org/10.1215/17358787-2019-0001


Received: 31 August 2018; Accepted: 7 January 2019; Published: July 2019
First available in Project Euclid: 27 February 2019

zbMATH: 07083758
MathSciNet: MR3978934
Digital Object Identifier: 10.1215/17358787-2019-0001

Primary: 81Q10
Secondary: 30E20 , 47B80 , 82B44

Keywords: Anderson-type Hamiltonian , discrete random Schrödinger operator , Krein–Lifshits spectral shift , rank-one perturbations

Rights: Copyright © 2019 Tusi Mathematical Research Group


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Vol.13 • No. 3 • July 2019
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