4 research outputs found
Dynamical Localization for the Random Dimer Model
We study the one-dimensional random dimer model, with Hamiltonian
, where for all and where the are i.i.d. Bernoulli
random variables taking the values . We show that, for all values
of and with probability one in , the spectrum of is pure point.
If and , the Lyapounov exponent vanishes only at the
two critical energies given by . For the particular value
, respectively , we show the existence of additional
critical energies at , resp. E=0. On any compact interval
not containing the critical energies, the eigenfunctions are then shown to be
semi-uniformly exponentially localized, and this implies dynamical
localization: for all and for all with sufficiently
rapid decrease:
Here , and is the spectral
projector of onto the interval . In particular if and
, these results hold on the entire spectrum (so that one can
take ).Comment: 14 page
Eigenvalue statistics for random Schrodinger operators with non rank one perturbations
We prove that certain natural random variables associated with the local
eigenvalue statistics for generalized lattice Anderson models constructed with
finite-rank perturbations are compound Poisson distributed. This distribution
is characterized by the fact that the Levy measure is supported on at most a
finite set determined by the rank. The proof relies on a Minami-type estimate
for finite-rank perturbations. For Anderson-type continuum models on , we
prove a similar result for certain natural random variables associated with the
local eigenvalue statistics. We prove that the compound Poisson distribution
associated with these random variables has a Levy measure whose support is at
most the set of positive integers
How much delocalisation is needed for an enhanced area law of the entanglement entropy?
We consider the random dimer model in one space dimension with Bernoulli
disorder. For sufficiently small disorder, we show that the entanglement
entropy exhibits at least a logarithmically enhanced area law if the Fermi
energy coincides with a critical energy of the model where the localisation
length diverges.Comment: 29 pages, changes in v3: correction of an error in the appendix in
Lemma A.2, typos correcte
Localization for One Dimensional, Continuum, Bernoulli-Anderson Models
We use scattering theoretic methods to prove strong dynamical and exponential
localization for one dimensional, continuum, Anderson-type models with singular
distributions; in particular the case of a Bernoulli distribution is covered.
The operators we consider model alloys composed of at least two distinct types
of randomly dispersed atoms. Our main tools are the reflection and transmission
coefficients for compactly supported single site perturbations of a periodic
background which we use to verify the necessary hypotheses of multi-scale
analysis. We show that non-reflectionless single sites lead to a discrete set
of exceptional energies away from which localization occurs.Comment: 32 page