Global multiplicity bounds and spectral statistics for random operators
13 Feb 2020-Reviews in Mathematical Physics (World Scientific Publishing Company)-Vol. 32, Iss: 09, pp 2050025
TL;DR: In this paper, the authors consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on ℝ.
Abstract: In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on ℝ. We show that spectral...
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References
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TL;DR: In this article, a simple model for spin diffusion or conduction in the "impurity band" is presented, which involves transport in a lattice which is in some sense random, and in them diffusion is expected to take place via quantum jumps between localized sites.
Abstract: This paper presents a simple model for such processes as spin diffusion or conduction in the "impurity band." These processes involve transport in a lattice which is in some sense random, and in them diffusion is expected to take place via quantum jumps between localized sites. In this simple model the essential randomness is introduced by requiring the energy to vary randomly from site to site. It is shown that at low enough densities no diffusion at all can take place, and the criteria for transport to occur are given.
9,647 citations
"Global multiplicity bounds and spec..." refers background or methods in this paper
...In the mid-fifties Anderson [1] proposed that for large disorder the models on l(Z) should exhibit localization....
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...In case of −∆, another method is to use smooth basis from L([0, 1]) and translate them using Z action to get the desired result....
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...d random variables following uniform distribution on [0, 1]....
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TL;DR: In this article, it was shown that the Green's function of the Anderson tight binding Hamiltonian decays exponentially fast at long distances on Ω v ≥ 0, with probability 1.
Abstract: We prove that the Green's function of the Anderson tight binding Hamiltonian decays exponentially fast at long distances on ℤ
v
, with probability 1. We must assume that either the disorder is large or the energy is sufficiently low. Our proof is based on perturbation theory about an infinite sequence of block Hamiltonians and is related to KAM methods.
731 citations
"Global multiplicity bounds and spec..." refers methods in this paper
...Several rigorous results on localization followed from the early eighties starting with the work of Fröhlich-Spencer [2] who formulated multi scale analysis....
[...]
TL;DR: In this paper, the authors presented a short proof of localization under the conditions of either strong disorder (λ > λ 0) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting inl 2 spaces.
Abstract: The work presents a short proof of localization under the conditions of either strong disorder (λ > λ0) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting inl
2 spaces. A prototypical example is the discrete Schrodinger operatorH=−Δ+U
0(x)+λV
x
onZ
d
,d≧1, withU
0(x) a specified background potential and {V
x
} generated as random variables. The general results apply to operators with −Δ replaced by a non-local self adjoint operatorT whose matrix elements satisfy: ∑
y
|T
x,y
|
S
≦Const., uniformly inx, for somes<1. Localization means here that within a specified energy range the spectrum ofH is of the pure-point type, or equivalently — the wave functions do not spread indefinitely under the unitary time evolution generated byH. The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elementsT
x, y
. Under rapid decay ofT
x, y
, the corresponding eigenfunctions are also proven to decay exponentially. The method is based on resolvent techniques. The central technical ideas include the use of low moments of the resolvent kernel, i.e. <|G
E
(x, y)|
s
> withs small enough (<1) to avoid the divergence caused by the distribution's Cauchy tails, and an effective use of the simple form of the dependence ofG
E
(x, y) on the individual matrix elements ofH in elucidating the implications of the fundamental equation (H−E)G
E
(x,x
0)=δ
x,x0
. This approach simplifies previous derivations of localization results, avoiding the small denominator difficulties which have been hitherto encountered in the subject. It also yields some new results which include localization under the following sets of conditions: i) potentials with an inhomogeneous non-random partU
0
(x), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms (T
x,y≈1/|x−y|(d+e)), and iv) localization produced by disordered boundary conditions.
701 citations
"Global multiplicity bounds and spec..." refers background in this paper
...There are several papers on local spectral statistics on discrete models such as [4, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]....
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...Some of the papers on localization for large disorder are [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and [14, 15, 16, 17]....
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TL;DR: In this paper, the authors considered a selfadjoint operator and a self-adjoint rank-one projection onto a vector, which is cyclic for A. They gave necessary and sufficient conditions for A + λ P to have empty singular continuous spectrum or to have only point spectrum for a.
Abstract: We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμAφ, we give necessary and sufficient conditions for A + λ P to have empty singular continuous spectrum or to have only point spectrum for a.e. λ. We apply these results to questions of localization in the one- and multi-dimensional Anderson models.
400 citations
"Global multiplicity bounds and spec..." refers background in this paper
...Some of the papers on localization for large disorder are [2], [44], [19], [16] [23], [13, 14], [1], [30] and [11] and [9]....
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TL;DR: In this paper, it was shown that for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues.
Abstract: We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrodinger operators in the continuum.
358 citations
"Global multiplicity bounds and spec..." refers methods in this paper
...localization followed from the early eighties starting with the work of Frohlich-Spencer [20] who formulated multi scale analysis. Some of the papers on localization for large disorder are [2], [44], [19], [16] [23], [13, 14], [1], [30] and [11] and [9]. We refer to any of [10], [18], [29], [45] and [4]. The next set ofquestions concern thesimplicity ofthe Lebesgue components of spectrum and in this d...
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