# Decorrelation estimates for random Schrödinger operators with non rank one perturbations

11 Feb 2021-

About: The article was published on 2021-02-11 and is currently open access. It has received None citations till now. The article focuses on the topics: Rank (linear algebra) & Compound Poisson distribution.

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TL;DR: In this paper, the authors present a general theory of Levy processes and a stochastic calculus for Levy processes in a direct and accessible way, including necessary and sufficient conditions for Levy process to have finite moments.

Abstract: Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

2,908 citations

### "Decorrelation estimates for random ..." refers methods in this paper

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TL;DR: In this paper, the Anderson tight binding model was considered and the exponential decay of the fractional moment of the Green function was shown to hold for complex energies near the energy where Anderson localization is expected.

Abstract: We consider the Anderson tight binding modelH=−Δ+V acting inl
2(Z
d
) and its restrictionH
Λ to finite hypercubes Λ⊂Z
d
. HereV={V
x
;x∈Z
d
} is a random potential consisting of independent identically distributed random variables. Let {E
j
(Λ)}
j
be the eigenvalues ofH
Λ, and let ξ
j
(Λ,E)=|Λ|(E
j
(Λ)−E),j≧1, be its rescaled eigenvalues. Then assuming that the exponential decay of the fractional moment of the Green function holds for complex energies nearE and that the density of statesn(E) exists atE, we shall prove that the random sequence {ξ
j
(Λ,E)}
j
, considered as a point process onR
1, converges weakly to the stationary Poisson point process with intensity measuren(E)dx as Λ gets large, thus extending the result of Molchanov proved for a one-dimensional continuum random Schrodinger operator. On the other hand, the exponential decay of the fractional moment of the Green function was established recently by Aizenman, Molchanov and Graf as a technical lemma for proving Anderson localization at large disorder or at extreme energy. Thus our result in this paper can be summarized as follows: near the energyE where Anderson localization is expected, there is no correlation between eigenvalues ofH
Λ if Λ is large.

261 citations

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TL;DR: In this paper, a family of finite-volume criteria is presented for the regime where the fractional moment decay condition holds for spectral band edges, provided there are sufficient LIFshitz tail estimates on the density of states.

Abstract: A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient “Lifshitz tail estimates” on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the “multiscale analysis”. In the converse direction, the analysis rules out fast power-law decay of the Green functions at mobility edges.

225 citations

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TL;DR: In this paper, the authors showed that there is no repulsion between energy levels of the Schrodinger operator and the corresponding wave functions, and that the repulsion of the wave functions is independent of the energy levels.

Abstract: Let
$$H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )$$
be an one-dimensional random Schrodinger operator in ℒ2(−V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x
t
), wherex
t
is a Brownian motion on the compact Riemannian manifoldK andF:K→R
1 is a smooth Morse function,
$$\mathop {\min }\limits_K F = 0$$
. Let
$$N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 $$
, where Δ∈(0, ∞),E
i
(V) are the eigenvalues ofH
V
. The main result (Theorem 1) of this paper is the following. IfV→∞,E
0>0,k∈Z
+ anda>0 (a is a fixed constant) then
$$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$
wheren(E
0) is a limit state density ofH
V
,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH
V
,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.

154 citations

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