Anderson model with decaying randomness: mobility edge

Kirsch, Werner; Krishna, M.; Obermeit, J.

doi:10.1007/s002090000136

Cited by 34 publications

(36 citation statements)

References 22 publications

(33 reference statements)

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“…To gain insight into this fundamental question, one may impose a decaying envelope on the ergodic random potential, and study the absolutely continuous spectrum for the new Schrödinger operator with random decaying potential as a step towards the understanding the original problem [Kr,KKO,B1,B2,RoS,2 A. FIGOTIN, F. GERMINET, A. KLEIN, AND P. MÜLLER De, BoSS, Ch]. Relaxing the decay conditions, one hopes to get an idea of the nature of the continuous spectrum for the original ergodic random Schrödinger operator.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Persistence of Anderson Localization in Schroedinger Operators With Decaying Random Potentials

Figotin¹,

Germinet²,

Klein³

et al. 2006

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Abstract. We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than |x| −2 at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as |x| −α at infinity, we determine the number of bound states below a given energy E < 0, asymptotically as α ↓ 0. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent α; (b) dynamical localization holds uniformly in α.

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Section: Introduction and Resultsmentioning

confidence: 99%

Persistence of Anderson Localization in Schroedinger Operators With Decaying Random Potentials

Figotin¹,

Germinet²,

Klein³

et al. 2006

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“…Recently, several works have been devoted to random decaying potentials. Of these we would like to mention Krishna [16] and Kirsch, Krishna, and Obermeit [15]. However, it seems that these investigations are very different from the present one, both in terms of their objectives as well as their techniques.…”

Section: )mentioning

confidence: 56%

“…However, it seems that these investigations are very different from the present one, both in terms of their objectives as well as their techniques. In fact, the results in [15,16] do not cover the potential (1.2), as they typically require that (E|V(x)| 2 ) 1/2 ≤ C|x| −1−ε .…”

Section: )mentioning

confidence: 99%

Untitled

Rodnianski¹,

Schlag²

2003

Internat. Math. Res. Notices

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The solution to that PDE is then used as a phase function of a parametrix U(t) to the time evolution e itH . More precisely, Hörmander defines U(t)φ (x) := e i[x·ξ−S(t,ξ)] φ(ξ)dξ (1.9)for Schwartz functions φ ∈ S(R d ) with supp( φ) ⊂ R d \ {0}. A novel feature of the scattering theory of (1.2) is the introduction of a time-dependent amplitude function in the parametrix. Since D 2 V(x) does not decay like |x| −2−ε , we can easily show that the parametrix (1.9) does not apply here. Rather, we definewith a suitable a(t, ξ).Generally speaking, the gain of 1/4 in terms of the conditions on the potential is achieved throughout this paper by systematically exploiting averaging in the potential.As an example, consider the Hamilton-Jacobi PDE (1.8). It is well known that initial conditions can be chosen such that DS(t, ξ) is a trajectory that escapes at a linear rate to infinity. Therefore, any integral like 2T T V DS(t, ξ) dt (1.11)is going to be significantly smaller (roughly by T −1/4 if β = 1/2) than the one with absolute values inside due to the mean-zero assumption on the randomness. Similarly, integrating the ODE (1.7) will lead to the expression ∞ t ∇V(x(s))ds. In contrast to the deterministic theory that only exploits the size of ∇V(x(s)) we need to invoke cancellations.It is well known that the existence of (modified) wave operators has strong spectral implications. More precisely, Ω + is an isometry that gives a unitary equivalence between and the restriction of H to a subspace of L 2 (R d ). The Weyl criteria implies that even the deterministic essential spectrum σ ess (H) of the Schrödinger operator with potential (1.2) coincides with the essential spectrum of the unperturbed operator −∆,. Therefore, we obtain that with probability onewhere σ a.c. denotes the absolutely continuous spectrum of H. It seems natural to believe at Cornell University Library on July 23, 2015 http://imrn.oxfordjournals.org/ Downloaded from at Cornell University Library on July 23, 2015 http://imrn.oxfordjournals.org/ Downloaded from Classical and Quantum Scattering 247 Our interest in this question arose in connection with a conjecture about square integrable potentials, see Simon's review [20]: if14) then − + V has a.c. spectrum essentially supported on [0, ∞). That conjecture would imply that the 1/2-model should have a.c. spectrum essentially supported on [0, ∞) for any realization of the ω n . In one dimension this has been shown by Deift and Killip [8]. In a series of several papers, Christ and Kiselev [6, 7] had previously settled the one-dimensional problem for potentials satisfying ∞ −∞ |V(x)| p dx < ∞, where 1 ≤ p < 2. Moreover, Christ and Kiselev have recently constructed wave operators in the onedimensional case under basically optimal conditions [5]. Another interesting approach to a.c. spectrum under optimal conditions is Molchanov, Novitskii, and Vainberg [17].This paper is organized as follows: in Section 2, we construct classical scattering trajectories for the 1/2-model with probability one, see Proposition 2.3 (th...

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“…Another proof of this result was recently given by Froese, Hasler, and Spitzer [57], and a related stability result for the absolutely continuous spectrum was given by Aizenman, Sims, and Warzel [5]. The existence of extended states and a mobility edge E m for small λ ≥ 0 for the lattice Anderson model, with decaying randomness, was also recently established by Kirsch, Krishna, and Obermeit [84] 1.5. Single Electron Models.…”

Section: Anderson Tight-bindingmentioning

confidence: 92%

Lectures on random Schrödinger operators

Hislop¹

2008

Contemporary Mathematics

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Abstract. These notes are based on lectures given in the IV Escuela de Verano en Análisis y Fisica Matemática at the Instituto de Matemáticas, Cuernavaca, Mexico, in May 2005. It is a complete description of Anderson localization for random Schrödinger operators on L 2 (IR d ) and basic properties of the integrated density of states for these operators. A discussion of localization for randomly perturbed Landau Hamiltonians and their relevance to the integer quantum Hall effect completes the lectures.

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Anderson model with decaying randomness: mobility edge

Cited by 34 publications

References 22 publications

Persistence of Anderson Localization in Schroedinger Operators With Decaying Random Potentials

Persistence of Anderson Localization in Schroedinger Operators With Decaying Random Potentials

Untitled

Lectures on random Schrödinger operators

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