New Variational Method with Applications to Disordered Systems

Luttinger, J. M.

doi:10.1103/physrevlett.37.609

Cited by 39 publications

(4 citation statements)

References 2 publications

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“…Since then, such a behavior has been called 'the Lifshitz tail'. For H 0 = −∆ (or −∆ with periodic potential) and for various classes of random potentials V ω it has been widely studied and rigourously proven in both continuous (Benderskii and Pastur [1], Friedberg and Luttinger [10], Luttinger [26], Nakao [29], Pastur [30], Kirsch and Martinelli [19,20], Mezincescu [27], Kirsch and Simon [21], Kirsch and Veselić [22]) and discrete (Fukushima [11], Fukushima, Nagai and Nakao [12], Nagai [28], Romerio and Wreszinski [34], Simon [37]) settings (both of these lists are far from being complete). Note in passing that these random Hamiltonians typically exhibit the so-called spectral localization (see e.g.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Lifshitz tail for continuous Anderson models driven by Lévy operators

Kaleta¹,

Pietruska-Pałuba²

2019

Preprint

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We investigate the behavior near zero of the integrated density of states ℓ for random Schrödinger operators1, where Φ is a complete Bernstein function such that for some α ∈ (0, 2], one has Φ(λ) ≍ λ α/2 , λ ց 0, and V ω (x) = i∈Z d q i (ω)W (x − i) is a random nonnegative alloy-type potential with compactly supported single site potential W . We prove that there are constants C, C, D, D > 0 such thatwhere Fq is the common cumulative distribution function of the lattice random variables q i . In particular, we identify how the behavior of ℓ at zero depends on the lattice configuration. For typical examples of Fq the constants D and D can be eliminated from the statement above.We combine probabilistic and analytic methods which allow to treat, in a unified manner, both local and non-local kinetic terms such as the Laplace operator, its fractional powers and the quasi-relativistic Hamiltonians.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Lifshitz tail for continuous Anderson models driven by Lévy operators

Kaleta¹,

Pietruska-Pałuba²

2019

Preprint

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“…The asymptotic decay at the lower spectral boundary coincides with the behaviour for B = 0, if (d =) 2 < α < 4 (= d + 2), that is, for slowly decaying single-site potential, see [37] or [38, Corollary 9.14], but it differs in the case α > 4 (= d + 2), see [36], [37] or [38,Theorem 10.2]. This is plausible from the Rayleigh-Ritz-like variational principle due to Luttinger [32] and Pastur [37], see also Equation (17.3) and Chapter 21 in [31]. For B = 0 and α > 4 the optimal wavefunction becomes too sharply localized so that the unperturbed (kinetic) energy begins to play a significant rôle.…”

Section: Remarks 24mentioning

confidence: 94%

“…The form of the latter asymptotic behaviour has been discovered by Lifshitz [29], [30], [31]. Convincing arguments for the validity of Lifshitz' conjecture were given by Friedberg and Luttinger [18], [32]. Its rigorous proof [15], [36], [37] relies on Donsker's and Varadhan's involved large-deviation results [15], [14,Section 4.3] about the long-time asymptotics of certain Wiener integrals.…”

Section: Introductionmentioning

confidence: 99%

The fate of lifshits tails in magnetic fields

et al. 1995

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We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.

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“…(ii) Convincing arguments for the validity of Lifshits' result (4.1) for the decay (D1) were also given in [54,102]. An alternative (rigorous) proof of the underlying long-time asymptotics is due to Sznitman who invented a coarse-graining scheme called the method of enlargement of obstacles [137].…”

Section: Lifshits Tailsmentioning

confidence: 99%

A Survey of Rigorous Results on Random Schrödinger Operators for Amorphous Solids

Leschke

Müller

Warzel

Interacting Stochastic Systems

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Electronic properties of amorphous or non-crystalline disordered solids are often modelled by one-particle Schrödinger operators with random potentials which are ergodic with respect to the full group of Euclidean translations. We give a short, reasonably self-contained survey of rigorous results on such operators, where we allow for the presence of a constant magnetic field. We compile robust properties of the integrated density of states like its self-averaging, uniqueness and leading high-energy growth. Results on its leading low-energy fall-off, that is, on its Lifshits tail, are then discussed in case of Gaussian and non-negative Poissonian random potentials. In the Gaussian case with a continuous and non-negative covariance function we point out that the integrated density of states is locally Lipschitz continuous and present explicit upper bounds on its derivative, the density of states. Available results on Anderson localization concern the almost-sure pure-point nature of the low-energy spectrum in case of certain Gaussian random potentials for arbitrary space dimension. Moreover, under slightly stronger conditions all absolute spatial moments of an initially localized wave packet in the pure-point spectral subspace remain almost surely finite for all times. In case of one dimension and a Poissonian random potential with repulsive impurities of finite range, it is known that the whole energy spectrum is almost surely only pure point.

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New Variational Method with Applications to Disordered Systems

Cited by 39 publications

References 2 publications

Lifshitz tail for continuous Anderson models driven by Lévy operators

Lifshitz tail for continuous Anderson models driven by Lévy operators

The fate of lifshits tails in magnetic fields

A Survey of Rigorous Results on Random Schrödinger Operators for Amorphous Solids

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