Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan

doi:10.2140/apde.2018.11.1049

Cited by 36 publications

(74 citation statements)

References 53 publications

(80 reference statements)

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“…Actually, Theorem 2.13 holds in a much more general setting, see [23]. We only mention here the (general) random breather model in which the characteristic functions of balls with random radii are replaced by random dilations of radially decreasing, compactly supported, bounded and positive function u:…”

Section: Theorem 212 ( [22 23])mentioning

confidence: 99%

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Peyerimhoff¹,

Täufer²,

Veselić³

2017

Nanosystems: Phys. Chem. Math.

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For the analysis of the Schrödinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space, quantitative forms of unique continuation imply Wegner estimates and regularity properties of the integrated density of states (IDS) of Schrödinger operators with random potentials. For discrete Schrödinger equations on the lattice, only a weak analog of the UCP holds, but it is sufficient to guarantee the continuity of the IDS. For other combinatorial graphs, this is no longer true. Similarly, for quantum graphs the UCP does not hold in general and consequently, the IDS does not need to be continuous.

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Section: Theorem 212 ( [22 23])mentioning

confidence: 99%

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Peyerimhoff¹,

Täufer²,

Veselić³

2017

Nanosystems: Phys. Chem. Math.

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“…By now there is a wealth of results pursuing the connection between unique continuation and spectral theory of random Schrödinger operators, see e.g. [6,8,2,7,26,4,21,22,3,16,28,18,19,29]. While the proofs of Theorems 2.1, 2.2 and 2.3 use in particular the guiding thread of [16], they show how this approach and the recent result of [1] complement each other in an efficient way.…”

Section: Introductionmentioning

confidence: 99%

Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential

Täufer

Tautenhahn

2017

Ann. Henri Poincaré

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We consider non-ergodic magnetic random Schrödinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of arguments from [16], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [1]. This generalizes Klein's result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold E 0 (∞) ∈ (0, ∞], it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian without magnetic field.

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“…More than this, there are several quantitative formulations of unique continuation which proved to be useful in a variety of applications, see e.g. [BK05,RL12,BK13,RMV13,NTTV16]. For instance, Bourgain and Kenig [BK05] showed that if ∆u = V u in R d , u(0) = 1 and u, V ∈ L ∞ (R d ) then for all x ∈ R d with |x| > 1 we have max |y−x|≤1 |u(y)| > c · exp −c ′ (log|x|)|x| 4/3 .…”

Section: Introductionmentioning

confidence: 99%

“…The full answer has been announced in [NTTV15], and full proofs have been given in [NTTV16]. There, the constant…”

Section: Introductionmentioning

confidence: 99%

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

Täufer¹,

Tautenhahn²

2017

Communications on Pure &Amp; Applied Analysis

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We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let Λ L = (−L/2, L/2) d and H L = −∆ L + V L be a Schrödinger operator on L 2 (Λ L ) with a bounded potential V L : Λ L → R d and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the typewhere φ is an infinite complex linear combination of eigenfunctions of H L with exponentially decaying coefficients, W δ (L) is some union of equidistributed δ-balls in Λ L and C sfuc > 0 an L-independent constant. The exponential decay condition on φ can alternatively be formulated as an exponential decay condition of the map λ → χ [λ,∞) The novelty is that at the same time we allow the function φ to be from an infinite dimensional spectral subspace and keep an explicit control over the constant C sfuc in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.

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Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

Cited by 36 publications

References 53 publications

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

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