Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials

Altmann, Robert; Henning, Patrick; Peterseim, Daniel

doi:10.1142/s0218202520500190

Cited by 28 publications

(38 citation statements)

References 55 publications

(37 reference statements)

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“…Section 18: Numerical methods of differential equations (a) periodic potential V1 (b) checkerboard potential V2 (c) random potential V3 In the special case δ = 0, equation (1) is linear and reduces to the linear Schrödinger eigenvalue problem. Quantitative localization results for this particular case have been derived in [1] using classical results from domain decomposition and the convergence theory of iterative solvers, cf. [4,5].…”

Section: Numerical Resultsmentioning

confidence: 99%

Localization studies for ground states of the Gross‐Pitaevskii equation

Altmann

Peterseim

Varga

2018

Proc Appl Math and Mech

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For oscillatory high-amplitude potentials it is known that the linear Schrödinger operator leads to exponentially localized ground states. This localization result can be quantified explicitly in terms of geometric parameters and the degree of disorder within the potential. In the present paper we study the influence of the amplitude and the importance of the periodicity in the nonlinear setting of the Gross-Pitaevskii equation which models ultracold bosonic gases. Gross-Pitaevskii equationQuantum-physical processes related to ultracold bosonic gases -so-called Bose-Einstein condensates -can in certain regimes be mathematically described by the Gross-Pitaevskii equation [7]. The corresponding eigenvalue problem is then related to the formation of ground states and excited states of such condensates. For oscillatory and disordered potentials surprising phenomena such as Anderson localization occur. We are particularly interested in such localization properties of the ground state, which is the minimal-energy solution of the problemFor simplicity, we consider homogeneous Dirichlet boundary conditions, i.e., u ∈ H 1 0 (Ω). The parameter δ ≥ 0 controls the non-linearity whereas V is a high-amplitude potential that oscillates on a small scale ε. Moreover, the potential V ∈ L ∞ (D) remains within the boundsProperties of the ground state are discussed in [6]. The aim of this paper is to study the influence of the non-linearity (in form of the parameter δ) and the amplitude of the potential (parameter β) on the possible localization of the ground state. Numerical resultsFor the numerical experiments, we introduce three prototypical potentials. The first potential V 1 is periodic and oscillates continuously between the values α and β. Since the influence of the lower bound is negligible, we fix α = 1 throughout all experiments. In two space dimensions the first potential reads V 1 (x, y) := α + β−α 2 1 + sin(2πx/ε) sin(2πy/ε) .The second and third potential are both piecewise constant (on an ε-mesh). The potential V 2 takes independently the value α or β with probability 1 2 each (coin toss). V 3 takes independent random values uniformly distributed in the interval [α, β], cf. Fig. 1. (a) periodic potential V1 (b) checkerboard potential V2 (c) random potential V3 Fig. 1: Realizations of the three potentials V1, V2, and V3.

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Section: Numerical Resultsmentioning

confidence: 99%

Localization studies for ground states of the Gross‐Pitaevskii equation

Altmann

Peterseim

Varga

2018

Proc Appl Math and Mech

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“…4] for the matrix case. One may even consider the scaling with the exact eigenvalue λ as done, e.g., in [EE07,AHP18]. Clearly, the latter is only of interest for theoretical observations rather than actual computations.…”

Section: Iterative Methods On Operator Levelmentioning

confidence: 99%

PDE eigenvalue iterations with applications in two-dimensional photonic crystals

Altmann

Froidevaux

2020

ESAIM: M2AN

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The first part of this paper is devoted to the approximative solution of linear and Hermitian eigenvalue problems where the differential operator satisfies a Gårding inequality. For this, known iterative schemes for the matrix case such as the inverse power or Arnoldi method are extended to the infinite-dimensional case. This formally allows one to apply different spatial discretizations in each iteration step and thus, justifies the use of adaptive methods. The second part considers eigenvalue problems as they appear in two-dimensional models of photonic crystals for the computation of band-gaps. If the permittivity of the material is frequency-dependent, then this leads to a nonlinear eigenvalue problem. For this, we consider two strategies. First, a linearization combined with the application of the inverse power method and second, a direct application of Newton's iteration.

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“…The energy of a function is characterized by the Rayleigh quotient, namely Piecewise constant potentials with two values α and β were considered in [AHP18]. For this, an operator preconditioner was constructed depending on the geometric structure of the potential, i.e., on the interaction of α-valleys and β-peaks.…”

Section: Schrödinger Eigenvalue Problemmentioning

confidence: 99%

“…In [AHP18] the localization of eigenfunctions was rigorously proven for the regime β ε −2 and certain statistical assumptions on the potential. The key ingredient was to prove the existence of spectral gaps in the presence of disorder in combination with a preconditioned block inverse iteration.…”

Section: Localization Of Eigenfunctionsmentioning

confidence: 99%

“…Assume that the potential V is given in form of a random checkerboard, i.e., the function values of V are either α or β, each with probability 0.5. Then, motivated by the theoretical findings in [AHP18], we may start with a number of hat functions distributed in the largest valleys. Here, a valley denotes a cube in which the potential is constant α.…”

Section: Largest Valleysmentioning

confidence: 99%

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Localized Computation of Eigenstates of Random Schrödinger Operators

Altmann¹,

Peterseim²

2019

SIAM J. Sci. Comput.

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This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schrödinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates localize in the sense of an exponential decay of their moduli. We propose a reliable numerical scheme which provides localized approximations of such localized states. The method is based on a preconditioned inverse iteration including an optimal multigrid solver which spreads information only locally. The practical performance of the approach is illustrated in various numerical experiments in two and three space dimensions and also for a nonlinear random Schrödinger operator.

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Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials

Cited by 28 publications

References 55 publications

Localization studies for ground states of the Gross‐Pitaevskii equation

Localization studies for ground states of the Gross‐Pitaevskii equation

PDE eigenvalue iterations with applications in two-dimensional photonic crystals

Localized Computation of Eigenstates of Random Schrödinger Operators

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