Localisation for Delone operators via Bernoulli randomisation

Abstract

Delone operators are Schrödinger operators in multi-dimensional Euclidean space with a potential given by the sum of all translates of a given "single-site potential" centred at the points of a Delone set. In this paper, we use randomisation to study dynamical localisation for families of Delone operators. We do this by suitably adding more points to a Delone set and by introducing i.i.d. Bernoulli random variables as coupling constants at the additional points. The resulting non-ergodic continuum Anderson model with Bernoulli disorder is accessible to the latest version of the multiscale analysis. The novel ingredient here is the initial length-scale estimate whose proof is hampered due to the non-periodic background potential. It is obtained by the use of a quantitative unique continuation principle. As applications we obtain both probabilistic and topological statements about dynamical localisation. Among others, we show that Delone sets for which the associated Delone operators exhibit dynamical localisation at the bottom of the spectrum are dense in the space of Delone sets.

changes in v2: more detailed exposition in the proofs of Theorem 3.6 and Lemma 3.8. To appear in Journal d'Analyse Mathématique

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Cited by in corpus (3)

• Localization and unique continuation for non-stationary Schrödinger operators on the 2D lattice
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Lifshitz asymptotics and localization for random breather models