Localisation for Delone operators via Bernoulli randomisation
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| Format: | Preprint |
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2020
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| _version_ | 1866916548593057792 |
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| author | Müller, Peter Rojas-Molina, Constanza |
| author_facet | Müller, Peter Rojas-Molina, Constanza |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2003_06325 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Localisation for Delone operators via Bernoulli randomisation Müller, Peter Rojas-Molina, Constanza Mathematical Physics Probability Spectral Theory 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. |
| title | Localisation for Delone operators via Bernoulli randomisation |
| topic | Mathematical Physics Probability Spectral Theory |
| url | https://arxiv.org/abs/2003.06325 |