Spectral approximation for substitution systems
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
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| _version_ | 1866910829637533696 |
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| author | Band, Ram Beckus, Siegfried Pogorzelski, Felix Tenenbaum, Lior |
| author_facet | Band, Ram Beckus, Siegfried Pogorzelski, Felix Tenenbaum, Lior |
| contents | We study periodic approximations of aperiodic Schrödinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schrödinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over $\mathbb{Z}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09282 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral approximation for substitution systems Band, Ram Beckus, Siegfried Pogorzelski, Felix Tenenbaum, Lior Spectral Theory Dynamical Systems 37B10, 37B52, 52C23, 81Q10 We study periodic approximations of aperiodic Schrödinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schrödinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over $\mathbb{Z}^d$. |
| title | Spectral approximation for substitution systems |
| topic | Spectral Theory Dynamical Systems 37B10, 37B52, 52C23, 81Q10 |
| url | https://arxiv.org/abs/2408.09282 |