On quantum ergodicity for higher dimensional cat maps
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916926314250240 |
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| author | Kurlberg, Pär Ostafe, Alina Rudnick, Zeev Shparlinski, Igor E. |
| author_facet | Kurlberg, Pär Ostafe, Alina Rudnick, Zeev Shparlinski, Igor E. |
| contents | We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm{Sp}}(2g,\mathbb Z)$, which we take to be ergodic. Under some natural assumptions, we show that there is a density one sequence of integers $N$ so that as $N$ tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant $N$ are uniformly distributed. For the two-dimensional case ($g=1$), this was proved by P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On quantum ergodicity for higher dimensional cat maps Kurlberg, Pär Ostafe, Alina Rudnick, Zeev Shparlinski, Igor E. Dynamical Systems Mathematical Physics Analysis of PDEs Number Theory We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm{Sp}}(2g,\mathbb Z)$, which we take to be ergodic. Under some natural assumptions, we show that there is a density one sequence of integers $N$ so that as $N$ tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant $N$ are uniformly distributed. For the two-dimensional case ($g=1$), this was proved by P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map. |
| title | On quantum ergodicity for higher dimensional cat maps |
| topic | Dynamical Systems Mathematical Physics Analysis of PDEs Number Theory |
| url | https://arxiv.org/abs/2411.05997 |