Resonances for Open Quantum Maps and a Fractal Uncertainty Principle
© 2017, Springer-Verlag Berlin Heidelberg. We study eigenvalues of quantum open baker’s maps with trapped sets given by linear arithmetic Cantor sets of dimensions δ∈ (0 , 1). We show that the size of the spectral gap is strictly greater than the standard bound max(0,12-δ) for all values of δ, which...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Springer Nature
2021
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| Online Access: | https://hdl.handle.net/1721.1/133911 |
| _version_ | 1826210516147634176 |
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| author | Dyatlov, Semyon Jin, Long |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Dyatlov, Semyon Jin, Long |
| author_sort | Dyatlov, Semyon |
| collection | MIT |
| description | © 2017, Springer-Verlag Berlin Heidelberg. We study eigenvalues of quantum open baker’s maps with trapped sets given by linear arithmetic Cantor sets of dimensions δ∈ (0 , 1). We show that the size of the spectral gap is strictly greater than the standard bound max(0,12-δ) for all values of δ, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus. |
| first_indexed | 2024-09-23T14:51:13Z |
| format | Article |
| id | mit-1721.1/133911 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:51:13Z |
| publishDate | 2021 |
| publisher | Springer Nature |
| record_format | dspace |
| spelling | mit-1721.1/1339112023-12-19T20:44:10Z Resonances for Open Quantum Maps and a Fractal Uncertainty Principle Dyatlov, Semyon Jin, Long Massachusetts Institute of Technology. Department of Mathematics © 2017, Springer-Verlag Berlin Heidelberg. We study eigenvalues of quantum open baker’s maps with trapped sets given by linear arithmetic Cantor sets of dimensions δ∈ (0 , 1). We show that the size of the spectral gap is strictly greater than the standard bound max(0,12-δ) for all values of δ, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus. 2021-10-27T19:57:11Z 2021-10-27T19:57:11Z 2017 2021-04-29T14:38:09Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/133911 en 10.1007/S00220-017-2892-Z Communications in Mathematical Physics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer Nature Springer |
| spellingShingle | Dyatlov, Semyon Jin, Long Resonances for Open Quantum Maps and a Fractal Uncertainty Principle |
| title | Resonances for Open Quantum Maps and a Fractal Uncertainty Principle |
| title_full | Resonances for Open Quantum Maps and a Fractal Uncertainty Principle |
| title_fullStr | Resonances for Open Quantum Maps and a Fractal Uncertainty Principle |
| title_full_unstemmed | Resonances for Open Quantum Maps and a Fractal Uncertainty Principle |
| title_short | Resonances for Open Quantum Maps and a Fractal Uncertainty Principle |
| title_sort | resonances for open quantum maps and a fractal uncertainty principle |
| url | https://hdl.handle.net/1721.1/133911 |
| work_keys_str_mv | AT dyatlovsemyon resonancesforopenquantummapsandafractaluncertaintyprinciple AT jinlong resonancesforopenquantummapsandafractaluncertaintyprinciple |