## Abstract

We study a linear cocycle over irrational rotation s?(x) = x+? of a circle T1. It is supposed that the cocycle is generated by a A? : T1 to SL(2, R) that depends on a small parameter ? « 1 and has the form of the Poincaré map corresponding to a singularly perturbed Schrödinger equation. Under assumption that the eigenvalues of A?(x) are of the form exp (±?(x)/?), where ?(x) is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter ?. We show that in the limit ? ? 0 the cocycle exhibits ED for the most parameter values only if it is exponentially close to a constant cocycle. In the other case, when the cocycle is not close to a constant one and, thus, it does not possess ED, the Lyapunov exponent is typically large.

Original language | English |
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Title of host publication | Proceedings of the International Conference Days on Diffraction 2020, DD 2020 |

Editors | O.V. Motygin, A.P. Kiselev, L.I. Goray, T.M. Zaboronkova, A.Ya. Kazakov, A.S. Kirpichnikova |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 38-43 |

Number of pages | 6 |

ISBN (Electronic) | 9780738142791 |

DOIs | |

State | Published - 25 May 2020 |

Event | 2020 International Conference Days on Diffraction, DD 2020 - ПОМИ РАН, St. Petersburg, Russian Federation Duration: 25 May 2020 → 29 May 2020 http://www.pdmi.ras.ru/~dd/download/DD20_program.pdf |

### Publication series

Name | Proceedings of the International Conference Days on Diffraction 2020, DD 2020 |
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### Conference

Conference | 2020 International Conference Days on Diffraction, DD 2020 |
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Country | Russian Federation |

City | St. Petersburg |

Period | 25/05/20 → 29/05/20 |

Internet address |

## Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Acoustics and Ultrasonics
- Instrumentation
- Atomic and Molecular Physics, and Optics
- Radiation