### Abstract

We propose a rigorous computational method for verifying the isolated eigenvalues of one-dimensional Schrödinger operator containing a periodic potential and a perturbation which decays exponentially at ±∞. We show how the original eigenvalue problem can be reformulated as the problem of finding a connecting orbit in a Lagrangian-Grassmanian. Based on the idea of the Maslov theory for Hamiltonian systems, we set up an integer-valued topological measurement, the rotation number of the orbit in the resulting one-dimensional projective space. Combining the interval arithmetic method for dynamical systems, we demonstrate a computer-assisted proof for the existence of isolated eigenvalues within the first spectral gap.

Original language | English |
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Title of host publication | Mathematical Challenges in a New Phase of Materials Science |

Editors | Yasumasa Nishiura, Motoko Kotani |

Publisher | Springer New York LLC |

Pages | 145-157 |

Number of pages | 13 |

ISBN (Print) | 9784431561026 |

DOIs | |

Publication status | Published - Jan 1 2016 |

Event | International Conference on Mathematical Challenges in a New Phase of Materials Science, 2014 - Kyoto, Japan Duration: Aug 4 2014 → Aug 8 2014 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 166 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | International Conference on Mathematical Challenges in a New Phase of Materials Science, 2014 |
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Country | Japan |

City | Kyoto |

Period | 8/4/14 → 8/8/14 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematical Challenges in a New Phase of Materials Science*(pp. 145-157). (Springer Proceedings in Mathematics and Statistics; Vol. 166). Springer New York LLC. https://doi.org/10.1007/978-4-431-56104-0_8