### Abstract

We consider a simple model which is a caricature of a crystal interacting with a radiation field. The model has two bands of continuous spectrum and the particle can pass from the upper one to the lower by radiating a photon, the coupling between the excited and deexcited states being of a Friedrichs type. Under suitable regularity and analyticity assumptions we find the continued resolvent and show that for weak enough coupling it has a curve-type singularity in the lower halfplane which is a deformation of the upper-band spectral cut. We then find a formula for the decay amplitude and show that for a fixed energy it is approximately exponential at intermediate times, while the tail has a power-like behaviour.

Original language | English |
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Pages (from-to) | 753-786 |

Number of pages | 34 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 56 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2004 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Journal of the Mathematical Society of Japan*,

*56*(3), 753-786. https://doi.org/10.2969/jmsj/1191334085