### 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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Journal of the Mathematical Society of Japan*,

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

**A model of interband radiative transition.** / Dittrich, Jaroslav; Exner, Pavel; Hirokawa, Masao.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 56, no. 3, pp. 753-786. https://doi.org/10.2969/jmsj/1191334085

}

TY - JOUR

T1 - A model of interband radiative transition

AU - Dittrich, Jaroslav

AU - Exner, Pavel

AU - Hirokawa, Masao

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=6344241174&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=6344241174&partnerID=8YFLogxK

U2 - 10.2969/jmsj/1191334085

DO - 10.2969/jmsj/1191334085

M3 - Article

AN - SCOPUS:6344241174

VL - 56

SP - 753

EP - 786

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -