### Abstract

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper, we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.

Original language | English |
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Article number | 1850011 |

Journal | Reviews in Mathematical Physics |

Volume | 30 |

Issue number | 5 |

Number of pages | 25 |

ISSN | 0129-055X |

DOIs | |

Publication status | Published - 2018 |

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### Keywords

- gapless adiabatic theorem
- adiabatic limit
- adiabatic pair creation
- mesoscopic transport
- resolvent expansion
- Feshbach method
- propagation estimates

### Cite this

*Reviews in Mathematical Physics*,

*30*(5), [1850011]. https://doi.org/10.1142/S0129055X18500113

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*Reviews in Mathematical Physics*, vol. 30, no. 5, 1850011. https://doi.org/10.1142/S0129055X18500113

**On the adiabatic theorem when eigenvalues dive into the continuum.** / Cornean, Decebal Horia; Jensen, Arne; Knörr, Hans Konrad; Nenciu, Gheorghe.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - On the adiabatic theorem when eigenvalues dive into the continuum

AU - Cornean, Decebal Horia

AU - Jensen, Arne

AU - Knörr, Hans Konrad

AU - Nenciu, Gheorghe

PY - 2018

Y1 - 2018

N2 - We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper, we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.

AB - We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper, we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.

KW - gapless adiabatic theorem

KW - adiabatic limit

KW - adiabatic pair creation

KW - mesoscopic transport

KW - resolvent expansion

KW - Feshbach method

KW - propagation estimates

U2 - 10.1142/S0129055X18500113

DO - 10.1142/S0129055X18500113

M3 - Journal article

VL - 30

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 5

M1 - 1850011

ER -