On the adiabatic theorem when eigenvalues dive into the continuum

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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 languageEnglish
Article number1850011
JournalReviews in Mathematical Physics
Volume30
Issue number5
Number of pages25
ISSN0129-055X
DOIs
Publication statusPublished - 2018

Fingerprint

Survival Probability
Bound States
Continuum
eigenvalues
theorems
continuums
Eigenvalue
Continuous Spectrum
Channel Model
continuous spectra
Quantum Dots
Energy
Theorem
Vanish
quantum dots
Scattering
Vary
Cover
Three-dimensional
energy

Keywords

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

Cite this

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title = "On the adiabatic theorem when eigenvalues dive into the continuum",
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.",
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author = "Cornean, {Decebal Horia} and Arne Jensen and Kn{\"o}rr, {Hans Konrad} and Gheorghe Nenciu",
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journal = "Reviews in Mathematical Physics",
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On the adiabatic theorem when eigenvalues dive into the continuum. / Cornean, Decebal Horia; Jensen, Arne; Knörr, Hans Konrad; Nenciu, Gheorghe.

In: Reviews in Mathematical Physics, Vol. 30, No. 5, 1850011, 2018.

Research output: Contribution to journalJournal articleResearchpeer-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

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JO - Reviews in Mathematical Physics

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