On the adiabatic theorem when eigenvalues dive into the continuum

Decebal Horia Cornean, Arne Jensen, Hans Konrad Knörr, Gheorghe Nenciu

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1 Citation (Scopus)


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
Issue number5
Number of pages25
Publication statusPublished - 2018


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

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