Adiabatic Theorems for Quantum Resonances |
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Authors: | Walid K Abou Salem Jürg Fröhlich |
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Institution: | (1) Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerland;(2) Department of Mathematics, University of Toronto, M5S 2E4 Toronto, ON, Canada |
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Abstract: | We study the adiabatic time evolution of quantum resonances over time scales which are small compared to the lifetime of the
resonances. We consider three typical examples of resonances: The first one is that of shape resonances corresponding, for
example, to the state of a quantum-mechanical particle in a potential well whose shape changes over time scales small compared
to the escape time of the particle from the well. Our approach to studying the adiabatic evolution of shape resonances is
based on a precise form of the time-energy uncertainty relation and the usual adiabatic theorem in quantum mechanics. The
second example concerns resonances that appear as isolated complex eigenvalues of spectrally deformed Hamiltonians, such as
those encountered in the N-body Stark effect. Our approach to study such resonances is based on the Balslev-Combes theory
of dilatation-analytic Hamiltonians and an adiabatic theorem for nonnormal generators of time evolution. Our third example
concerns resonances arising from eigenvalues embedded in the continuous spectrum when a perturbation is turned on, such as
those encountered when a small system is coupled to an infinitely extended, dispersive medium. Our approach to this class
of examples is based on an extension of adiabatic theorems without a spectral gap condition. We finally comment on resonance
crossings, which can be studied using the last approach. |
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