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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 018 024 610
Sigal, I. M.; Vasilevic, B.:
Mathematical theory of quantum tunneling decay at positive temperatures
Ann. Henri Poincare 3, No. 2, 347-387 (2002).
47N50Applications in quantum physics
47D08Schroedinger and Feynman-Kac semigroups
47N55Applications in statistical physics
Primary Classification:
82C10Quantum dynamics and nonequilibrium statistical mechanics general
Secondary Classification:
81Q10 47D08 Sorry no match found within MSC 2000.
quantum tunneling; nondecay probability; rigorous estimate; nonzero temperature; weakly coupled reservoir; separable Hamiltonians; complex deformations of Hamiltonians; Feynman-Kac theorem for resonances; free energy for resonances; semiclassical bound;

Perturbative and, predominantly, power-series approximations are one
of the basic tools in quantum theory but a true challenge only
emerges when one has to deal with the much (i.e., typically,
exponentially) smaller corrections. This type of task often emerges
in connection with tunneling, and the paper deals with it on a high
level of mathematical rigor. Its main goal is formulated as an
initiation of the mathematical (i.e., more than just hand-waving)
study tunneling at positive (physicists would say `finite', meaning
`non-zero', with a certain logarithmic-scale under-tone)
The authors offer the first rigorous treatment of several aspects of
the problem of the error estimate in the common exponential formula
for non-escape probability. Their ``ab initio" treatment of the
estimates based on the time-evolution of the density matrix is
pioneering and impressively consequent, proceeding, roughly speaking,
through deformations (i.e., complexifications) and Feynman-Kac
formula extended to the case of resonances.
The task was difficult requiring a number of new ideas. Thus, the
selection of the reasonable assumptions combines the technical
feasibility (by keeping, say, the class of Hamiltonians ``simple
enough") with the insightful reduction of the model (treating, e.g.,
the interaction with the external reservoir with an appropriate
care). The effort is rendered successful via new notion of a ``free
resonance energy" which is a function of temperature and whose
imaginary part defines the line-width. The latter quantity is, by the
way, assigned also a new semi-classical estimate as a byproduct.
Remarks to the editors:

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