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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 018 024 610  
Author(s):  
Sigal, I. M.; Vasilevic, B.:  
Shorttitle:  
Mathematical theory of quantum tunneling decay at positive temperatures  
Source:  
Ann. Henri Poincare 3, No. 2, 347387 (2002).  
Classification:  
 
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
quantum tunneling; nondecay probability; rigorous estimate; nonzero temperature; weakly coupled reservoir; separable Hamiltonians; complex deformations of Hamiltonians; FeynmanKac theorem for resonances; free energy for resonances; semiclassical bound;  
Review:  
Perturbative and, predominantly, powerseries 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 handwaving) study tunneling at positive (physicists would say `finite', meaning `nonzero', with a certain logarithmicscale undertone) temperatures. \par The authors offer the first rigorous treatment of several aspects of the problem of the error estimate in the common exponential formula for nonescape probability. Their ``ab initio" treatment of the estimates based on the timeevolution of the density matrix is pioneering and impressively consequent, proceeding, roughly speaking, through deformations (i.e., complexifications) and FeynmanKac formula extended to the case of resonances. \par 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 linewidth. The latter quantity is, by the way, assigned also a new semiclassical estimate as a byproduct.  
Remarks to the editors:  