Quantum tunneling at positive temperature (Q2752780)

This work addresses the problem of tunneling at positive temperature. The mathematical framework used rests on the notion of free resonance energy, \(F(\beta)\) \((\beta\) being the inverse temperature), which plays the role of the resonance energy at zero temperature. Since the notion of temperature pertains to equilibrium states of systems with infinite degrees of freedom, while tunneling is obviously a nonequilibrium process, and a quantum particle has only three degrees of freedom, the author starts by clarifying what is here meant by tunneling at positive temperature: this refers to a particle being in contact with a reservoir (as a photon or phonon gas), which at time \(t=0\) is at thermal equilibrium at temperature \(T\) or which is initially thermalized, in which case the temperature is introduced through the initial condition only, and has no other effect on the dynamics of the system. The author extends the definition of the partition function to unstable systems and, correspondingly, the Feynman-Kac theorem to resonances. Finally, it is proven in the paper that the probability of escape of a particle from a trap (or well) due to tunneling is given by \(P(t)= 1-\exp(\Gamma t/ \hbar)(1+ {\mathcal O}( \Gamma)\), with \(\Gamma=-2 \text{Im} (F\beta)\), which can be interpreted as the ``width'' of the free energy. The paper provides a sort of justification for this formula, which is widely employed in condensed matter physics and cosmology. A difference with previous works is that the definition of \(F(\beta)\) is here given in terms of the Schrödinger operator itself, rather than through its spectral characteristics, as done in previous analysis.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00038].