Abstract. We study a microscopic Hamiltonian model describing an N -level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at inverse temperature β. Assuming that the coupled system S + R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S → R and R → S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted P S,λ,t , concentrated on the set of energy differences sp(H S )−sp(H S ) (H S is the Hamiltonian of S, t is the length of the time interval during which the measurement of the energy transfer is performed, and λ is the strength of the interaction between S and R). The second FCS, P R,λ,t , describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit t → ∞ of these two measures followed by the weak coupling limit λ → 0 and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of P R,λ,t by quantum transfer operators.

Abstract. The dynamic reflection probability of [DS] and the spectral reflection probability of [GNP,GS4] for a one-dimensional Schrödinger operator H "´∆`V are characterized in terms of the scattering theory of the pair pH, H 8 q where H 8 is the operator obtained by decoupling the left and right half-lines R ď0 and R ě0 . An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of [BRS]. This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black box model and follows a strategy parallel to [JLPa].

Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient, a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turns provides a short proof of an important result of [1]. These developments parallel those recently obtained for Jacobi matrices [10].

We consider a finite dimensional quantum system S in an arbitrary initial state coupled to an infinitely extended quantum thermal reservoir R in equilibrium at inverse temperature β. The coupling is given by a bounded perturbation of the dynamics and the coupling strength is controlled by a parameter λ. We assume the system S + R has Contents

No abstract