ABSTRACT. The differential expression Lm = −∂ 2 x + (m 2 − 1/4)x −2 defines a self-adjoint operator Hm on L 2 (0, ∞) in a natural way when m 2 ≥ 1. We study the dependence of Hm on the parameter m, show that it has a unique holomorphic extension to the half-plane Re m > −1, and analyze spectral and scattering properties of this family of operators.

A quantum system S interacts in a successive way with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling between S and E. We show that the system, initially in any state close to a reference state, approaches a repeated interaction asymptotic state in the limit of large times. This state is τ -periodic in time and does not depend on the initial state. If the reference state is chosen so that S and E are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.