On the existence of the subdynamics concept

Abstract

The concept of subdynamics projectors Π introduced by the Brussels group in the treatment of systems in statistical mechanics remains central in the construction of the non-unitary transformation theory or in the theory of complex spectral representations. Originally only one of such subdynamics, II, was supposed to persist asymptotically. When the evolution of the systems presents temporal behaviours with long tails, the usefulness of this subdynamics concept, and even its existence, is to be questioned. To catch the nature of the problem, we analyze simple systems with a non exponential decay. We indicate how the corresponding operators may effectively be constructed and we examine the validity of the asymptotic projection.