Construction of Minimal Cocycles

Blending methods of Topological Dynamics and Control Theory, we develop a new technique to construct compact-Lie-group-valued minimal cocycles arising as fundamental matrix solutions of linear diierential equations with recurrent coeecients subject to a given constraint. The precise requirement on the coee-cients is that they belong to a speciied closed convex subset S of the Lie algebra L of the Lie group. Our result is proved for a very thin class of cocycles, since the dimension of S is allowed to be much smaller than that of L, and the only assumption on S is that L 0 (S) = L, where L 0 (S) is the ideal of L(S) generated by the diierence set S ? S, and L(S) is the Lie subalgebra of L generated by S. This covers a number of diierential equations arising in Mathematical Physics, and applies in particular to the widely studied example of the Rabi oscillator. x1. Introduction We develop a technique to construct cocycles that have desired dynamical properties and arise as the fundamental matrix solutions to linear diierential equations of a given speciic form. Our motivation comes from recent developments in Mathematical Physics regarding stability questions in the evolution of quantum systems, which turn out to be intimately related to the dynamical properties of certain ows 10]. As an example of the general situation to be studied here, consider the so called \Rabi oscillator," i.e. the system governed by the equation (1.1) i dd dt = f(t) f(t) ? ; 2 C 2 : This is the Schrr odinger equation for the dynamics of a \two level atom" or a spin 1=2 particle moving under external magnetic eld f(t).