Biunitary transformations and ordinary differential equations - III

In two previous papers, the authors have introduced the concept of binormal differential equations. It has been shown that invariants of the Courant-Snyder type are associated to the scalar products of the column vector associated to an ordinary differential equation and to its binormal. In this paper, we show the equivalence of the above invariant and the Lewis form. We also introduce a density matrix for a second-order differential equation and clarify the geometrical meaning of the Twiss parameters. The importance of the above results in the analysis of quantum problems such as, e.g., the evolution of squeezed states is finally stressed