A grassmannian approach to scattering systems

In mathematical physics the scattering matrix S(??) is a unitary-matrix valued function of real frequency variable ??. In linear system theory, the transfer matrix T(s) is an operator-valued function of a complex variable s = ei??. In recent theory the "scattering matrix" is the cayley transform of the "transfer matrix": S( ) = 1 + iT(ei??)/1 - iT(ei??) (1) In [1] Matrix and I have described how it is useful to consider s ?? T(s) not as a matrix-valued function of s, but as a curve in a Grassmann manifold ??(s) = {(u, T(s)u): u ?? input space U} (2) The relation expressed by formula 1 between "T" and "s" then means, geometrically, that they are basically the same geometric-physical object, but expressed in a different coordinate chart. The classical prototype is in the theory of functions of a complex variable: Analytic functions can be considered as the unit circle the upper half plane. In [2] I have surveyed various methods for describing the "scattering matrix" and investigated their relation to standard system theory. What seems to be ultimately involved is an extension of the theory of pseudo- and Fourier-integral operators. I will describe some of these ideas in my lecture. What seems particularly important is the geometric interpretation of the "transfer matrix" and "scattering matrix" as the symbol attached to such operators, and the associated bundleology.