On the relation of scattering matrix, Riccati equation and boundary conditions

Abstract

A physical process concerning transmission and reflection was first attacked by G. G. Stokes [I] as early as 1862. He analyzed the optical behavior of a set of identical glass plates. At the turning of the century, Schmidt [2] obtained a Riccati equation for reflection by extending Stokes’ original approach to the case in which the added layer may be arbitrarily thin. Ambarzumian [3] related this idea to the theory of radiative transfer around 1943. Later, Chandrasekhar [4] extended and generalized Ambarzumian’s analysis which he called the “principle of invariance,” to a powerful mathematical technique for the problem of radiative transfer. The name “invariant imbedding” introduced by R. Bellman [5] in 1956, suggests a formulation for the properties of a given configuration that does not depend on the medium in which that configuration may be imbedded. Indeed the local response of a part of an obstacle is often determined as if it were imbedded in a free space. The method of “invariant imbedding” is a technique to establish differential equations for operators that describe the transmission and reflection by means of adding an arbitrary thin obstacle. During the same period, Redheffer [6,7] worked on transmission line theory that led to some general operators equations for scattering processes. He considered an obstacle extended from x toy as imbedded in a nonreflective homogeneous medium, and he obtained a pair of differential equations for the scattering matrix S(x,y), with S(x, X) = E = the identity matrix. Wang [8] obtained the same results by a different technique, by assuming the continuity of coefficients. The continuity condition can be easily replaced by piecewise continuous. The study of the solution with an arbitrary initial value, S(x, X) = S, , was done by McCarty [9], Redheffer [lo], and Reid [7]. McCarty and Reid considered the finite-dimensional case. Much of their results can be generalized to an infinite-dimensional case, Redheffer used the