  A Schur-class function in $d$ variables is defined to be an analytic
contractive-operator valued function on the unit polydisk. Such a function is
said to be in the Schur--Agler class if it is contractive when evaluated on any
commutative $d$-tuple of strict contractions on a Hilbert space. It is known
that the Schur--Agler class is a strictly proper subclass of the Schur class if
the number of variables $d$ is more than two. The Schur--Agler class is also
characterized as those functions arising as the transfer function of a certain
type (Givone--Roesser) of conservative multidimensional linear system. Previous
work of the authors identified the Schur--Agler class as those Schur-class
functions which arise as the scattering matrix for a certain type of (not
necessarily minimal) Lax--Phillips multievolution scattering system having some
additional geometric structure. The present paper links this additional
geometric scattering structure directly with a known reproducing-kernel
characterization of the Schur--Agler class. We use extensively the technique of
formal reproducing kernel Hilbert spaces that was previously introduced by the
authors and that allows us to manipulate formal power series in several
commuting variables and their inverses (e.g., Fourier series of elements of
$L^2$ on a torus) in the same way as one manipulates analytic functions in the
usual setting of reproducing kernel Hilbert spaces.
