  The theory of symmetric, non-selfadjoint operators has several deep
applications to the complex function theory of certain reproducing kernel
Hilbert spaces of analytic functions, as well as to the study of ordinary
differential operators such as Schrodinger operators in mathematical physics.
Examples of simple symmetric operators include multiplication operators on
various spaces of analytic functions such as model subspaces of Hardy spaces,
deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators
(including Schrodinger operators from quantum mechanics), Toeplitz operators,
and infinite Jacobi matrices.
  In this paper we develop a general representation theory of simple symmetric
operators with equal deficiency indices, and obtain a collection of results
which refine and extend classical works of Krein and Livsic. In particular we
provide an alternative proof of a theorem of Livsic which characterizes when
two simple symmetric operators with equal deficiency indices are unitarily
equivalent, and we provide a new, more easily computable formula for the Livsic
characteristic function of a simple symmetric operator with equal deficiency
indices.
