CANONICAL SYSTEMS WITH RATIONAL SPECTRAL DENSITIES - EXPLICIT FORMULAS AND APPLICATIONS

This paper solves explicitly the direct spectral problem of canonical differential systems for a special class of potentials. For a potentia l from this class the corresponding spectral function may have jumps a nd its absolutely continuous part has a rational derivative possibly w ith zeros on the real line. A direct and self-contained proof of the d iagonalization of the associated differential operator is given, inclu ding explicit formulas for the diagonalizing operator;and the spectral function. This proof also yields an explicit formula for the solution of the inverse problem. As an application new representations are der ived for a large class of solutions of nonlinear integrable partial di fferential equations. The method employed is based on state space tech niques and uses the idea of realization from mathematical system theor y.