Canonical Systems with Rational Spectral Densities: Explicit Formulas and Applications

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Abstract
This paper solves explicitly the direct spectral problem of canonical differential systems for a special class of potentials. For a potential from this class the corresponding spectral function may have jumps and its absolutely continuous part has a rational derivative possibly with zeros on the real line. A direct and self‐contained proof of the diagonalization of the associated differential operator is given, including 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 derived for a large class of solutions of nonlinear integrable partial differential equations. The method employed is based on state space techniques and uses the idea of realization from mathematical system theory.