The Inverse Spectral Problem for Perturbed Canonical Systems

Henrik Winkler, TU Dresden

We consider a singular two-dimensional canonical system Jy' = -zHy on [0, ) with Weyl’s limit point case at . Here H is a measurable, real and nonnegative definite matrix function with tr H = 1, called Hamiltonian. It follows from results of L. de Branges that each Nevanlinna function Q is the Titchmarsh-Weyl coefficient of a canonical system with a unique Hamiltonian H. Unfortunately, a recipe for the explicit reconstruction of H from any given Q is still unknown. In our talk we present how the Hamiltonian of a canonical systems changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes a result of H. Dym and N. Kravitsky for so-called vibrating strings and a formula of I.M. Gelfand and B.M. Levitan for Sturm-Liouville systems.