Small perturbations of canonical systems

We consider a singular two-dimensional canonical system Jy' = -zHy on [0, infinity) such that at oo Weyl's Limit point case holds. Here H is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with tr H = 1 and the class of Nevanlinna functions. In this note we show how the Hamiltonian H of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of H. Dym and N. Kravitsky for so-called vibrating strings, in particular a generalization of a construction principle of I.M. Gelfand and B.M. Levitan can be shown.