The inverse monodromy problem

The inverse monodromy problem for m × m canonical differential systems y^′_t (λ) = iλyt(λ)H(t)J on a finite interval [0, d] is to recover the Hamiltonian H(t) of the differential system from the monodromy matrix, i.e., the value of the matrizant (fundamental solution) of the system at the right-hand end point d of the interval. This problem does not have a unique solution unless extra constraints are imposed. A number of known results are reviewed briefly. Special classes of monodromy matrices for which the solutions of the inverse monodromy problem may be parametrized by identifying the matrizant with the resolvent matrices of a class of bitangential extension problems are discussed. The exposition makes extensive use of two classes of reproducing kernel Hilbert spaces of vector-valued entire functions that originate in the work of Louis de Branges and the interplay between them. Some new subclasses of these spaces are introduced and their role in the inverse monodromy problem are discussed.