Our main result is the stability of the absolutely continuous spectrum
of two model examples of strings under rather wide classes of
perturbations. In particular, this enables us to prove that the
absolutely continuous spectrum of the isospectral problem associated
with the conservative Camassa-Holm flow in the dispersive regime is
essentially supported on the interval [1/4,\infty) for the entire
natural phase space. If time permits, applications to quantum graphs
and 1D Hamiltonians with \delta'-interactions will be discussed as
well.

The talk is based on joint work with J. Eckhardt (Loughborough, UK).