We use the dressing transformation in order to reconstruct one-dimensional Hamiltonians starting from their spectra. Whenever the given spectrum departs from oscillator-like local behaviour the resulting potential is fractal. An estimate of this fractal dimension is presented.

We present an analysis of various integration schemes which are applied to the numerical integration of, first, a one-dimensional Hamiltonian system and, second, Painlevé equations I and II. Both systems are well-known integrable ones. Integrable integrators are best suited for the simulation...