On the original Ulam's problem and its quantization

In this paper we show that under general resonance the classical piecewise linear Fermi-Ulam accelerator behaves substantially different from its quantization in the sense that the classical accelerator exhibits typical recurrence and non-escaping while the quantum version enjoys quadratic energy growth in general. We also describe a procedure to locate the escaping orbits, though exceptionally rare in the infinite-volume phase space, for the classical accelerators, which in particular include Ulam's very original proposal and the linearly escaping orbits therein in the existing literature, and hence provide a complete (modulo a null set) answer to Ulam's original question. For the quantum accelerators, we reveal under resonance the direct and explicit connection between the energy growth and the shape of the quasi-energy spectra.