In 1961, Stanislaw Ulam proposed a simple model: a particle bounces between a fixed wall and a wall that oscillates according to a piecewise-linear law. It seemed like just another billiard. But behind this simplicity lay a question that troubled physicists for decades: can the particle accelerate to infinity?

The answer turned out to be paradoxical. In the classical version, almost never. Accelerating orbits exist, but they are so rare that they form a set of measure zero. Almost every particle returns infinitely often to its initial energy level. The quantum version, however, behaves differently: energy grows quadratically. The quantum ball accelerates where the classical one stays put.

It is as if a classical billiard ball aimed at a pocket almost always returned to its starting position, while a quantum one flew off into the sky. The difference between the two worlds is not in details but in the very nature of motion. Quantization radically changes the conclusion: classical typical recurrence gives way to quantum acceleration.

Before this work, it was unknown how almost all orbits behave in the piecewise-linear Fermi–Ulam model and why the quantum version accelerates. Now it is proven: classical accelerating orbits form a set of measure zero, and quantum energy grows quadratically under generic resonance. The authors reduced the dynamics to a skew product over an interval exchange and computed the quasi-energy spectrum via finite matrices. This could be useful in designing dynamic traps for cold atoms and in mesoscopics, where boundaries change over time.

The non-resonant case remains open: there the skew product disappears, and the dynamics on invariant circles becomes an infinite interval exchange, an independent unsolved problem. For the quantum version, it is expected that for Diophantine or Liouville resonance numbers, the absolutely continuous spectrum degenerates into a singularly continuous or point spectrum, but analysis of large matrices still hits computational difficulties.