On the original Ulam's problem and its quantization

TL;DR

This paper compares classical and quantum versions of Ulam's problem, showing classical systems typically recur while quantum systems exhibit quadratic energy growth, and provides methods to identify escaping orbits.

Contribution

It demonstrates the fundamental differences between classical and quantum behaviors in the Fermi-Ulam accelerator under resonance conditions and offers a procedure to locate escaping orbits.

Findings

Classical accelerators show recurrence and non-escaping behavior.

Quantum accelerators exhibit quadratic energy growth.

A method to locate escaping orbits in classical systems is provided.

Abstract

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.