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<p>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 spectrum.</p>

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On the Original Ulam’s Problem and Its Quantization

  • 期刊论文

  • 发表日期：

    2026年5月2日

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  • Changguang Dong,
  • Jing Zhou

摘要

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 spectrum.