According to https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem#Quantum_mechanical_version the most recent paper is Schulman, L. S. (1978). "Note on the quantum recurrence theorem". Phys. Rev. A 18 (5): 2379–2380.

which assumes only discrete energy levels.

If there is scattering state, the energy level is typically continuous. My question is, in such system, does the Poincaré recurrence theorem still hold? Why?

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    $\begingroup$ The Poincare recurrence theorem does not hold for unbound systems. In bound systems it's pretty much useless, I would say. Far more important would be ergodicity and results like the KAM-theorem. $\endgroup$ – CuriousOne Aug 2 '16 at 6:00
  • $\begingroup$ See this paper. $\endgroup$ – Count Iblis Aug 2 '16 at 6:23

We just had a pedagogical paper on Poincare recurrence:


Yes, it does not hold if there are scattering states. Actually, classically, a free particle in one dimension with a finite momentum will not return to its initial state. For the Poincare recurrence to occur, the motion of the system should be bounded. This is apparent in the proof of the theorem.


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