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Title: Why chaotic quantum spectra follow random-matrix statistics
Plain: Energy levels of quantum systems whose classical motion is chaotic are observed to be spaced like eigenvalues of large random matrices. The mechanism and the exact conditions under which this holds are not established.
Precise: For a quantum system whose classical limit is fully chaotic (Anosov flow or ergodic billiard), the unfolded level statistics as $\hbar \to 0$ are expected to match GOE (time-reversal invariant) or GUE (no time-reversal symmetry). Determine the mechanism and the exact class of systems; known exceptions such as arithmetic surfaces show Poisson statistics. Answer: a characterization with proof; the sharper child problems are where work proceeds.
Settled by: A theorem identifying a class of chaotic systems with random-matrix spectral statistics and explaining the arithmetic exceptions.
Kind: phenomenon
Answer type: mechanism
References: (empty)
Literature status: open
Status note: Periodic-orbit theory reproduces the random-matrix form factor to all orders in $\tau$ non-rigorously (Muller, Heusler, Braun, Haake and Altland, 2004).
Posed since: 1984