Why chaotic quantum spectra follow random-matrix statistics
In plain words
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 statement
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.
What would settle it
A theorem identifying a class of chaotic systems with random-matrix spectral statistics and explaining the arithmetic exceptions.
Status in the literature
Periodic-orbit theory reproduces the random-matrix form factor to all orders in $\tau$ non-rigorously (Muller, Heusler, Braun, Haake and Altland, 2004).
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See also
- Related Quantum unique ergodicity on negatively curved surfaces
- Related Which states produce Feshbach resonances in ultracold atom-molecule collisions?
- Related Energy scale below which ETH matrix elements become random-matrix-like
- Related Random-matrix bulk statistics for random regular graphs of fixed degree
- Related Poisson spacing statistics for generic integrable systems
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