Quantum chaos and the Riemann zeros

7 problems

• Poisson spacing statistics for generic integrable systems

  Partially resolved

  For systems with regular classical motion, quantum levels should be spaced like uncorrelated random numbers. Even for a flat rectangular drum this is only partly proven.

• Random-matrix pair correlation for a specific chaotic billiard

  Open

  Prove for one concrete chaotic shape, such as a stadium-shaped drum, that its high vibration frequencies have random-matrix spacing statistics. No single example is proven.

• Why chaotic quantum spectra follow random-matrix statistics

  Open

  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.

• A self-adjoint operator whose eigenvalues are the Riemann zeros

  Open

  If the Riemann zeros were the energy levels of a quantum system, they would all be real numbers, which is the Riemann hypothesis. The task is to find such a system.

• Montgomery's GUE pair correlation of the Riemann zeros for all ranges

  Open

  Montgomery showed that the spacing between Riemann zeros matches random-matrix theory for some range of distances, assuming the Riemann hypothesis. Extending this to all distances is open.

• Quantum unique ergodicity on negatively curved surfaces

  Open

  On a curved surface where every classical orbit is chaotic, every high-frequency standing wave should spread evenly over the surface. This is proven only for special surfaces with number-theoretic symmetry.

• Gaussian value distribution of chaotic eigenfunctions

  Open

  Berry proposed that a high-frequency standing wave of a chaotic system looks locally like a random superposition of plane waves, so its values follow a bell curve. Even a much weaker consequence, that no eigenfunction has unusually high peaks, is unproven.