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Arithmetic quantum chaos. (English) Zbl 0831.58045

Piatetski-Shapiro, Ilya (ed.) et al., The Schur lectures (1992). Ramat- Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 8, 183-236 (1995).
Arithmetic groups have been found useful in the physics literature [E. B. Bogomolny, B. Georgeot, M.-J Giannoni, and C. Schmit, Phys. Rev. Lett. 69, No. 10, 1477-1480 (1992)]while investigating symptoms of chaos (energy level spacing distributions) in quantum versions of classically chaotic Hamiltonian systems.
The paper presents a review of topics pertaining to Hamiltonians which are geodesic flows on an arithmetic hyperbolic manifold. Recent numerical computations of the eigenvalues and eigenfunctions of the Laplacian for such manifolds are reported. Their fine structure (level spacing distribution) is studied by relating these problems to ones about Riemann’s zeta functions and the \(L\)-functions (generalized zeta functions).
The principal mathematical material with outlines of the proofs is contained in sections devoted to arithmetic manifolds, and \(L\)-functions. Classical Lindelöf, Ramanujan and Sato-Tate conjectures are discussed in this context.
For the entire collection see [Zbl 0821.00011].

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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