Arithmetic quantum chaos. (English) Zbl 0831.58045

Sarnak, Peter

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].

Reviewer: P.Garbaczewski (Wrocław)

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

Keywords:

quantum chaos; arithmetic groups; Riemann zeta function; Hamiltonians; geodesic flows; hyperbolic manifold; eigenvalues; eigenfunctions; Laplacian