IAS/Park City Mathematics Series
Volume 12, 2002Arithmetical Quantum Chaos
Audrey Terr as
Abstract
Physicists have long studied spectra of Schrodinger operators and random ma-
trices thanks to the implications for quantum mechanics. Analogously number
theorists and geometers have investigated the statistics of spectra of Laplacians
on Riemannian manifolds. This has been termed by Sarnak "arithmetic quantum
chaos" when the manifolds a re quotients of a symi.netric space modulo an arithmetic
group such as the modular group SL(2, Z). Equivalently one seeks the statistics
of the zeros of Selberg zeta functions. Parallels with the statistics of the zeros of
the Riemann zeta function have been evident to physicists for some t ime. Here we
survey what may be called "finite quantum ch aos" seeking connections with the
continuous theory. We will also discuss discrete analogue of Selberg's trace formula
as well as Ihara zeta functions of graphs.
(^1) Math. D ept., U.C.S.D., San Diego, Ca 92093 -0112
E-mail address: aterras©math. ucsd. edu.
333
@200 7 American Math e matical Society