- Introduction
Lecture 1. Finite models
This is a story of a tree related to the spectral theory of operators on Hilbert sp aces.
The tree h as three branches as in Figure l. The left branch is that of quantum
physics: the statistics of energy levels of quantum mechanical systems; i.e. the
eigenvalues of the Schrodinger operator £ ¢ 11 = >. 11 ¢ 11 • The middle branch is that
of geometry and number theory. In the middle we see the spectrum of the La place
operator .C = /:::,. on a Riemannian manifold NI such as the fundamental domain of
the modula r group SL(2, Z) of 2 x 2 integer matrices with determinant l. This is a n
example of what Peter Sarnak has called "arithmetic quantum ch aos." The right
branch is that of graph theory: the statistics of the spectrum of .C = t he adjacency
operator (or combinatorial Laplacian) of a Cayley graph of a finite matrix group.
\Ve call this subject "finite quantum chaos."
What about the roots of the tree? These roots represent zeta functions and
trace formulas. On the left is the Gutzwiller trace formula. In the middle is the
Selberg trace formula and the Selberg zeta function. On the right is the Ihara zeta
function of a finite graph and discrete analogues of the Selberg trace formula. The
zeros of the zeta functions correspond sometimes mysteriously to the eigenvalues of
the operators at the top of the tree.
Here we will emph asize the right branch and root of the tree. But we will
discuss the connections with the other tree parts. The articles [84], [85], [86] are
closely related to these lectures. A good website for quantum chaos is that of
Matthew '\Ill. \Vatkins http://www.maths.ex.ac.uk;-mwatkins.
We quote Oriol Bohigas and Marie-Joya Gionnoni [11], p. 14: "The question
now is to discover the stochastic laws governing sequences having ve ry different
origins, as illustrated in ... [Figure 2]. There are displayed six spectra, each co n-
taining 50 levels ... " Note that the spectra h ave b een rescaled to the same vertical
axis from 0 to 49.
In Figure 2, column (a) represents a Poisson spectrum, meaning that of a
random variable with spacings of probability density e-x. Column (b) represents
primes between 7791097 and 7791 877. Column ( c) represents the resonance energies
of the compound nucleus observed in the reaction n+^166 Er. Column ( d) comes from
eigenvalues corresponding to transverse vibrations of a membrane whose boundary
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