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Physicists have long studied spectra (or eigenvalues)
of Schroedinger operators and random matrices, thanks to the
implications for quantum mechanics. Analogously number theorists
and geometers have investigated the spectra of the differential
operators known as Laplacians associated to certain surfaces with
a Riemannian distance. For surfaces with symmetries coming from
number theory, this has been termed "arithmetic quantum chaos"
by Peter Sarnak (Princeton U.). Here we survey (using a picture
gallery) the connections between continuous quantum chaos and what
may be called "finite quantum chaos". In the finite theory, the
differential operators are replaced with finite nxn matrices of zeros
and ones - the adjacency matrices of finite upper half plane graphs.
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