The dynamics of a unitary operator is closely related to questions in
harmonic analysis. Interesting is for example, to study the behavior of the
Fourier transforms of singular continuous, i.e. fractal measures.
(The Picture shows the Fourier transform of the standard Cantor set
on the circle). Such questions are studied since Wiener and Wintner at the
beginning of the century.
The topological entropy of a unitary dynamical system is zero
(Reports on Mathematical Physics, 40: 91-95, 1997).
It is an interesting question whether some nonlinear systems which are
integrable in the sense that they can be written as a Lax pair, can
have positive topological entropy in infinite dimensions.
The spectral properties of quantum systems can be computed numerically
quite effectively .
For bounded operators, this trick gives a convenient tool for experimentally
testing the existence of eigenvalues or measuring the dimension of the
spectral measures. Experiments predict for
example that on a discrete two dimensional lattice with random magnetic field,
the Hamiltonian has purely singular continuous spectrum. The experiments
also predict that for operators appearing from twist maps the spectrum should
in general contain some eigenvalues.
See Slides of Western States talk