Unitary Dynamics

Fourier transform of the Cantor set
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


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