
Many properties of the orbit are determined by the spectral measures which are probability measures on the real line. Dynamical properties of the system depend on the nature of the spectral measure because the Fourier transform of is . For example, a Hamiltonian L with absolutely continuous spectrum leads by the RiemanLebesgue lemma to the transient behavior . On the other hand, if an operator L has purely discrete spectrum, then is almost periodic a fact which is responsible for recurrent behavior of the dynamics.
In this note, we use the fact that it is often irrelevant, whether we evolve with the Hamiltonian L or if we use a deformed Hamiltonian f(L), where f is an invertible smooth real function. The reason is that the spectral measures of f(L) are only distorted versions of the spectral measures of L. It is therefore natural, to look for a function f such that the discrete time step can be easily computed. Properties of the spectrum, which are unchanged by a replacement can now be determined faster through numerical experiments because iterating a map is more simple than integrating a differential equation.
If where L has been rescaled such that aL
has norm smaller or equal to 1, the
time evolution can be computed by iterating the map
on .
This is the time one map for the unitary evolution
of . This allows an efficient determination of the Fourier coefficients
with
of measures on the circle, which
determine the spectral measures of L.
Some properties of the quantum evolution are not affected by the change
because the spectral measures of
and the spectral measure of L are related by
for every interval I.
One possibility to test for discrete spectrum of an operator is to determine numerically the Wiener averages . We illustrate this method for a tight binding model of an electron in a constant or random magnetic field in the plane. For random magnetic fields, where the existence of point spectrum is not known, we made numerical experiments on a grid of size up to .
We also illustrate the theoretical usefulness of the discrete time evolution by providing a relation between quantum mechanical return probabilities of the generator for a random walk on a graph and the return probability of the classical random walk on a graph: continuity properties of spectral measures with respect to the dimensional Hausdorff measures are related to powerlaw decays of averaged return probabilities of the random walk.