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##

Dependence on Initial Conditions

Two different mechanisms of localization of a quantum state have to be
distinguished.
First, there are the quantum manifestations of classical properties
leading to localization, such as
invariant lines
in classical phase space
not yet
having broken up und thus being obstacles to global dynamics in phase
space. This is the dominant feature of the quantum dynamics displayed
in figures
5.9 and
5.13;
from a classical point of view it is to be expected and therefore less
interesting.
Second, there are the genuinely quantum effects that are leading to
localization as visible in the
remaining
figures in subsection
5.2.1
(and in section C.4 of the appendix)
and that are to be explained in section 5.3.
Quite naturally, these quantum effects are in the focus of attention in
the present chapter.

The quantum dynamics for , while at first sight dominated by
classical localization only, is also affected by the quantum localization
mechanism in a crucial way.
This can be seen by starting the dynamics with different initial
conditions. For figures
5.9 and
5.13,
the
have been chosen to be localized *within*
the
closed invariant lines, leading to the classical localization result.
For figure 5.16,

on the other hand, the same initial states have been chosen (here for
and only), but shifted to
the point ,
well beyond
the outermost invariant line:
.
(Note that the topmost classical initial point in figure 1.4 is
,
too,
making comparison between the classical and quantum
cases
easier.)
The resulting dynamics obviously differs from the case with initial
conditions near : the quantum dynamics remains localized, while
the classical dynamics becomes diffusive for large enough .
To summarize,
depending on the initial condition, one may have
quantum localization for classical or quantum reasons, or for a
combination of both.

The localization phenomena that
have been demonstrated in
section 5.2
using numerical means are given a
theoretical foundation
in the following section 5.3.

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Martin Engel 2004-01-01