Localization Established

As discussed in the previous subsection,
both
discrete SCHRÖDINGER equations (5.94a, 5.94b)
in a natural way
give rise
to the
approximating tight-binding equation

In terms of the transfer matrix formalism of subsection
5.1.2, the definitions

in conjunction with the equation of motion

seem to suggest themselves for modelling equation (5.101).

However, this formulation is disadvantageous for several reasons.
First, it rests on the assumption that
*for all* ,
because makes up the denominators of two of
the matrix elements of the transfer matrix (5.102).
Unfortunately, this assumption cannot be checked
analytically for
all combinations
of and ,
because
a closed solution of the integral in equation (5.86)
is
not
available.
Already a single
with
could spoil the whole
theory.
Even more,

The workaround
for the case of a vanishing (second-) nearest neighbour matrix element
sketched in
the footnote
on page only works
in the context of the kicked rotor,
because of the translation invariance of the discrete
SCHRÖDINGER equation
for that model.
This is
in contrast to
the translation
*non*invariance
of the discrete SCHRÖDINGER equation
of the kicked harmonic oscillator, i.e.
the explicit
-dependence of the oscillator's hopping matrix element for
the interaction with the nearest neighbour at the -th site.
Divisions by occur in *all* transfer matrix models of
the kicked harmonic oscillator, including those to be discussed below,
and, as a matter of principle, cannot be avoided.
In which sense
the condition (5.105)
can be taken to be satisfied is addressed in some more detail in
subsection 5.3.4.

A second, and more important, disadvantage is that the transfer matrices
of equation (5.102)
do not satisfy the unimodularity requirement of FURSTENBERG's theorem:

(5.78) |

Finally, backpropagation
towards the left
cannot simply be accomplished as in the case of the rotor
by using the same transfer matrices
as for propagation to the right
and setting

The last-mentioned disadvantage is shared by all transfer matrix models of the kicked harmonic oscillator. Essentially, this is again a consequence of the asymmetry (5.87).

In the light of these shortcomings of the model specified by the matrix (5.102), the question needs to be addressed if and how the localization results of subsection 5.1.2 carry over to the oscillator.

One way to approach this problem
is to define the
more general vectors

are suitable unimodular transfer matrices if the satisfy the recurrence relation

the choice of the initial conditions and is free. Using , and choosing and ,

is obtained. No explicit closed formula for the can be given because the same is true for . In this way I obtain

and

Equations (5.113,
5.114)
and the dynamical equation

For proving localization in the kicked harmonic oscillator it suffices to consider the transfer dynamics in positive -direction, because the lattice on which the discrete SCHRÖDINGER equation of the oscillator is defined is infinite towards the right only, as opposed to the bi-infinite lattice in the case of the rotor. So localization towards the left is automatically built into the system by the condition , once FURSTENBERG's theorem has been shown to apply. Nevertheless it is instructive to establish the analogy with the conventional tight binding models as far-reaching as possible. In fact, using the models discussed below, exponential localization is typically found in both directions -- cf. figures 5.12a, 5.12b. What is more, a discrete point spectrum of energy eigenvalues describing the localized dynamics can only be expected in the case of exponential localization in both directions -- cf. subsection 5.1.2.

In the rotor case,
the same transfer matrix can be used for right- and leftward
dynamics on the lattice, provided the two different sets of vectors
, of equations (5.62)
are used.
For the oscillator, on the other hand, the nontriviality of the
off-diagonal matrix elements of the
necessitates the
introduction of
another set of
transfer matrices for leftward transfer.
No vectors
,
could be
constructed that allow to model the
tight binding equation (5.101)
using the same matrix for
propagation into both directions.
In a sense,
the situation in the oscillator case is reversed, as compared to the
rotor: the simplest formulation of backpropagation is
obtained by
using the same vectors
as for propagation in positive -direction, but
another set of
transfer matrices, namely
(quite naturally)
the inverses of the
:

Alternatively, if for the sake of analogy one wants to use the vectors

then one has

with

Summarizing, the transfer matrices , and can be used to describe the right- and leftward tight-binding dynamics on the lattice in all but the pathological cases. Up to this point no statement has been made concerning the respective localization properties that might follow from this description. In the following I discuss just this aspect of the dynamics.

The above construction of a tight-binding model for the kicked harmonic oscillator is made to be as close to the corresponding model for the kicked rotor as possible. However, comparing equations (5.114, 5.117, 5.120) with their rotor counterpart (5.61) makes it clear that the harmonic oscillator case is considerably more intricate, due to the presence of and in the matrix elements. This makes the application of FURSTENBERG's theorem more difficult than in the canonical rotor case.

In many cases, the transfer matrices defined in equations (5.114, 5.117, 5.120) satisfy the assumptions of FURSTENBERG's theorem: the are all unimodular by construction, the respective groups generated by the sets are noncompact and irreducible in the sense of appendix B, and all available numerical evidence supports the assumption that the considered give rise to ``sufficiently well-behaved'' measures as defined by the integrability condition (B.2).

The last remark may be made more explicit by
discussing
in some more detail
the assumptions of FURSTENBERG's theorem
for the considered here.
From appendix B,
the integrability condition is

where is the matrix group to which the belong, is the measure according to which the are distributed in this group, and is the maximum norm of the matrix . For more details on FURSTENBERG's theorem see appendix B.

In order to establish the validity of the condition (B.2) with respect to the maximum norm, the distributions of values of the matrix elements of the need to be checked. The off-diagonal matrix elements of , and are determined by the nearest neighbour interactions , for which no closed formula is available, as discussed earlier. Nevertheless, the corresponding distribution of values can be studied numerically. Figures 5.21b, 5.21c and

5.22b, 5.22c present distributions of and for and two different values of . Obviously, the distributions are nicely peaked, and make it thus plausible that, by integration in the sense of (B.2), they contribute to satisfying the integrability condition. It would be desirable to strengthen this argument by considering other parameter combinations as examples as well, but unfortunately the numerical effort for this task is very high; already the computation of the matrix elements needed for figures 5.21 and 5.22 required a considerable amount of computer time. Still, it is natural to assume that a similarly unproblematic behaviour of the is given for other values of and as well. A more stringent reasoning leading to (B.2) being satisfied is presented in subsection 5.3.3 below.
It remains to
investigate
the nontrivial diagonal matrix elements of the ,
namely

Summarizing, although a rigorous proof is lacking, there is good reason -- including some convincing numerical evidence -- to assume that typically the condition (B.2) is satisfied for the matrices , and modelling the kicked harmonic oscillator.

As a result, I have shown that at least for some combinations of values of the parameters, FURSTENBERG's theorem can be applied to the transfer matrix formulation of the kicked harmonic oscillator in a way which is largely analogous to the conventional procedure in the case of the kicked rotor; it may be assumed that similar results probably hold for many other values of the parameters. By the same reasoning as in subsection 5.1.3 it can be concluded that the norms of the vectors and generated by the respective equations of motion decay exponentially, which in turn implies the same for the absolute values of the expansion coefficients of the quasienergy states discussed in subsection 5.3.1. This finally establishes the result that the nonresonant kicked harmonic oscillator exhibits ANDERSON localization, provided the parameters are such that the conditions are met which I have discussed above. When this is the case then generic sequences of states generated by the quantum map (5.72) consist of exponentially localized states. The states are localized with respect to the basis of harmonic oscillator eigenstates, and therefore they are localized in phase space as well. Figure 5.12 stresses the first aspect of localization; figures 5.9 through 5.11 and the examples in section C.4 of the appendix demonstrate localization in phase space.

As discussed in subsection 5.1.2, a lower bound for
the speed of the decay of
is established via the
LIAPUNOV exponent of the corresponding transfer matrices. From this
point of view it
does not come as a surprise
that in the case of the oscillator the transfer matrices
and
or
do not coincide as they do in the case of the rotor.
This is again a consequence of the asymmetry expressed by the inequality
(5.87) and can lead to different speeds of convergence to
zero of
on the right and on the left.
But note that FURSTENBERG's theorem provides a lower bound on the speed
of convergence only; the speed may be actually larger, and
in special cases
it
*can*
coincide on the right and on the left.

Inevitably, the above discussion of the distribution of values of the matrix elements -- especially of the off-diagonal matrix elements -- of the transfer matrices is to some degree heuristic, and therefore the same is true, unfortunately, with respect to the distribution of the transfer matrices. In particular the arguments in favour of nicely distributed off-diagonal matrix elements leading to FURSTENBERG-integrability are but qualitative. This problem is addressed -- and solved -- in subsection 5.3.3 by considering a different class of transfer matrices.

- ... condition
^{5.11} - Note that implies , because .
- ....
^{5.12} - Note that due to the intricacy of the numerical evaluation of the , making it very computer time consuming, only a much smaller number of values of the could be taken into account here than in figure 5.17a ( values versus values), thus giving rise to the quite coarse-grained histograms in figure 5.21 and 5.22.