Localization in All but the Pathological Cases

In the previous subsection implicitly the ``classical'' approach to the
modelling by means of transfer matrices was
followed:
only the first
component of the vector equations
(5.115, 5.116, 5.119)
was constructed to be equivalent to the tight binding equation
(5.101), while the second component
was made to be a *trivial* identity by construction;
in addition, only the three amplitudes
,
and
were allowed to contribute for any given .
In the present
subsection I use a
more general
starting point for deriving transfer matrices
which are
more suitable for proving localization than
,
and
.

Generalized or *higher order transfer matrix models* reformulating
the tight binding equation (5.101) can be
introduced
by considering -dimensional transfer matrices
(rather than the only two-dimensional transfer matrices as used up to this
point) with some suitable :

The importance of condition (iii) is easily underestimated.
It avoids the construction of models that are only *consistent* with
the tight binding equation, without being *equivalent* to it. Such a
situation occurs, for example, if one specifies the parameters in such a
way that
the only nontrivial subequation of (5.122c) is a
nontrivial
linear combination of
(5.101);
then it is not guaranteed that the
solutions
of such a model also satisfy the tight
binding equation.

This scheme allows to construct a multitude of different transfer matrix models. Once a model has been defined in this way, the remaining conditions of FURSTENBERG's theorem need to be checked, thus possibly leading to a proof of localization, if the model has been constructed properly.

Following this scheme, one can discuss the transfer matrix model with
and , for example.
I have tried to choose the 18
parameters of this model in such a way that the resulting expressions
are as simple as possible. After some
(lengthy)
algebra, the
transfer matrix

The
system (5.123) is suited for
propagation in positive -direction.
Backpropagation can be achieved by using

It is easy to see -- by explicitly writing down all three subequations --
that the first subequation of the equation of motion
(5.123c) contains the tight
binding equation *three* times (in the sense of (iv)), the second
subequation is equivalent to the tight-binding equation (satisfying
condition (iii)), and the third subequation is just a trivial identity.

The do not share the worst disadvantage of the discussed in the previous subsection. Most matrix elements take on the trivial values 0 or 1, which are uncritical with respect to the integrability condition (B.2); the only nontrivial matrix elements, and , are characterized by Lorentzian distributions of values, as figure 5.23 shows.

In fact, for both values of considered in this figure, the approximating Lorentzians are nearly identical to those in figures 5.21a and 5.22a, respectively, where they approximate the distributions . This indicates that the quotient of two consecutive nearest neighbour matrix elements -- which marks the only difference between and -- does not take on values large enough to make and differ significantly, an observation that is supported by figure 5.24, which displays typical values of for several values of and . Apparently the distribution of all nontrivial matrix elements of the is governed essentially by the Lorentzian distribution generated by the (quasi-) random number generator (5.81).
Summarizing, for the parameters considered explicitly, and most probably
for *all* parameter combinations belonging to the nonpathological
systems with
for all ,
and
satisfy
the integrability condition (B.2)
and therefore give rise to
quantum localization of the -nonresonant kicked harmonic oscillator;
the localization mechanism is identified as that of classical
ANDERSON localization.