Dynamical Localization of the Quasi-Periodic Schrödinger Operators ()
1. Introduction
The spectral theory of Schrödinger operators with random or almost periodic potentials has been an area of very active study since the late 1970’s. From the beginning, it has been understood and emphasized that these two classes of models share an important property, namely that the potentials can be generated dynamically. On the one hand, this makes a unified proof of basic spectral results possible, such as the almost sure constancy of the spectrum and the spectral type, since they hold as soon as the dynamical framework is fixed and an ergodic measure is chosen. On the other hand, by the very nature of the dynamical definition of the potentials, it comes as no surprise that tools from dynamics will enter the spectral analysis of these operators.
1.1. Quasi-Periodic Schrödinger Operator on
Consider the one-dimensional quasi-periodic Schrödinger operator
(1)
where
is a real number and v is a smooth function on
.
We may assume the following on the data:
- Diophantine condition on the frequency
: That is:
(2)
for some constants
and
.
- v is a function of class
, satisfies:
(3)
1.2. Anderson Localization
We say that an operator satisfies Anderson localization if it has pure point spectrum with exponentially decaying eigenfunctions.
1.3. Dynamical Localization
Another localization criterion stronger than Anderson localization, this is called dynamical localization. Consider the evolution equation in time associated to
,
(4)
where
. We say that
satisfies,
The dynamical localization (D. L), if for a.e
,
(5)
The strong dynamical localization (Strong D.L), if,
(6)
The main result of this paper is the following:
Under the assumptions ((2) and (3)), we prove the following:
Theorem 1. 1) Assume that
and v are as above, then there exists a constant
such that:
If
then
is pure point with a set of exponential decaying eigenfunctions which form an orthonormal basis of
for all
.
2) Assume that (2) and (3) are hold, then for a.e
the operator
satisfies the strong dynamical localization (D.L) for all
.
Remark.
This result improves in some way the previous one by Eliasson, such that dynamical localization is proven with an appropriate potential. To my knowledge, there are no results on the spectral properties of Schrödinger operators with discontinuous potential. The ideas presented in this paper can be used to obtain new results for several models with discontinuities.
Let us review now, some of the results in the literature that are most relevant to this paper:
In [1] , L.H. Eliasson considered the operator
given by (1) with frequency
satisfying a Diophantine condition and the function v satisfying a Gevrey-class regularity and a transversality condition. Under these assumptions, he proved using KAM methods that for
where
depends on the function v and on the Diophantine condition on
the operator
has pure point spectrum for a.e.
. Moreover, this implies, using Kotani’s theory (see [2] ) that the Lyapunov exponent is nonzero for a.e. energy E. The author has also suggested that the argument could be modified to obtain exponential decay of the eigenfunctions, but without proof.
J. Bourgain and M. Goldstein considered (see [3] ) the operator
given by (1) where
satisfies a Diophantine condition and v is a non-constant analytic function. They also assumed that the Lyapunov exponent is positive for a.e.
and for all E. The authors proved that the operator
satisfies Anderson localization with exponential decay of the eigenfunctions at almost Lyapunov rate for every
and for a.e.
. Their result is nonperturbative -the constant
depends only on the potential v. In this paper we use the KAM approach which is a perturbative method―the constant
depends on v and
-with different conditions on v, also we prove the Dynamical localization which is stronger than Anderson localization.
For the quasi-periodic model, and unlike Anderson’s case, there were fewer results that were found for this kind of localization. However, several results on the (D.L.) were published for the random model, for more references see [4] [5] .
In the case of quasi-periodic models, this localization phenomenon (D.L) implies Anderson localization, and which also implies by the RAGE theorem that the spectrum is purely punctual (see [6] ). In view of this, these models are natural candidates for (D.L). In this context, F. Germinet and S. Jitomirskaya (see [7] ), have improved the results of [8] and [9] , by proving the strong (D.L) of the operator
, for all
and diophantine
. Later, in 2004, J. Bourgain and S. Jitomirskaya announced (without demonstration) this result for the quasi-periodic Schrödinger operators, see [10] for more details.
Quasi-periodic operators have been heavily studied over the years; we direct the reader to the survey [11] for a guide on the literature.
1.4. Idea of Proof
1.4.1. KAM Theory
KAM theory is the perturbative theory, initiated by Kolmogorov, Arnold and Moser in the 1950s, of quasi-periodic motions in conservative dynamical systems. This theory deals with the persistence, under perturbation, of quasi-periodic motions in Hamiltonian dynamical systems. An important example is given by the dynamics of nearly integrable Hamiltonian systems. In general, the phase space of a completely integrable Hamiltonian system of n degrees of freedom is foliated by invariant n-dimensional tori (possibly of different topology). KAM theory shows that, under suitable regularity and non-degeneracy assumptions, most (in measure theoretic sense) of such tori persist (slightly deformed) under small Hamiltonian perturbations. The union of persistent n-dimensional tori (Kolmogorov set) tends to fill the whole phase space as the strength of the perturbation is decreased. The major technical problem arising in this context is due to the appearance of resonances and small divisors in the associated formal perturbation series.
1.4.2. Application to the Schrödinger Operators
The method of proof is a refinement of an already refined KAM method developed by Eliasson in a series of fundamental papers in the theory of quasi-periodic Schrodinger operators (especially [1] ). The method consists of an infinite sequence of transformations aiming at conjugating the infinite dimensional matrix defined by the operator on
:
to a diagonal matrix
, by an orthogonal matrix made up of a complete set of eigenvectors. An iterative procedure that permits us to construct a such matrix,
that conjugate
closer and closer to a diagonal matrix
.
In the perturbative regime, these matrices are perturbations of diagonal matrices and the problem is to diagonalize them completely or partially, i.e. to show that they have some point spectrum. The unperturbed matrices have a dense point spectrum so that their eigenvalues are, up to any order of approximation, of infinite multiplicity, which is a very delicate situation to perturb. For matrices with strong decay of the off-diagonal elements, this difficulty can be overcome if the eigenvectors are sufficiently well clustering. One way to handle this is to control the almost multiplicities of the eigenvalues. The eigenvalues are given by functions of one or several parameters and in order to control the almost multiplicities it is necessary that these functions are not too flat. If the parameter space is one-dimensional and if the quasi-periodic frequencies satisfy some Diophantine condition, then it turns out that this control of the derivatives of eigenvalues is not only necessary, but also sufficient for the control of the almost multiplicities. If the parameter space is higher-dimensional this control is more difficult to achieve and not yet well understood.
2. Iterative Study
This section is organized in the following way:
- A first part devoted to the study of the first step of the iteration described in the previous paragraph. Under some conditions on v and
we construct the matrices
and
which satisfy the estimates of Lemma 2.
- In the second part and after a suitable choice of parameters, an inductive proposition, Proposition 3 is introduced in order to prove the first result in Theorem 1, which is a simple consequence of Lemma 4.
- At the end we give the proof of Theorem 1(2).
Consider now the symmetric infinite-dimensional matrix that depends on the parameter
,
with,
.
For the formulation of the first step of iteration we shall assume the following:
- The rotation number
and the potential v satisfy (2) and (3).
-
.
- Consider the equation:
(7)
where the matrices X,
and
are defined in the following way:
Let
for
.
1) The matrix X is defined by
and satisfies the equation:
(8)
where
.
2)
.
3)
.
Lemma 2. Let
and
.
If
then:
1)
a)
.
b)
.
c)
where
.
2)
a)
.
b)
.
3)
a)
where
.
b)
.
c)
.
4)
.
Proof.
1(a): Let
and
, we have:
therefore
since
, then it follows that
1(b): Using 1(a) we obtain:
(9)
Thus from the generalized Young inequality [12] (page 9), (9) implies an estimate of X in the operator norm on
.
1(c): By lemma
(Eliasson [13] ) and for all
we deduce that:
hence
.
2(a): Let
then we have:
.
Now we have to estimate the elements of all matrices which constitute the matrix
.
(*)1
where
.
In the same way we get:
(*)2
.
(*)3
.
(*)4
.
(*)5
This gives
2(b):
.
therefore
(10)
It follows from the generalized Young inequality [12] (page 9) that (10) implies the desired estimate of
in the operator norm on
.
3(a): Since
then
therefore
thus
3(b): In order to estimate
, we have to find an upper bound of
. We have
then
, which implies:
(*)1
In the same way we get:
(*)2
(*)3
(*)4
.
Since
is bounded for all
then
therefore
and
where all constants
and
depend on
and
. It follows that
3(c):
4) By construction of
the result follows immediately.
3. Induction
Let a, b such that
and consider
,
,
,
,
,
and for all
we define the sequences
These parameters are defined in an iterative way and it is with which we will be able to define the matrices
and
satisfying
(11)
where the matrices
,
and
are defined in the following way:
1) The matrix
is defined by
and satisfies the equation
(12)
where
.
2)
.
3)
. and satisfying the property
described in the following proposition.
Proposition 3. Let
. If
,
then the following property
is holds.
Proof. A direct application of Lemma 2 allows us to obtain the desired result for each n.
4. Study of Convergence
Now we will deal with the study of the convergence of our iteration. We will therefore look at the conditions and the size of
with which we will have the convergence, this will be the goal of the next lemma. Finally, we conclude with the proof of Theorem 1 which is a simple deduction of Proposition 3 and Lemma 4.
Lemma 4. Suppose that
Then for
we have for all n:
1)
.
In particular
.
2)
.
Proof: 1) The result is holds for
. Suppose that the result remain holds for
thus
. Now we shall prove that the result is also true for
.
Let
, we have
Since
, hence
, which proves that
.
2) By 1) we have
then
, thus
hence
.
Remark. 1) One can assume without loss of generality that
and we have the same result, in fact the operators
and
have the same spectral properties.
2) The real b exists and satisfying all conditions.
Proof of Theorem 1. 1) The operator
is identified to matrix
with
. Then for
and
we have the existence of matrices
and
for all
such that for all
,
, where
is a diagonal matrix,
,
and
.
Therefore
and
with
is a diagonal matrix. All convergence are fulfilled for all
.
On the other hand
in norm and for all
with
is an orthogonal matrix. In fact: Let
we have
converges if and only if
converges, now since
then we have the existence of U for all
. Moreover from Lemma 4 and for
the matrix
is pure point with finite-dimensional eigenvectors for all
and the measure of
goes to 0 as
. The eigenvectors of
are formed by the columns of U.
2) Let
for
, so by the exponential decaying of eigenfunctions we can easily deduce that
5. Conclusion and Suggestion
The result in Theorem 1 improves the previous one by Eliasson, such that dynamical localization is proven with an appropriate potential. To my knowledge, there are no results on the spectral properties of Schrödinger operators with discontinuous potential. The ideas presented in this paper can be used to obtain new results for several models with discontinuities. For example, one can consider the quasi-periodic operator defined for
in the torus
with a piecewise smooth potential or even functions in a piecewise Gevrey class, and see if similar localization results can be obtained.