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\begin{document}
\title{Random Schr\"odinger operators with Wavelet interactions}
\author{W~Kirsch$^1$, ~~M~Krishna$^2$, ~~P~Hislop$^3$}
\maketitle
\begin{abstract}
In this lecture, we report on a work in spectral theory of
Random Schr\"odinger operators with independent random interactions.
We study a Schr\"odinger operator on $L^2 ( R )$ with 
a potential that is an infinite sum of rank one perturbations
constructed from a wavelet basis, with random, independent coupling
constants. An interval of dense pure point
spectrum and exponentially decaying eigenfunctions
are shown to exist almost surely for such models.
\end{abstract}

In this lecture we report on a work on random Schr\"odinger operators
with interactions diagonal in  a wavelet basis with the diagonal
entries independent random variables with a common probability
distribution $\mu$ supported on $\RR$.  We show that there is dense pure
point spectrum in such models with exponentially decaying
eigenfunctions.

The study of Schr\"odinger operators with random potentials originated
in the theory of transport in disordered systems in condensed matter
physics and a model with independent randomness potential living on a
lattice was proposed and Anderson  originally argued that there is
dense localized states in such models.  Such a result was proved
in a mathematical framework in the early 80 s by Simon-Wolff,
Fr\"ohlich-Spencer-Martinelli-Scoppola  and Delyon-Levy-Souillard
 using an estimate of Fr\"ohlich-Spencer and 
subsequently several improvements were made of the proofs of this estimate.   
We refer to the review of Simon \cite{bs} and the books of 
Carmona-Lacroix \cite{cl}, Cycon-Froese-Kirsch-Simon \cite{cfks}
and Figotin-Pastur \cite{fp} for more information about these works.

Recently a great simplification of the proof of pure point spectrum in
such models were obtained by Aizenman-Molchanov \cite{am} and Aizenman
\cite{ma}.  All
these results were on the lattice.   For the case of Schr\"odinger 
operator Combes-Hislop \cite{ch} obtained a parallel formulation.

The techniques of Aizenman-Molchanov \cite{am} while being simple are not
suitable for random perturbations of unbounded operators and that is
our motivation to study this problem, to see if there are models where
similar simpler techniques are valid.  We find that the model we
present below is amenable to the analysis in the framework of the work
of Aizenman \cite{ma}.   

\section{The Model}

The model considered is the following.  We take the separable Hilbert
space $L^2(\RR)$ and consider the operators
\begin{equation}
\label{eqn1.1}
H^\omega = -\frac{d^2}{dx^2} + V^\omega, ~~ V^\omega = \sum_{n \in
\ZZ^2} q^\omega(n) P_n,
\end{equation}
self adjoint on a dense domain there. Here we have taken $P_n$ to be 
the orthogonal projection on to the one dimensional
subspace generated by the function $\psi_{n}(x)$ 
when $\psi_n(x)$ is an element of a (orthonormal) wavelet basis 
\begin{equation}
\label{eqn1.2}
\psi_n(x) = \sqrt{2^{-n_1}} \psi(2^{-n_1}x - n_2), ~~ n = (n_1, n_2)
\in \ZZ^2.
\end{equation}
We choose the wavelet $\psi$ to satisfy the following, with $\widehat{\psi}$
denoting its Fourier transform. 
\begin{hyp}
\label{hyp1.1}
Let $\psi$ be a wavelet with 
\begin{enumerate}
\item $\widehat{\psi} \in \cc^4(\RR)$.
\item $\widehat{\psi}$ has compact support in $\RR \setminus
\{0\}$.
\end{enumerate}
\end{hyp}

\noindent
We also assume that $\int | \psi |^2 dx = 1$.
We assume the distribution of randomness to satisfy the hypothesis
\begin{hyp}
\label{hyp1.2}
Let $q^\omega(n), ~n\in\ZZ^2$ be independent and identically distributed random
variables with their common probability distribution $\mu$ being
absolutely continuous and of compact support in $\RR$. 
\end{hyp}

Since $\mu$ has compact support, each $q^\omega(n)$ takes values in a
compact set, so the operator $V^\omega$ is bounded for each $\omega$.
Then, the operator $H^\omega$ is self adjoint on the same domain as 
$-d^2/dx^2$, which is the collection of functions
$$
\{f \in L^2(\RR) : f, f^{\prime} ~\text{are absolutely continuous and} 
~~ f^{\prime\prime} \in L^2(\RR) \}.
$$

Then we prove the following theorem in Kirsch-Krishna-Hislop
\cite{kkh}.

\begin{thm}
Let $H^\omega$ be a family of Random Schr\"odinger operators as in 
equations (\ref{eqn1.1}, \ref{eqn1.2}),  
with the wavelet $\psi$ satisfying the Hypothesis (\ref{hyp1.1}) and 
the random variables $q^\omega(n)$ satisfying Hypothesis
(\ref{hyp1.2}).  Then there is a number $E(\mu) < 0$ such that 
the spectrum of $H^\omega$ in ($-\infty$, $E(\mu)$) is pure point 
with exponentially decaying eigenfunctions for almost every $\omega$.
\end{thm}

Let us note that Hypothesis 1.2 implies that the spectrum of
$H_\omega$ is semibounded. The theorem is nontrivial when the
support of the measure $\mu$ has nontrivial intersection with the
negative real axis.

The idea of proof is the following.  First there is a criterion  
of Aizenman \cite{ma} based on the Simon-Wolff \cite{sw} criterion for
identifying the pure point spectra of rank one perturbations, 
which he used for a class of discrete Schr\"odinger
operators on the lattice $\ell^2(\ZZ^d)$.  
We prove the variant of this in the following form.

\begin{lemma}
\label{lem1.1}
Suppose $A_0$ is a self-adjoint operator on a separable Hilbert space 
and $P_n$ is a collection of orthogonal projections, each of one dimensional
range, and $\sum_n P_n = I$.  Suppose  $A^\omega$ is a family of 
self-adjoint operators given by
$$
A^\omega = A_0 + \sum_n q^\omega(n) P_n,  ~~ a. e. \omega,
$$ 
with $q^\omega(n)$ independent random variables with distribution 
$\mu_n$ satisfying $B_n \equiv \int |x| ~ d\mu_n(x) < \infty$.  Let 
$B = \sup_{n} B_n$.  Consider a constant $K_s$ given by
$$
\int \frac{|x|^s}{|x - \alpha|^s} ~ d\mu_n(x) \leq 
K_s \int \frac{1}{|x - \alpha|^s} ~ d\mu_n(x), ~~ \forall ~~ n, 
$$ 
for some $0 < s < 1$ and 
suppose  $(a, b)$ is an interval in the spectrum  
spectrum of $A^\omega$,  such that for all E in $(a, b)$, the condition
\begin{equation}
\label{eqn1.3}
K_s \sup_{n} \sum_{m} \|P_n (A_0 - E)^{-1}P_m\|^s  < 1, 
\end{equation}
is satisfied.  Then the spectrum of $A^\omega$ in (a, ~b) is pure
point almost every $\omega$. 
\end{lemma}

We remark that firstly when $A_0$ is unbounded
and $\mu$ is not of compact support, we have
to show that the family of operators $A^{\omega}$ is self-adjoint, which in itself
is a hard problem, while for $\mu$ of compact support the operator
$A^\omega$ is self-adjoint for each $\omega$.  Secondly, such a constant 
$K_s$ always exists if each $\mu_n$ is
of compact support and the union of supports of $\mu_n$ is contained
in a compact set.   

We briefly sketch the proof of the lemma in order to illustrate the
method.
The standard second resolvent identity easily leads one to the following
inequality:
\begin{eqnarray} 
\lefteqn{ \EE \{ \; \| P_n ( A^\omega - E)^{-1} P_m \|^s \; \} }
   \nonumber  \\
 & \leq &  \| P_n ( A_0 - E)^{-1} P_m \|^s  + \nonumber \\
 &      &  \sum_k \EE \{ \; |q_k|^s \;  \| P_k ( A^\omega - E)^{-1}
        P_m \|^s  \} \; \| P_n ( A_0 - E)^{-1} P_k \|^s .
\end{eqnarray} 
In order to evaluate the expectation of the product of the random
variables $|q_k|^s$ and $\|  P_k ( A_k^\omega - E)^{-1} P_m \|^s$,
we use a decoupling lemma. This requires that we write a similar
resolvent identity comparing $A^\omega$ with $A_k^\omega \equiv
A^\omega - q_k | \psi_k \rangle \langle \psi_k |$. This easily leads
to the formula
$$
| \langle \psi_k , ( A^\omega - E)^{-1} \psi_m \rangle | =
\frac{ | \langle \psi_k , ( A_k^\omega - E)^{-1} \psi_m \rangle | }{
1 + q_k \; |\langle \psi_k , ( A_k^\omega - E)^{-1} \psi_k \rangle |
} .
$$
Consequently, we see that
\bea 
\lefteqn{ \EE \{ \; |q_k|^s \;  \| P_k ( A_k^\omega - E)^{-1}
        P_m \|^s  \} } \\ 
 \leq & \EE \left\{ \frac{ |q_k|^s }{ 1 + q_k \; |\langle \psi_k , (
A_k^\omega - E)^{-1} \psi_k \rangle | } \; | \langle \psi_k , (
A_k^\omega - E)^{-1} \psi_m \rangle | \right\} .
\eea
This form is similar to that appearing in the definition of $K_s$.
Indeed, we first take the expectation with respect to the
$q_k$-variable to obtain
$$
\int \; \frac{ |q_k|^s }{ 1 + q_k \; |\langle \psi_k , ( 
A_k^\omega - E)^{-1} \psi_k \rangle | } \; d \mu_k \leq K_s \int
\frac{ d \mu_k }{ 1 + q_k \; |\langle \psi_k , ( 
A_k^\omega - E)^{-1} \psi_k \rangle | } ,
$$
and then combine this result with the expectation over the remaining
variables to obtain
\bea 
\lefteqn{ \EE \{ \; |q_k|^s \;  \| P_k ( A_k^\omega - E)^{-1}
        P_m \|^s  \}   } \\ 
 & \leq &  K_s \EE \{ \| P_k ( A_k^\omega - E)^{-1} P_m
\|^s  \} .
\eea
Substituting this back into equation (4), we obtain
\begin{eqnarray}
\lefteqn{ \EE \{ \; \| P_n ( A^\omega - E)^{-1} P_m \|^s \; \} }
   \nonumber  \\
 & \leq &  \| P_n ( A_0 - E)^{-1} P_m \|^s  + \nonumber \\
 &      &  K_s \; \left\{ \sum_k \EE \{  \| P_k ( A^\omega - E)^{-1}
        P_m \|^s  \} \right\} \; \| P_n ( A_0 - E)^{-1} P_k \|^s .
\end{eqnarray}
In order to obtain estimate (3), we define 
$$
W_E \equiv \sup_m \; \sum_n \; \EE \{ \| P_n ( A^\omega - E)^{-1} P_m
\|^s \} ,
$$
and similarly for the unperturbed operator $A_0$,
$$
W_E^0 \equiv \sup_m \; \sum_n \; \EE \{ \| P_n ( A_0 - E)^{-1} P_m
\|^s \} .
$$
Our equation now reads
$$
W_E ( 1 - K_s W_E^0 ) \leq W_E^0 .
$$
The invertibility of the factor $( 1 - K_s W_E^0 )$ requires the
condition in the lemma. We remark that the exponential decay of the
Green's function follows from iterating the equation above.

In order to apply the lemma to models, we
must verify inequality (\ref{eqn1.3}).
This inequality may not be valid for
arbitrary $P_n$ for any $0 < s < 1$, since the quantity
$$
 \|P_n (A_0 - E)^{-1}P_m\| 
$$ 
is only square summable in m for each fixed n.  Our choice of $P_n$
coming from the wavelets as in the Hypothesis (\ref{hyp1.1}) is used
to achieve this estimate. 

We now give a sketch of the proof of the estimate (\ref{eqn1.3})
for the model (\ref{eqn1.1}-\ref{eqn1.2}).
First we note that
$$
\|P_n (A - E)^{-1}P_m\|  = |<\psi_n, (A - E)^{-1} \psi_m>|, 
$$
if $\psi_n$ is the normalized vector determining $P_n = 
| \psi_n \rangle \langle \psi_n | $.
as in (\ref{eqn1.2}).  Let us set
$$
T(n, m, ; E) = |<\psi_n, (-d^2/dx^2 - E)^{-1} \psi_m>|, ~~n, m \in \ZZ^2, ~~ E
\in (-\infty, 0).
$$

We now compute $T(n, m, ; E)$ by first 
using the Plancherel Theorem to
re-express the matrix element in terms of the
Fourier transform. We use the fact that
$-d^2/dx^2$ is unitarily
equivalent to multiplication by the function $\xi^2$ acting on 
$L^2(\RR)$ under the unitary map given by the Fourier transform. 
\begin{equation}
\label{eqn1.4}
\begin{split}
T(n, m, ; E) &= |\int_\RR d\xi ~~ 
\overline{\widehat{\psi_n}(\xi)}\widehat{\psi_m}(\xi)
\frac{1}{(\xi^2 - E)}|\\
T(n, m; E) &= |2^{\frac{n_1+m_1}{2}}\int_\RR d\xi ~~ 
e^{i(2^{n_1}n_2 - 2^{m_1}m_2)\xi}
\overline{\widehat{\psi}(2^{n_1}\xi)}\widehat{\psi}(2^{m_1}\xi)
\frac{1}{(\xi^2 - E)}|\\
&\leq C(a, E, \psi)
\begin{cases}
 &\frac{2^{4N}}{|n_2 - m_2 2^{m_1 - n_1}|^4}, 
~~~ n_2 \neq m_2 2^{m_1 - n_1}\\
&2^{4N}, ~~~n_2 = m_2 2^{m_1 - n_1},
\end{cases}
\end{split}
\end{equation} 
with 
$$
C(a, E, \psi)
\leq C(a, \psi)
~\text{max}\{\frac{1}{|E|^j}, ~~ j=1, \cdots, 5\}.
$$
In the above the constant $C(a, \psi)$ depends upon the $L^2$
norms of $\{\psi^{(j)}, j = 0, \cdots, 4\}$.

In obtaining the above inequality we have used the fact that 
$$
\widehat{\psi_n}(\xi) = 2^{\frac{n_1}{2}} e^{-in_2
2^{n_1}\xi}\widehat{\psi}(2^{n_1}\xi), ~~ n = (n_1, n_2) \in \ZZ^2,
$$
and performed integration by parts four times, using the fact that 
$\widehat{\psi}$ has compact support away from zero to obtain the
estimate after some work.  We also had to do the estimates differently
for $n_1 >0$ and $n_1 <0$. In the latter case, we first make a change of
variable $2^{n_1}\xi \rightarrow \xi^{\prime}$, and then use the fact that
$|\xi^\prime|$ is bounded below by a positive number, since $\widehat{\psi}$ 
has support away from 0.  ( When $n_1 < 0$, we will need to make
an estimate of the form
$$
|\frac{2^{-n_1}}{((\xi2^{-n_1})^2 - E)}| \leq 
\frac{1}{|\xi|}\frac{|2^{-n_1}\xi|}{|((\xi2^{-n_1})^2 - E)|} \leq
\frac{1}{|a|\sqrt{|E|}} 
$$ 
which uses that $1/|\xi|$ is bounded above).
The N used above is the smallest number such that 
$$
\text{supp}(\widehat{\psi}(\xi)) \cap \text{supp}(\widehat{\psi}(2^{n_1 -
m_1}\xi)) = \emptyset, ~~ if ~~|n_1 - m_1| > N.
$$

Consequent to the above fact, we also have, 
$T(n, m E) = 0, ~~|n_1 - m_1| > N$.  
Therefore when taking sum of $|T(n. m, E)|^s$ over $m_1$ only 
finitely many terms
contribute for each fixed $n_1$ and when taking the sum over
$m_2$, for each fixed $n_2$, we can absorb the integer part of the 
$n_2 2^{n_1 - m_1}$ into $m_2$ and do the sum.  If we do this we
will get a contribution from the fractional part of $m_2 -
n_2^{n_1 - m_1}$ in the denominator.  This is bounded by a ,
perhaps large, number that depends only on N.  Therefore we will
get an estimate of the form, for $1/4 < s < 1$, for some
absolute constant D(N, s) depending upon N and s, 
$$
\sum_{m \in \ZZ^2} |T(n, m , E)|^s \leq C(a, \psi)^s ~  D(N, s)  
~ \text{max}\{\frac{1}{|E|^{js}}, ~ j = 1, \cdots, 4\}.
$$

Once the above estimate is done we find that for any $1/4 < s < 1$,
the condition in equation (\ref{eqn1.3}) is satisfied.  Using this one
obtains a proof of the theorem using the lemma, since the constant
$K_s$ in the lemma being finite, taking $E$ large enough one would
have
$$
K_s \sup_{n} \sum_{m} \|P_n (A_0 - E)^{-1}P_m\|^s  < \frac{K_s ~C(a, \psi)
D(N, s)}{|E|^s}
< 1. 
$$  
In fact one can also make $K_s$ as small as one wants by taking
appropriate measure $\mu$ with large support but small moment
$\int |x| ~ d\mu(x)$, but we will not discuss that here.

Finally we give examples of wavelets that satisfy the Hypothesis
(\ref{hyp1.1}).  Lemar\'e-Meyer \cite{pa} wavelets have mother functions in 
the Schwarz space and their Fourier transforms have
compact support in $[-\frac{8\pi}{3}, - \frac{2\pi}{3
}] \cup  [\frac{2\pi}{3},  \frac{8\pi}{3}]$, \cite{pa} theorem 7.  

One starts with functions $\psi_1$ and $\psi_2$, whose supports are in
$[-\pi/3, \pi/3]$ and $[-2\pi/3, 2\pi/3]$ respectively, with the
normalization
$$
\int d\eta ~ \psi_i(\eta) = \frac{\pi}{2}, ~~ i = 1,2.
$$
Then one forms the functions
$$
\theta_i(\xi) = \int_{-\infty}^\xi d\eta ~ \psi_i(\eta), ~~ i = 1,2,
$$
and the function
$$
b(\xi) = \sin(\theta_1(\xi - \pi))\cos(\theta_2(\xi - 2\pi)).
$$
When $\xi$ is positive but below $2\pi/3$, $\theta_1(\xi - \pi)$ is
zero, while for $\xi$ bigger than $8\pi/3$, $\theta_2(\xi - 2\pi)$ is 
$\pi/2$, hence b is zero.  Similarly one can see that it is non-zero
only in the set $[-8\pi/3, -2\pi/3]$.  So such a b has compact support
away from zero.  Then the mother function $\psi$ of the wavelet has
the Fourier transform 
$$
\widehat{\psi}(\xi) = \frac{\text{Sgn}\xi}{\sqrt{2\pi}}e^{i\xi/2}b(|\xi|),
$$
and gives rise to a wavelet basis
$$
\psi_{n}(x) =  2^{-n_1/2} \psi(2^{-n_1}x - n_2), ~~ n \in \ZZ^2. 
$$

We note here that when the supports of  $\widehat{\psi_{n}}$ and
$\widehat{\psi_m}$ are disjoint, their orthogonality is trivial.
Even otherwise, for $n \neq m$, their orthogonality comes
from the cancellations induced by the functions defining $\psi$.

Other such examples using local sine and cosine functions with the
function b above having compact support away from zero
were constructed by Aucher-Weiss-Wickerhauser \cite{pa}.  Their
wavelets are indexed by the set $\ZZ \times \ZZ^+$ and give
orthonormal bases for $L^2(\RR)$ having the properties stated in the
Hypothesis (\ref{hyp1.1}).  

{\noindent \bf Acknowledgment:} We thank Prof P K Jain for an
invitation to one of us (MK) to present this work in this
conference.


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\endthebibliography

\vspace{1cm}

\leftline{$^1$Institute f\"ur Mathematik, Ruhr-Universit\"at-Bochum,
Germany}
\leftline{$^2$Institute of Mathematical Sciences, Chennai, India}
\leftline{$^3$Department of Mathematics, University of Kentucky, Lexington,
USA}

\end{document}

