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Spectral Theory, Discrete Dirac Operator, Hausdorff dimension
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\begin{document}

\title{{Sparse 1D  discrete Dirac operators II: Spectral properties}}

\author{S. L. Carvalho\thanks{
Supported by FAPESP under grant \#2010/10236-3. Email: \texttt{silas@dm.ufscar.br}},
\ C. R. de Oliveira\thanks{Partially supported by CNPq.
Email: \texttt{oliveira@dm.ufscar.br}} \ \\
\small Departamento de Matem\'atica -- UFSCar, \\
\small S\~{a}o Carlos, SP,  13560-970 Brazil \\
 \author{ R. A. Prado\thanks{ 
Email:\texttt{\ robertoprado@fct.unesp.br}}} \\ R. A. Prado\thanks{Partially
supported by PROPe/UNESP and FUNDUNESP. Email: \texttt{robertoprado@fct.unesp.br}}
\\ 
\small Departamento de Matem\'{a}tica, Estat\'{\i}stica e Computa\c{c}\~ao -- UNESP,
\\ \small Presidente Prudente, SP,
19060-900 Brazil}
\date{}
\maketitle

\begin{abstract}
We study spectral properties of some discrete Dirac operators 
with nonzero potential only at some sparse and suitably randomly distributed 
positions. As observed in the corresponding Schr\"o\-ding\-er operators, we
determine the 
Hausdorff dimension of its spectral measure and identify  a sharp spectral
transition from  point to singular continuous.
\end{abstract}



\section{Introduction\label{2}}

\setcounter{equation}{0} \setcounter{theorem}{0}

Sparse potentials have played a special role in the context of discrete (also called
``tight-binding'') Schr\"o\-ding\-er operators, particularly due to the possibility of 
rather detailed spectral and dynamical analysis, which makes such investigations
especially interesting (see, for instance,
\cite{CMW,CombesMantica,JitLast,Krutikov,MWGA,Tcherem,Zla}).  Pearson \cite{Pearson}
was the first to recognize the
utility of this kind of 
potential on the construction of Schr\"o\-ding\-er operators with singular
continuous spectrum (see  Chapter~13 in~\cite{ISTQD} for dynamical interpretations
of different spectral types). There are many works that deal with sparse potentials,
related to distinct sparseness conditions and `barriers' with distinct growing rate 
(see the above mentioned references and~\cite{L2} for a brief discussion and a
collection of results).

In the works \cite{OP,OP1}, two of the present  authors have proposed a discrete
version of
the unidimensional Dirac operator, now defined on $l^2(\mathbb{Z},\mathbb{C}^2)$, 
which can be considered a relativistic version of the usual discrete Schr\"o\-ding\-er
operator (with Planck's constant $\hbar\equiv 1$)
\begin{equation}\label{HNRe}
(H_S\psi)_n=-\frac{1}{2m}(\Delta\psi)_n+V_n\psi_n=\frac{1}{2m}\left(
\psi_{n+1}+\psi_{n-1}-2\psi_n\right)+V_n\psi_n\; ,
\end{equation}
\noindent
 where $(V_n)$ is the potential given by a real sequence and $m>0$ denotes the
particle mass. Explicitly, the action of the discrete Dirac operator is given by
\begin{equation}\label{MDir}
(H_D)(m,c)=H_0(m,c)+V\,{\mathrm I_2}=c\mathcal{B}+mc^2\sigma_3+V\,{\mathrm I_2}\;,
\end{equation}
\noindent
where $c>0$ represents the speed of light,
\begin{equation*}
\mathcal{B}=\left(\begin{array}{cc}
0 & d^\ast \\
d & 0 
\end{array}\right),
\end{equation*}
\noindent
$\sigma_3$ is the Pauli matrix $\left(\begin{array}{cc}1 & 0\\0 & -1 
\end{array}\right)$, ${\mathrm I_2}$ is the $2\times2$ identity matrix, and~$d$ is
the finite difference operator (a discrete version of the first derivative)
\begin{equation*}
(d\psi)_n=\psi_{n+1}-\psi_n\;,
\end{equation*}
\noindent
with its adjoint given by $(d^\ast\psi)_n=\psi_n-\psi_{n-1}$; thus, $H_0(m,c)$ 
is a bounded self-adjoint operator  with the mass parameter taking values $m\ge0$,
an important difference with respect to nonrelativistic models.

The resulting Dirac evolution equation can be written in the compact form
\begin{eqnarray}\label{HRel}
i\frac{\partial\Psi_n}{\partial t}=(H_D(m,c)\Psi)_n=\left(\begin{array}{cc} mc^2
+V_n & cd^\ast \\ cd & -mc^2 +V_n\end{array}\right)\Psi_n\;,
\end{eqnarray}
\noindent
with the ``spinorial'' wave function $\Psi=(\Psi_n)$, with
$\Psi_n=\left(\begin{array}{c}\psi_{1,n} \\ 
\psi_{2,n}
\end{array}\right)$, where $\psi_{1,n}$ and $\psi_{2,n}$ are the solutions associated,
respectively, with 
positive and negative energy values (see, for instance, Chapter~3 in~\cite{Sa}
or~\cite{thaller}).  It is possible to show  \cite{OP,OP1} that the nonrelativistic
limit ($c\rightarrow\infty$)
of such Dirac operator is the corresponding Schr\"o\-ding\-er operator
\eqref{HNRe}, as one would expect.

In a previous work \cite{deOP1}, some lower bounds of the dynamics generated by
$H_D(m,c)$ with sparse potentials have been obtained. In  the present work we
continue our investigations on the discrete Dirac operator with sparse potentials by
presenting a detailed analysis of spectral properties of~\eqref{MDir} defined in
$l^2(\mathbb{Z}_+,\mathbb{C}^2)$ (here
$\mathbb{Z}_+$ represents the set of all nonnegative integers~$n\ge0$), satisfying
the boundary condition,
\begin{equation}
\psi_{2,-1}\cos\phi-\psi_{1,0}\sin\phi=0\,,
\label{bouconphi}
\end{equation}
with $\phi\in[0,\pi]$, and subject to randomly sparse perturbations   
composed of infinitely many vertical `barriers' whose distances from consecutive
barriers 
are rapidly growing. These barriers 
may assume distinct sizes, that grow, diminish or remain constant. 
The 
randomness in the distribution will play a fundamental role in the 
determination of the exact Hausdorff dimension of the spectral measure. We will
consider a set $\{a_j\}_{j\ge 1}$ of a rapidly increasing sequence of natural
numbers, so that the potential $V_n=0$ if $n\notin\{a_j\}_{j\ge 1}$ and nonzero
$V_{a_j}$; see ahead for precise statements.




We, nevertheless, emphasize the results obtained by Zlato\v{s} \cite{Zla} in the
context of Schr\"o\-ding\-er operators, who
has proved the 
existence of a transition between pure point and singular continuous spectra
for a class of sparse operators such that $\{V_{a_j}\}_{j=1}^\infty$ is a constant
sequence. Depending on the sparseness and
on the intensity of the barriers, it is possible
 to have pure point or singular continuous spectrum, and with an explicitly
determination of the  Hausdorff dimension of the spectral measure.

A  similar deterministic model was proposed  in~\cite{MWGA}, but with off-diagonal
perturbations. 
Carvalho \textit{et. al.} \cite{CMW} extended this operator to the strip
$\Lambda:=\mathbb{Z}_+\times\{0,1,\ldots,L-1\}$ of width $L$ in the $\mathbb{Z}^
2_+$ plane, obtaining the Hausdorff dimension associated with its spectral 
measure with arbitrary precision.
In \cite{CMW1}, the same authors have dealt with a modification of the
(unidimensional) Schr\"o\-ding\-er model proposed in~\cite{MWGA}, in which they have
considered some randomness in the position
of the barriers, and this has allowed the determination of the exact Hausdorff 
dimension of the spectral measure.

Hence, by taking into account our previous study~\cite{deOP1}, it is natural to try
to extend these spectral results obtained for Schr\"o\-ding\-er operators to the Dirac
model. In order to achieve our goal, we apply and extend to this
new situation the main techniques discussed in~\cite{CMW,CMW1}. This work is to be
considered a natural continuation of~\cite{deOP1}.

Despite  some results obtained here are valid for a rather general class of
potentials, we deal essentially with the sequence $(V_n(\omega))$ of
barriers given by
\begin{eqnarray}
V_n = \left\{ \begin{array}
            {c@{\quad \mathrm{if} \quad}l}
             v   &  n=a_j^{\omega} \in \mathcal{A} \, ,\\
             0 & n \not\in \mathcal{A} \, ,
             \end{array} \right.
\label{Ve}
\end{eqnarray}
\noindent
where $v\neq 0, -\infty<v<\infty$ and $\mathcal{A}=(a_j^\omega)_{j\ge 1}$ 
is a random sequence of natural numbers written in the form $a_j^\omega = a_j +
\omega_j$ such that
$a_j$ satisfies
\begin{eqnarray}
a_j-a_{j-1} \ge 2 \;, \qquad \qquad j=2,3,\ldots
\label{spacon1}
\end{eqnarray}
\noindent
and
\begin{eqnarray*}
\lim_{j\rightarrow\infty}\frac{a_{j+1}}{a_j} = \beta >1 \;;
\end{eqnarray*}
\noindent
$\omega=(\omega_1,\omega_2,\ldots)$ 
represents a sequence of independent random variables defined on a probability
space $(\Xi,\mathcal{B},\nu)$, so that there is some $\eta>0$ with~$\omega_j$
uniformly distributed over the finite set
$\left\{0,1,2,\ldots,\left[j^\eta\right]\right\}$,  for all~$j
$ (here, $[x]$ denotes the integer part of $x\in\mathbb{R}$).


Condition~\eqref{spacon1} makes each barriers placed at a unique point, 
with $\beta$ the ``sparseness parameter.'' In order to simplify our analysis, we
follow~\cite{MWGA} and restrict the
separation between barriers by the identity
\begin{eqnarray}\label{spacon}
a_j - a_{j-1} = \beta^j \;, \qquad \qquad j=2,3,\ldots
\end{eqnarray} 
\noindent
with $a_1+1=\beta \ge 2$ an integer (we might have considered any real
number $\beta>1$ 
and rewrite~\eqref{spacon} as $a_j - a_{j-1} = \left[\beta^j\right]$,  
$j=2,3,\ldots$). Summing up, the distance between (average) consecutive nonzero
potential positions grows exponentially with an additional power-law randomness;
thus, there is no superpositions among the barriers, for any $\eta>0$ (this 
is an improvement with respect to Zlato\v{s}'s work that has considered $\eta=1$). 


\DEFI \label{defiHvphi} Denote by $H_{v,\phi}(m,c)$ the Dirac operator~\eqref{MDir}
acting on $l^2(\mathbb{Z}_+,\mathbb{C}^2)$, with potential $(V_n(\omega))_{n\ge 0}$
satisfying \eqref{Ve}--\eqref{spacon}, and the 
$\phi $-boundary condition~\eqref{bouconphi} at~$n=-1$. This will be the main object
of study here.
\DEFF


%\OBSI Note that we whereas the distances between the average values (in
%the distribution $\nu$) of the positions of the barriers grow 
% exponentially (thanks to condition~\eqref{spacon}), the uncertainty of these
%positions (due to $\omega_j\in\{0,\dots,[j^\eta]\}$) grows only as a power law with
%the sparseness index; so, there is no superpositions among the barriers. 
%\OBSF

Besides the verification of a huge amount of technical details and suitable
adaptations, the main difficulties we have found in such extension to the (Dirac)
relativistic framework were:
\begin{itemize}
\item The nonlinearities in the Dirac transfer matrices entries (as functions of
energy and potential values) when compared with the Schr\"o\-ding\-er case; see
Eq.~\eqref{mt}.
\item In contrast to the block-Jacobi matrices in~\cite{CMW}, the spectral matrix
$\Omega$ in the Dirac case (see
Proposition~\ref{TAC}) reflects the impossibility of decoupling the upper and lower
spinor components.
\item Finding explicit expressions for the Green's function; see Eq.~\eqref{FGreen}
(it has
involved a trial and error process while handling the exact matrix form and
indices!).
\item Finding a  formula for variation of parameters (Lemma~\ref{LFVP}) was 
nontrivial and it also involved a trial and error process, and this was fundamental
for an appropriate version of the Jitomirskaya-Last inequalities for Dirac operators
(Theorem~\ref{JLineq}). 
\item Checking the validity of the spectral criteria of Last-Simon (see
Proposition~\ref{PROPO} and Corollary~\ref{C4.4a} for examples of application of
these
criteria), whose long details are not reported here. 

\end{itemize}



We note that in the process of adaptation of known results to the relativistic
setting, we will avoid  repeating the arguments and just present references when a
proof  turns out to be quite similar to the corresponding one for Schr\"o\-ding\-er
operators (e.g., the spectral criteria of Last-Simon just mentioned).


\OBSI
\label{rank1}The operator $H_{v,\phi}(m,c)$ with $\phi $-phase
boundary condition~\eqref{bouconphi}  may be treated as a rank-one 
perturbation of the same Dirac operator with Dirichlet boundary condition
\begin{equation}
\psi_{2,-1}=0\;.
\label{boucond}
\end{equation}
As a matter of fact, if $H_{v,0}(m,c)$ represents the operator which satisfies
\eqref{boucond}, we have
\begin{equation*}
H_{v,\phi }(m,c)=H_{v,0}(m,c)+E_{0} \tan \phi\;,  \label{Eqdir}
\end{equation*}
where $E_{0}$ is an operator on $l^{2}(\mathbb{Z}_{+},\mathbb{C}^2)$ 
given by $(E_0\Psi)_n=\left(\begin{array}{c} 0 \\ \psi_{2,n}\delta_{n,0}
\end{array}\right)$,  $\delta_{n,0}=1$ if $n=0$ and zero otherwise.
\OBSF





\subsection{Main Results}In this subsection we state the main results and
conclusions of this work. Their proofs will be the discussed in the next sections.
Such results can be shortly described as follows:  determination of the essential
spectrum 
of~$H_{v,\phi}(m,c)$ along with the exact Hausdorff dimension of the spectral 
measure, and proving the existence of a spectral transition.  

What happens with the essential spectrum of $H_{0,0}(m,c)$ (obtained in 
Proposition \ref{SUPO}) when we consider the
sparse barriers? This question is settled by the following theorem, whose proof is
presented in Subsection \ref{SFN2}.

\begin{theorem}
\label{tee} Let $H_{v,\phi}(m,c)$ be the Dirac operator in
Definition~\ref{defiHvphi} and let
\begin{equation}\label{Hprime} 
H^\prime_{0,0}(m,c)=H_{0,0}(m,c)+v\delta_0{\mathrm I_2}\;,
\end{equation}
\noindent
where $(\delta_0\psi)_n=\delta_{n,0}\psi_n$ for any $\psi\in l^2(\mathbb{Z}_+,
\mathbb{C})$. Then,
$\sigma_{\mathrm{ess}}(H_{v,\phi}(m,c))=\sigma(H^\prime_{0,0}(m,c))$.
\end{theorem}


For the Hausdorff dimension, we have

\begin{theorem}
\label{thethe1}  Let~$\Omega$ be the spectral 
matrix measure of $H_{v,\phi}(m,c)$, given by Definition \ref{defiHvphi}, and the sparseness parameter~$\beta\ge2$. Then $\Omega$ is 
absolutely continuous with respect to the spectral measure~$\rho$ introduced in~\eqref{TBor}. Given $\varepsilon >0$ and a
closed interval of 
energies 
\begin{equation}
L\subset I= \left[-\sqrt{m^2c^4+4c^2},-mc^2\right]\cup\left[mc^2,\sqrt{m^2c^4
+4c^2}\right]\;,  
\label{INTI}
\end{equation}
for almost every $\phi\in[0,\pi]$ and almost every~$\omega\in\Xi$, the spectral
measure $\rho$ restrict to~$L$ has
 Hausdorff dimension
\begin{equation}
\alpha _{\rho}(E)=\max\left\{0,1-\frac{\ln r}{\ln \beta }\right\}\;,  
\label{HDim}
\end{equation}
with $r=r(v,E)$ given by~\eqref{erre}.
\end{theorem}

With respect to the existence of spectral transitions, now we state a result
parallel to Theorem~2.4 in~\cite{CMW1}.

\begin{theorem}
\label{tmarwre} Write
\begin{equation}
I_{1}\equiv \left\{ E \in \sigma(H_{0,0}(m,c))\setminus A:r<\beta\right\}  
\label{I1}
\end{equation}
\noindent with  $\beta \in \mathbb{N}$, $\beta \geq 2$ and $A$ as in~\eqref{defAeq}.
Then, for $\nu$-almost every $\omega\in\Xi$, there exists a set $A_1$ of Lebesgue
measure zero such that: 

(a)  the spectrum of~$H_{v,\phi}(m,c)$
restricted to the set $I_1\backslash A_1$ is purely singular continuous;

(b) the spectrum of~$H_{v,\phi}(m,c)$ is pure point when restricted to
$\sigma(H_{0,0}(m,c))
\setminus I_1$ for almost every $\phi \in \lbrack 0,\pi ]$.
\end{theorem}

Theorem~\ref{tmarwre} shows that there exists a sharp transition between
singular continuous and pure point spectrum. Note from~\eqref{HDim} that the
condition for the Hausdorff dimension to be positive is the same for the existence
of singular continuous spectrum, i.e., $\beta>r$. In fact, the set of energies
for which the Hausdorff dimension is zero coincides with the set where the 
pure point spectrum is supported. This result, due to Theorem~\ref{tmarwre} is,
nevertheless, far from trivial. 
Note also that there is no absolutely continuous spectrum for this class of
sparse potentials. 

The proofs of Theorems \ref{thethe1} and \ref{tmarwre} are both presented in
Subsection~\ref{HDST}. Despite some obvious differences due to the
intrinsic nature of the Schr\"o\-ding\-er and Dirac 
tight-binding models, if we compare the results just stated with those obtained in
\cite{CMW,CMW1,
Zla} for sparse potentials satisfying relations 
\eqref{Ve}--\eqref{spacon}, we have got quite similar statements; this does not seem
to be clear from the actions of the corresponding operators and the different forms
of their transfer matrices. In any event, recall
that only in the relativistic case it is meaningful to consider~$m=0$.




\section{The Spectrum of $H_{v,\phi}(m,c)$
\label{SFN}}

\setcounter{equation}{0} \setcounter{theorem}{0}

\subsection{Green function and the spectral matrix}

\label{SFN1}

It is well known that the spectral properties of  any other self-adjoint operator 
are directly related to the behavior of the Green function
$\mathcal{G}(z)$, for $z=E+i\varepsilon$, in the limit $\varepsilon\downarrow 0$.

In the case of the discrete Dirac operator, the expression of the Green function
$\mathcal{G}(z)$  we introduce here is nothing but the $2\times2$ matrix
$\mathcal{G}(0,0;
z)$, whose elements are given  by
\begin{eqnarray}
\mathcal{G}_{m,n}(0,0;z)=\langle e_m\otimes\delta_0,(H_{v,0}-z{\mathrm I_2})^{-1}e_n
\otimes\delta_0\rangle\;,
\label{Magreen}
\end{eqnarray}
\noindent
$m,n=1,2$, where the sequence $\{\delta_n\}_{n\ge 0}$ represents the canonical 
basis for $l^2(\mathbb{Z}_+,\mathbb{C})$ , while $e_1=\left(\begin{array}{c} 1\\
0\end{array}\right)$, $e_2=\left(\begin{array}{c} 0\\1 \end{array}\right)$ is 
the canonical basis for $\mathbb{C}^2$; thus, the sequence $\{e_1\otimes
\delta_n,e_2\otimes\delta_n\}_{n\ge 0}$ constitutes a basis for $l^2(\mathbb{Z}_+,
\mathbb{C}^2)$, a linear space with scalar product denoted by $\langle\cdot,
\cdot\rangle$.
\OBSI The results presented in this subsection refer to the operator $H_{v,0}(m,
c)$, with Dirichlet boundary condition \eqref{boucond}. See Remark \ref{Hvphi}
for a discussion of the results regarding the operator $H_{v,\phi}(m,c)$.
\OBSF

 
Explicitly, $\mathcal{G}(z)$ is given by
\begin{eqnarray}
\mathcal{G}(z)=\frac{1}{c}\left(\begin{array}{cc} u_{1,0}^D(z)\chi_{1,0}(z) & 
u_{1,0}^D(z)\chi_{2,0}(z)\\ u_{1,0}^D(z)\chi_{2,0}(z) & u_{2,0}^D(z)\chi_{2,0}(z)
\end{array}\right)\;,
\end{eqnarray}
\noindent
where $\chi_n(z)=\left(\begin{array}{c}\chi_{1,n}(z)\\\chi_{2,n}(z)\end{array}
\right)= -u_n^N(z)+m(z)u_n^D(z)$ is a $l^2(\mathbb{Z}_+,\mathbb{C}^2)$ solution
to the Dirac equation 
\begin{equation}
H_{v,0}\Psi=z\Psi\;;
\label{eqdir}
\end{equation}
\noindent
for some fixed $z$, the set of all solutions to~\eqref{eqdir} is a linear space 
of dimension 2 (the Dirac equation~\eqref{eqdir} is in fact a system of 2 
differential 
equations of first order), whose appropriate basis is given by the functions
$u^D(z)$ and $u^N(z)$ satisfying the initial conditions
\begin{eqnarray}
\begin{array}{l}
u^D_{2,-1} = 0 \,, \qquad u^D_{1,0} = 1 \,, \\
u^N_{2,-1} = 1 \,, \qquad u^N_{1,0} = 0\,,
\end{array}
\label{DNboucon}
\end{eqnarray}
\noindent
which correspond to Dirichlet and Neumann boundary conditions, respectively;
finally, $m(z)$ is
the well-known Weyl-Titchmarsh function~\cite{CodLev}. 

Equation~\eqref{Magreen} is in fact a special case derived from the general 
formula we have found:
\begin{eqnarray}
\mathcal{G}(i,j;z)=\frac{1}{c}\left\{
        \begin{array}{c@{\qquad \mathrm{if} \qquad}l}\left(\begin{array}{cc} 
u_{1,j}^D(z)\chi_{1,i}(z) & 
u_{2,j}^D(z)\chi_{1,i}(z)\\ u_{1,j}^D(z)\chi_{2,i}(z) & u_{2,j}^D(z)\chi_{2,i}(z)
\end{array}\right) & j+1\le i\;, \\
\left(\begin{array}{cc} 
u_{1,i}^D(z)\chi_{1,j}(z) & 
u_{1,i}^D(z)\chi_{2,j}(z)\\ u_{2,i}^D(z)\chi_{1,j}(z) & u_{2,i}^D(z)\chi_{2,j}(z)
\end{array}\right) & j\ge i\;,\\
\left(\begin{array}{cc} 
u_{1,i}^D(z)\chi_{1,i}(z) & 
u_{1,i}^D(z)\chi_{2,i}(z)\\ u_{1,i}^D(z)\chi_{2,i}(z) & u_{2,i}^D(z)\chi_{2,i}(z)
\end{array}\right) & j= i\;
 \end{array} \right.
\label{FGreen}
\end{eqnarray}    
\noindent
(note that the matricial function defined above is continuous at $j=i$). 

\OBSI
The
matrix $\mathcal{G}(i,j;z)$ is nothing but the integral kernel of the 
resolvent operator $(H_{v,0}-z{\mathrm I_2})^{-1}$; therefore, it must satisfy
the identity
\begin{equation*}
\left(\sum_{k=0}^\infty \left(H_{v,0}(m,c)\right)_{ik}\mathcal{G}(k,j;z)\right)_{lm}
=\delta_{ij}\delta_{lm}\;,
\end{equation*}
\noindent
$i,j\in\mathbb{Z}_+$, $l,m=1,2$. This assertion can be checked by direct substitution.
\OBSF

If $\{P_\Gamma\}$ is a family of spectral projections with respect
to the Borel subset $\Gamma\subset \mathbb R$, it follows by the spectral theorem 
 that 
\begin{equation*}
\mathcal{G}(z)=\int_{-\infty}^{\infty}\frac{d\Omega(\lambda)}{\lambda-z}\;,
\end{equation*}
\noindent
where $\Omega_{m,n}=\langle e_m\otimes\delta_0,P_E e_n\otimes\delta_0
\rangle$, $m,n=1,2$ are the elements of the spectral matrix $\Omega$ of $H_{v,0}$. 
Since $e_1\otimes
\delta_0=\left(\begin{array}{c} \delta_0\\ 0\end{array}\right)$, $e_2
\otimes\delta_0=\left(\begin{array}{c} 0\\\delta_0 \end{array}\right)$ are 
cyclic vectors in $l^2(\mathbb{Z}_+,\mathbb{C}^2)$, it is sufficient to deal 
with this spectral measure, since any other will be absolutely continuous
with respect to $(\Omega_{m,n})_{m,n=1}^2$. 

Explicitly, we have
\begin{eqnarray*}
\mathcal{G}(z)=\frac{1}{c}\left(\begin{array}{cc} m(z) &-1+m(z)\frac{mc^2-z+V_0}
{c}\\ -1+m(z)\frac{mc^2-z+V_0}{c} & \frac{mc^2-z+V_0}{c}\left(-1+
m(z)\frac{mc^2-z+V_0}{c}\right)
\end{array}\right)\;,
\end{eqnarray*}
\noindent
and the imaginary part of $\mathcal{G}(z)$ is
\begin{eqnarray}
\label{Geze}
\Im\mathcal{G}(z)=\frac{1}{c}\left(\begin{array}{cc} \Im m(z) &-\frac{\Im z}{c}
\Re m(z)+\Re a\Im m(z)\\ -\frac{\Im z}{c}\Re m(z)+\Re a\Im m(z)
 & -\frac{\Im z}{c} + \Re m(z)\Im a^2+\Im m(z)\Re a^2
\end{array}\right)\;,
\end{eqnarray}
\noindent
with $a\equiv (mc^2-z+V_0)/c$. Hence, we conclude that the most important spectral
properties of 
$H_{v,0}(m,c)$ depend on the boundary behavior of the function $\Im
m(E+i\varepsilon)$ as $\varepsilon\downarrow 0$ (since $\lim_{\varepsilon
\downarrow 0}\Im a=\lim_{\varepsilon\downarrow 0}\Im z=0$).

It follows from the definition of $m(z)$ and  Theorem~3.1 in Chapter~9
of~\cite{CodLev} that
\begin{equation}\label{TBor}
m(z)=\int_{-\infty}^\infty\frac{d\rho(\lambda)}{\lambda-z}\;,
\end{equation}
with $\rho(\lambda)$ denoting a nondecreasing function of bounded variation,
continuous to
the right and that satisfies the limits $\lim_{\lambda\rightarrow -\infty}\rho
(\lambda)=0$ and $\lim_{\lambda\rightarrow \infty}$ $\rho(\lambda)=1$. The next
result is crucial to our analysis.
\begin{proposition}\label{TAC}
The spectral matrix measure $\Omega$ is absolutely continuous with respect
to the Borel-Stieltjes measure $\rho$. Furthermore, there exists a symmetric
and nonnegative matrix $Y(E)$ such that
\begin{equation}\label{RadNik}
d\Omega(E)=Y(E)d\rho(E)\;,
\end{equation}
\noindent
with 
\begin{eqnarray}\label{Y}
Y(E)=\frac{1}{c}\left(\begin{array}{cc} 1 & \frac{mc^2-E+V_0}
{c}\\ \frac{mc^2-E+V_0}{c} & \left(\frac{mc^2-E+V_0}{c}\right)^2
\end{array}\right)\;.
\end{eqnarray}
\end{proposition}
\begin{proof} 
By the Radon-Nikodym theorem, $\Omega$
is absolutely continuous with respect to $\rho$ if, and only if, there exists a
symmetric integrable matrix $Y(E)$ such that~\eqref{RadNik} is valid for
every $E$. Moreover, the elements of the matrix~$Y(E)$ are given by
\begin{equation}\label{Yij}
Y_{ij}(E)=\frac{d\Omega_{ij}(E)}{d\rho(E)}=\lim_{
\epsilon \downarrow 0}\frac{\Im\mathcal{G}_{ij}(E+i\epsilon )}{\Im\, m(
E +i\epsilon )}\;,
\end{equation}
\noindent
where the last equality comes from the Borel-Stieltjes inversion formula 
applied to $\Omega$ and $\rho$ (see the Appendix of~\cite{LASEIEU}). 

We obtain~\eqref{Y} simply plugging~\eqref{Geze} into~\eqref{Yij}. Observe that
$Y(E)$ is a symmetric integrable matrix (i.e., $Y(E)\in {\LL}^1(\mathbb{R}
,d\rho)$). This concludes the proof of the
proposition.
\end{proof}
\newline  

Proposition~\ref{TAC} shows that the most important features of the spectral
matrix measure~$\Omega$, such  its continuity or singularity with respect
to Lebesgue and Hausdorff measures, are entirely determined by the 
Borel-Stieltjes measure $\rho$. 

To deal with the spectral multiplicity issues, we need the following 
\DEFI
The sesquilinear form 
\begin{equation}
\langle \mathbf{f},\mathbf{g}\rangle =\int_{\mathbb{R}
}\sum_{i,j=1}^{2}f_{i}^{\ast }(E )g_{j}(E)d\Omega_{ij}(E
)\;,  \label{FSES}
\end{equation}
 defined for $f_{i},g_{i}\in {\LL}^{2}(\mathbb{R},\mathbb{C},d\rho)$ is 
nonnegative provided $\Vert \mathbf{f}\Vert ^{2}=\langle \mathbf{f},\mathbf{f}
\rangle \geq 0$ holds in this space.
\DEFF

Let us consider the sesquilinear form~\eqref{FSES}. Suppose that $
\mathbf{f}$ is a simple function, that is, $\mathbf{f}(E )=\displaystyle
\sum_{l=1}^{n}\chi _{B_{l}}(E )(f_{l,1},f_{l,2})$, where $
(f_{l,1},f_{l,2})\in \mathbb{C}^{2}$ and $B_{l}$ are disjoint
Borel sets. For this class of functions, \eqref{FSES} is a nonnegative
sesquilinear form. If $\mathbf{f}$ is such that $f_{i}\in {\LL}^{2}(\mathbb{R},
\mathbb{C},d\rho)$, we can approximate $\mathbf{f}$ by simple
functions in order to obtain $\left\Vert \mathbf{f}\right\Vert \geq 0$. We
have, as a consequence, a separable Hilbert space
${\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega )$ with scalar product~\eqref{FSES}.

Next, since the operator $H_{v,0}(m,c)$ satisfies Weyl's limit point case (see
also~\cite{LASEIEU}), there exists a unitary 
transformation 
$\tilde{U}:l^{2}(\mathbb{Z}_{+},\mathbb{C}^2)\longrightarrow {\LL}^{2}(\mathbb{R},
\mathbb{C}^2,d\Omega)$ such that $H_{v,0}(m,c)=\tilde{U}^{-1}\tilde{H}_{v,0}
(m,c)\tilde{U}$, where $\tilde{H}_{v,0}(m,c):{\LL}^{2}(\mathbb{R},\mathbb{C}^{2},
d\Omega)\longrightarrow {\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega)$ is a 
multiplication operator (see, e.g., Theorem 2.12 of \cite{LASEIEU}).

As for Schr\"o\-ding\-er operators, we have the following result, whose proof is a
direct
extension of Lemma~B.13 of~\cite{LASEIEU}:

\begin{lemma}
\label{lema} The set $\sigma (\rho):=\left\{ E \in \mathbb{R}
:\rho((E -\varepsilon ,E +\varepsilon ))>0\text{ for all }
\varepsilon >0\right\} $ is precisely the spectrum $\sigma (\tilde{H}_{v,0 }
(m,c))$ of the multiplication operator $\tilde{H}_{v,0}(m,c)\mathbf{f}(E)=
E \mathbf{f}(E)$, with domain $\mathcal{D}(\tilde{H}_{v,0}(m,c))=\{\mathbf{f}\in
{\LL}^{2}(\mathbb{R},\mathbb{C}
^{2},d\Omega)$ $\mid E\mathbf{f}(E)\in {\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega)
\}$.
\end{lemma}

Next, there exists a measurable unitary matrix $F(E)$ which diagonalizes $Y(E)$;
 that is,
\begin{eqnarray*}\label{F}
Y(E)=F^{-1}(E)\left(\begin{array}{cc} y_1(E) & 0 \\ 0 & y_2(E)\end{array}\right)
F(E)\;,
\end{eqnarray*}
\noindent
where $y_i(E)$, $i=1,2$, are the integrable eigenvalues of $Y(E)$. The matrix
$F(E)$ provides an unitary operator 
\begin{equation*}
{\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega )\rightarrow
{\LL}^{2}(\mathbb{R},y_{1}d\rho)\oplus {\LL}^{2}(\mathbb{R},y_{2}d\rho)\;,\qquad
\qquad 
\mathbf{f}(E )\mapsto F\mathbf{f}(E )\;,
\end{equation*}
\noindent which leaves $\tilde{H}_{v,0}(m,c)$ invariant (with respect to the
scalar product defined in ${\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega)$). This
permits us to investigate the spectral multiplicity of $\tilde{H}_{v,0}(m,c)$.

\begin{lemma}
\label{Lmult} The spectral multiplicity of $\tilde{H}_{v,0}(m,c)$ is 1.
\end{lemma}
\begin{proof} Evaluating $y_{1}(E)$ and $y_{2}(E)$ explicitly, we obtain $y_1(E)
=0$ and $y_2(E)=(1+a^2)/c$, where $a=(mc^2-E+V_0)/c$. Thus,
\begin{equation*}
{\LL}^{2}(\mathbb{R},\mathbb{C}^{2},d\Omega )=\{0\}\oplus {\LL}^{2}(\mathbb{R},y_{2}
d\rho)= {\LL}^{2}(\mathbb{R},y_{2}d\rho)\;.
\end{equation*}

Hence, $\tilde{H}_{v,0}(m,c)$ is equivalent to  multiplication by $E$ on
${\LL}^{2}(\mathbb{R},d\rho)$, since $y_2(E)d\rho(E)$ and $d\rho(E)$ are mutually
absolutely continuous.
\end{proof}

\OBSI\label{Hvphi}
It is possible to show that we obtain similar results for the operator
$H_{v,\phi}(m,c)$, i.e., with boundary condition \eqref{bouconphi}. For this
problem, we must adopt as a basis of the space of solutions the functions $u^\phi(z)$
and 
$u^{\phi^\ast}(z)$ (with ${\phi^\ast}=\phi+\pi/2$), which satisfy the initial
conditions
\begin{eqnarray*}
\begin{array}{l}
u^\phi_{2,-1} = \sin\phi \,, \qquad u^\phi_{1,0} = \cos\phi \,, \\
u^{\phi^\ast}_{2,-1} = \cos\phi \,, \qquad u^{\phi^\ast}_{1,0} = -\sin\phi\,.
\end{array}
\end{eqnarray*}
The result of Lemma \ref{lema} extends naturally, as well as the result of 
Proposition \ref{TAC}, except for the Radon--Nikodym
derivative, now given by
\begin{eqnarray}\label{Y1}
Y(E)=\frac{1}{c}\left(\begin{array}{cc} \cos^2\phi & \cos\phi\left(
\frac{mc^2-E+V_0}{c}+\cos\phi\sin\phi\right)\\ \cos\phi\left(
\frac{mc^2-E+V_0}{c}+\cos\phi\sin\phi\right) &
\left(\frac{mc^2-E+V_0}{c}\cos\phi+\sin\phi\right)^2
\end{array}\right)\;;
\end{eqnarray}
\noindent
the same situation holds for Lemma \ref{Lmult}, with the eigenvalues of \eqref{Y1}
given now by $y_1=0$ and
$y_2=[{\cos^2\phi+\left((mc^2-E+V_0)/c+\sin\phi\right)^2}]/{c}$.
\OBSF



\subsection{The essential spectrum}
\label{SFN2}

Before we deal with the essential spectrum of $H_{v,\phi}(m,c)$, it would be 
interesting to characterize the spectrum of the free operator $H_{0,0}(m,c)$
\cite{OP1} and its spectral matrix measure.
\begin{proposition}
The spectrum of the free operator is given by the intervals
\begin{equation}
\sigma({H}_{0,0}(m,c))=
\left[ -\sqrt{m^2c^4+4c^2},-mc^2\right] \cup 
\left[mc^2 ,\sqrt{m^2c^4+4c^2}\right] .
 \label{SUPO}
\end{equation}
Moreover, the spectral matrix measure $\Omega$ is purely absolutely continuous 
(with respect to the Lebesgue measure $\ell$) in these intervals.
\end{proposition}
\begin{proof} In order to prove this proposition, we will determine the exact 
behavior of the function $m(E+i\varepsilon)$ as $\varepsilon
\downarrow 0$. 

After some manipulations, it follows by~\eqref{HRel}  that
\begin{equation*}
\left(-\Delta-\frac{m^2c^4-z^2}{c^2}\right)\psi_{j,n}=0\,,\quad n\in\mathbb{Z}_+,
\end{equation*} 
\noindent
 where $\psi_{j}, j=1,2,$ are the components of the spinor $\Psi$. 

Since $m(z)$ is uniquely defined imposing that $\Psi=-u^N+m(z)u^D$ is $l^2(
\mathbb{Z}_+,\mathbb{C}^2)$, it is simple to obtain
\begin{equation*}
m(z)=-\frac{w}{2}+\sqrt{\frac{w^{2}}{4}-1}\;,
\label{eme}
\end{equation*}
\noindent
with $w=[{m^2c^4-z^2+2c^2}]/{c^2}$. Now, put 
\begin{equation*}
\Im\, m(E)=\limsup_{\varepsilon \downarrow 0}\Im\, m(z)\;,\quad z=E +i\varepsilon ,
\end{equation*}
 and let $L(\rho )$ be the set of all $E \in 
\mathbb{R}$ for which this limit exists. It is known (see Appendix~B
of~\cite{LASEIEU}) that the minimal (or essential) supports $\mathcal{M}$, $
\mathcal{M}_{\mathrm{ac}}$ and $\mathcal{M}_{\mathrm{s}}$ of~$\rho$, the
absolutely continuous part $\rho_{\mathrm{ac}}$ and the singular part
$\rho_{\mathrm{s}}$ of $\rho$, with respect to the Lebesgue measure in $\mathbb{R}$,
are, respectively, given by $E \in L(\rho )$ such that $0<\Im
\,m(E)\leq \infty $, $0<\Im\, \,m(E)<\infty $ and $\Im\, \,m(E)=\infty $. These 
criteria can be obtained by de la Vall\'{e}e-Poussin's
decomposition theorem~\cite{Saks}, the Radon-Nikodym theorem and
the following application of Lemma~3 in~\cite{GP}:
\begin{lemma}
\label{RadNikl} If the Radon-Nikodym derivative $(d\rho/d\ell)(E)$  exists and takes
values in~$[0,\infty]$, then $\Im\, m(E)$ also exists and $
(d\rho/d\ell)(E)=(1/\pi)\Im\, m(E)$ ($\ell$ is the Lebesgue measure on $
\mathbb{R}$).
\end{lemma}

Since
\begin{equation*}
\lim_{\varepsilon\downarrow 0}\Im\, m(E +i\varepsilon)=
\begin{cases} 
   \displaystyle\frac{\sqrt{(E^2-m^2c^4)(m^2c^4+4c^2-E^2)}}{2c^2} & \text{if} 
\;\; E^2\in[m^2c^4,m^2c^4+4c^2] ,\\
   0 & \text{otherwise}\;, 
  \end{cases}
\end{equation*}
\noindent
it follows by the above criteria and Proposition~\ref{TAC} that the spectral
matrix measure $\Omega$ is purely absolutely continuous on the interval defined
by~\eqref{SUPO}; moreover, the essential spectrum of $H_{0,0}(m,c)$ coincides
with its own spectrum. This conclude the proof of the proposition. \end{proof}




\OBSI\label{REE}
As we have discussed in Remark~\ref{rank1}, the operator $H_{v,\phi}(m,c)$ with 
boundary condition~\eqref{bouconphi} at $n=-1$  may be written 
as a rank-one perturbation of $H_{v,0}(m,c)$; hence,
$\sigma_{\mathrm{ess}}(H_{v,\phi}(m
,c))=\sigma_{\mathrm{ess}}(H_{v,0}(m,c))$, by Weyl's criterion (see, for instance,
Section~11.3 of~\cite{ISTQD}). Thus, it is sufficient to deal with $H_{v,0}(m,c)$ in
order
to determine the essential spectrum of $H_{v,\phi}(m,c)$.
\OBSF

We are now ready to prove Theorem~\ref{tee}.

\noindent \begin{proof} (Theorem~\ref{tee}) By taking into account the above
results and Remark \ref{REE}, the proof of Theorem~\ref{tee} reduces to a direct
extension of Theorem~3.13 in~\cite{CFKS} to the discrete Dirac operator.
\end{proof}

The eigenvalues of $H^\prime_{0,0}(m,c)$ are isolated points of the essential
spectrum of $H_{v,0}(m,c)$; hence, they cannot belong to the continuous spectrum
of $H_{v,0}(m,c)$, neither be eigenvalues of infinite multiplicity (since we
have a unidimensional problem). Thus, they must be accumulation points of the
discrete spectrum of $H_{v,0}(m,c)$. 

Since the operator $H^\prime_{0,0}(m,c)$ is a
rank-one perturbation of $H_{0,0}(m,c)$, it follows that
its spectrum is the union of the interval~\eqref{SUPO} with the possible addition of
a finite number of isolated points. Thus, it follows by Theorem~\ref{tee} that the
essential 
spectrum of $H_{v,\phi}(m,c)$ has the same structure.








\section{Transfer matrices and Pr\"ufer variables}
\label{TMPV}
\zerarcounters

In order to determine the exact Hausdorff dimension of the spectral matrix 
measure $\Omega$ and, consequently, the spectral nature of $H_{v,\phi}(m,c)$,
we study the exact 
asymptotic behavior of the solutions to the Dirac eigenvalue equation 
\begin{equation}\label{eqdirre} 
H_{v,\phi}(m,c)\Psi=E\Psi\;,
\end{equation}
\noindent
with $E\in\mathbb{R}$; this is an important step in our approach. It is here
that the concepts of transfer matrix and 
Pr\"ufer variables play a fundamental role. What follows is an adaptation of 
the material presented in Sections~3 and~4 of~\cite{MWGA} to the Dirac operator
setting.

For $E\in \mathbb{C}$, let 
\begin{equation}
T(n,n-1;E)=\left( 
\begin{array}{cc}
1+\displaystyle\frac{m^2c^4-(E-V_n)^2}{c^2} & \displaystyle\frac{mc^2+E-V_n}{c}
\vspace{2mm} \\ 
\displaystyle\frac{mc^2-E+V_n}{c} & 1
\end{array}
\right)  \label{mt}
\end{equation}
\noindent be the $2\times 2$ transfer matrix associated with the
$l^{2}(\mathbb{Z}_{+},\mathbb{C}^2)$ solution to the Dirac equation~\eqref{eqdirre}
(see \cite{OP,OP1}). 

The equation 
\begin{equation*}
\left( 
\begin{array}{c}
\psi_{1,n+1} \\ 
\psi_{2,n}
\end{array}
\right) =T(n,n-1;E)\left( 
\begin{array}{c}
\psi_{1,n} \\ 
\psi_{2,n-1}
\end{array}
\right)  \label{uuTuu}
\end{equation*}
\noindent holds for $n\geq 0$, with ${\dbinom{\psi_{1,0}}{\psi_{2,-1}}}={
\dbinom{\cos\phi }{\sin \phi }}$ satisfying~\eqref{bouconphi} for some 
$\phi \in \lbrack 0,\pi ]$. Another important tool is the product of the $n+1$ 
first transfer matrices, denoted by 
\begin{equation}
T(n;E)=T(n,n-1;E)T(n-1,n-2;E)\ldots T(0,-1;E)\;.
\label{Tn}
\end{equation}
 
Given the values~\eqref{Ve} of the potential $V_{n}$ and the sparseness condition 
\eqref{spacon}, only two different $2\times 2$ matrices appear on the r.h.s.\
of~\eqref{Tn}: that is,
\begin{equation*}
T_{v}(E)=\left( 
\begin{array}{cc}
1+\displaystyle\frac{m^2c^4-(E-v)^2}{c^2} & \displaystyle\frac{mc^2+E-v}{c}
\vspace{2mm} \\ 
\displaystyle\frac{mc^2-E+v}{c} & 1
\end{array}\right)  
\end{equation*}
and
\begin{equation*} 
T_{0}(E)=\left( 
\begin{array}{cc}
1+\displaystyle\frac{m^2c^4-E^2}{c^2} & \displaystyle\frac{mc^2+E}{c}
\vspace{2mm} \\ 
\displaystyle\frac{mc^2-E}{c} & 1
\end{array}
\right)  \label{TvT0ad}
\end{equation*}
occur depending on the entry $n$ in~\eqref{mt} being or not $a_{j}^{\omega }\in 
\mathcal{A}$. 

Let $E=\pm\sqrt{m^2c^4+2c^2(1-\cos\varphi)}$, with $\varphi\in[0,\pi)$, be 
parametrizations of the intervals~\eqref{SUPO}. Now  note that, for such energies,  
the free matrix $T_{0}(E)$ is similar to a purely clockwise rotation $R(\varphi)$,
that is,
 \begin{equation}
UT_{0}(E)U^{-1}=\left( 
\begin{array}{cc}
\cos \varphi & \sin \varphi \vspace{2mm} \\ 
-\sin \varphi & \cos \varphi
\end{array}
\right) =R(\varphi )\;,  \label{R}
\end{equation}
where 
\begin{equation*}
U\equiv \left( 
\begin{array}{cc}
-\displaystyle\frac{mc^2-E}{c\sin\varphi} & -\displaystyle\frac{1-\cos\varphi}
{\sin\varphi} \vspace{2mm} \\ 
0 & 1
\end{array}
\right) ~.\;  \label{U}
\end{equation*}
\noindent\ Note also that $U$ is not uniquely defined since any other matrix $
U^{\prime }=HU$, with $H$ commuting with $R$, satisfies~\eqref{R}.

Since the product of rotation matrices is also a rotation, we obtain 
\begin{equation}
UT(n;E)U^{-1}=R((n-a_{k}^{\omega })\varphi )P(E)R((a_{k}^{\omega }-a_{k-1}^{\omega })
\varphi )\cdots P(E)R((a_{1}^{\omega }+1)\varphi )  \label{UTn}
\end{equation}
\noindent as the conjugation of~\eqref{Tn} by $U^{-1}$, for every $n\in 
\mathbb{N}$ and $\omega_{j}\in \{0,, 1 , 2,\ldots ,[j^\eta]\}$, $j\geq 1$; $k$ is an
integer
 such that $a_{k}^{\omega }\leq n<a_{k+1}^{\omega }$; $P(E)$ is defined by 
{\small{\begin{eqnarray}\label{Pe}
\nonumber P(E)R(\varphi ) & = & UT_{v}(E)U^{-1}\\
&=&\left(
\begin{array}{cc} 
1+\frac{vE_+}{2c^2}-\frac{v^2}{c^2} & -\frac{v}{2c^2}\sqrt{\frac{E_+E_-}
{{F}}}\left(E_+-\frac{4c^2}{E_+}-2v
\right)\\
-\frac{v}{2c^2}\sqrt{\frac{E_+{F}}{{E_-}}} & 1-
\frac{vE_+}{2c^2}
\end{array}\right)\;
\end{eqnarray}}}
where $E_+=E+mc^2$, $E_-=E-mc^2$ and $F= m^2c^4+4c^2-E^2$.

Next we consider the following change of
variables, known as {EFGP} transform:
\begin{eqnarray}\label{EFGPT}
\mathbf{v_k}:=\left(\begin{array}{c}
R_k\cos\theta_k^\omega\\
R_k\sin\theta_k^\omega\end{array}\right)=U\left(\begin{array}{c}
\psi_{1,k}\\
\psi_{2,k-1}\end{array}\right)=\left(\begin{array}{c}
\frac{(E-mc^2)\psi_{1,k}/c-(1-\cos\varphi)\psi_{2,k-1}}{\sin\varphi} \\
 \psi_{2,k-1}\end{array}\right)\;,
\end{eqnarray}
\noindent
The variables $R_k$ and $\theta_k^\omega$ are called, respectively, Pr\"ufer 
radii and angles. Observe that the Pr\"{u}fer angles are random variables, since
$\omega_j$ are randomly distributed.
 The Pr\"{u}fer radii, on the other hand, are random variables only as a 
function of the Pr\"{u}fer angles, and their dependence on $\omega $ will be 
omitted. 

\OBSI
The convention we have
employed here differs from the convention in~\cite{KLS}, since
 the functions  $\cos\theta_k$ and $\sin\theta_k$ are exchanged. Nevertheless,
the behavior of the Pr\"ufer variables are identical in both conventions.
\OBSF

We nonetheless use a slightly different expression of~\eqref{EFGPT}, more
adequate to relation~\eqref{UTn}. Given the vectors 
\begin{equation*}
\mathbf{v}_{k}=\left(R_{k-1}\cos \theta_{k}^{\omega},R_{k-1}\sin\theta_{k}^{\omega}
\right), \qquad \tilde{\mathbf{v}}_{k}=\left(R_{k}\cos\tilde\theta_{k}^{\omega},R_{k}
\sin\tilde\theta_{k}^{\omega}\right)\;,
\label{vk}
\end{equation*}
 the Pr\"{u}fer variables $\left(R_k,\theta_k^\omega\right)_{k
\ge 0}$ satisfy a recurrence relation induced by 
\begin{equation}
\mathbf{v}_{k}=R((a_{k}^{\omega}-a_{k-1}^{\omega})\varphi)\tilde{\mathbf{v}}_
{k-1}
\label{vk1}
\end{equation}
\noindent and 
\begin{equation}
\tilde{\mathbf{v}}_{k}=P(E)\mathbf{v}_{k}\;,
\label{vkt1}
\end{equation}
with $\mathbf{v}_{1}=R((a_{1}^{\omega }+1)\varphi)
\tilde{\mathbf{v}}_{0}$,
\begin{eqnarray*}
\tilde{\mathbf{v}}_0(\theta_0)=R_0\left(\begin{array}{c}
\cos\theta_0\\
\sin\theta_0\end{array}\right)=U\left(\begin{array}{c}
\cos\phi\\
\sin\phi\end{array}\right)=\left(\begin{array}{c}
\frac{(E-mc^2)/c\cos\phi-(1-\cos\varphi)\sin\phi}{\sin\varphi} \\
 \cos\phi \end{array}\right)\;,
\label{e3.7MW1}
\end{eqnarray*} 
\[
R_0^{2}=\left(\frac{((E-mc^2)\cos\phi)/c-(1-\cos\varphi)\sin\phi}
{\sin\varphi}\right)^2+{\cos}^2\phi.
\] 

Thus, if $\mathbf{\psi}_n=\left(
\psi_{1,n},\psi_{2,n-1}\right)$ represents a solution to~\eqref{eqdirre} 
satisfying the initial conditions $\mathbf{u}_0=\left(\cos\phi,\sin\phi
\right)$, then 
\begin{eqnarray*}\label{e3.1MW1}
R(\varphi)\tilde{\mathbf{v}}_k=U\mathbf{u}_{a_{k}+1}\;.
\end{eqnarray*}

By equivalence of norms, the growth of $T(n;E)$ may be controlled by
the euclidean norm (see Section 4 of \cite{MWGA} for details) 
\begin{equation*}
\Vert UT(n;E)U^{-1}v_{0}\Vert ^{2}=\Vert UT(a_{N}^{\omega }+1;E)U^{-1}
v_{0}\Vert^{2}=R_{N}^{2}\;,  \label{R2N}
\end{equation*}
\noindent where the equality holds for any unit vector $v_{0}=(\cos 
\theta_{0},\sin \theta_{0})$ and for each $n$ such that $a_{N}^{\omega }\leq
n<a_{N+1}^{\omega }$. Thus, from equations~\eqref{vk1},~\eqref{vkt1} and~\eqref{Pe},
$R_N^2$
can be written as
\begin{eqnarray}\label{Rnj}
\left(R_N\right)^2=\left(R_0\right)^2\displaystyle\prod_{n=1}^N\left(\frac{R_n}
{R_{n-1}}\right)^2=\left(R_0\right)^2\left(\exp\left\{\frac{1}{N}\sum_{n=1}^N
\ln f\left(\theta^\omega_n,\varphi\right)\right\}\right)^N\;.
\end{eqnarray}
\noindent
with
\begin{eqnarray}\label{efe}
f(\theta^\omega,\varphi):=\left(A(\varphi)\cos\theta^\omega+B(\varphi)\sin
\theta^\omega\right)^2+\left(C(\varphi)\cos\theta^\omega+D(\varphi)\sin
\theta^\omega\right)^2\;,
\end{eqnarray}
\noindent
where $A(\varphi)=1+\frac{v(E+mc^2)}{2c^2}-\frac{v^2}{c^2}$, $D(\varphi)=
1-\frac{v(E+mc^2)}{2c^2}$, \[
B(\varphi)=-
\frac{v}{2c^2}\sqrt{\frac{E^2-m^2c^4}
{m^2c^4+4c^2-E^2}}\left(E+mc^2-\frac{4c^2}{E+mc^2}-2v\right)
\] and \[
C(\varphi)=-\frac{v}{2c^2}\sqrt{\frac{(E+mc^2)(m^2c^4+
4c^2-E^2)}{E-mc^2}}
\] are the entries of $P(E)$.

The Pr\"{u}fer angles $(\theta_{k}^{\omega})_{k\geq 1}$ are, on the other hand, 
obtained recursively by
\begin{equation}
\theta _{k}^{\omega }=\tan ^{-1}\left(\frac{C+ D\tan\theta _{k-1}^{\omega }}{A+ B
\tan\theta _{k-1}^{\omega }}
\right) -(\beta ^{k}+\omega_{k}-\omega _{k-1})\varphi  \label{PA}
\end{equation}
\noindent for $k>1$, with $\theta _{1}^{\omega }$ given by 
$
\theta _{1}^{\omega }=\theta _{0}-(a_{1}+\omega _{1})\varphi$.

Hence, the determination of the exact asymptotic behavior of the sequence
$\left(R_n(\theta_0)\right)_{n\ge 1}$ involves an estimate of the Birkhoff-like
sum  
\begin{eqnarray}\label{SomBir}
\frac{1}{N}\sum_{n=1}^N \ln f\left(v,\theta^\omega_n\right)
\end{eqnarray}
\noindent
for $N$ large, which, on the other hand, depends on the distribution properties
 of the sequence $\left(\theta_n^\omega\right)_{n\ge 1}$ of the Pr\"ufer
 angles. The tool that intertwines these elements is the following theorem 
(recall that a sequence  $w=(x_n)_{n\ge 1}$ is said to be uniformly
distributed modulo $\pi$ ($\ud\pi$) if it is equally distributed, in 
fractional portions, over half open subintervals of $[0,\pi)$).

\begin{theorem}[Theorem~1.1 in~\cite{KN}]\label{T1.1KN}
The sequence $w=(x_n)_{n\ge 1}$ of real numbers $x_n\in[0,\pi)$ is $\ud$ 
$\pi$
if, and only if, for every continuous real function $h$ defined on the 
closed interval $I=[0,\pi]$, we have
\begin{eqnarray}\label{sud2}
\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^Nh(\{x_n\})=\frac{1}{\pi}
\int_0^\pi h(x)dx\;.
\end{eqnarray}
\end{theorem}

Theorem~\ref{T1.1KN} provides a criterion that permits the substitution,
on the asymptotic limit $N\rightarrow\infty$, of the  average~\eqref{SomBir}, by the
integral
\begin{eqnarray*}\label{IntBir}
\frac{1}{\pi}\int_0^\pi \ln f(v,\theta)d\theta\;,
\end{eqnarray*} 
\noindent
in case  the sequence $\left(\theta_n^\omega\right)_{n\ge 1}$ of the 
Pr\"ufer angles is $\ud\pi$ and  $\ln f(v,\theta)$, with 
$f(v,\theta)$ given by~\eqref{efe}, is a periodic Riemann integrable
function of period $\pi$. 

\begin{lemma}\label{Lefe}
The function $h(\theta):=\ln f(v,\theta)$ is a periodic Riemann integrable
function of period $\pi$, which average is given by
\begin{eqnarray}\label{e4.17MW}
\frac{1}{\pi}\int_0^\pi h(\theta)d\theta = \ln r(v,E)\;,
\end{eqnarray}
\noindent
with 
\begin{equation}\label{erre}
r(v,E)=1+\frac{1}{({m^2c^4+4c^2-E^2})}\,\frac{v^2}{c^2}\left[\frac{(E^2-m^2c^4)^2+4m^2c^6}{(E^2-m^2c^4)
}-4vE+ 2v^2\right]\;.
\end{equation}
\end{lemma}
\begin{proof} The proof of the lemma is an adaptation of 
some results presented in Section~4 in~\cite{MWGA}. \end{proof}

Lemma~\ref{Lefe} and Theorem~\ref{T1.1KN} provide a precise estimate 
for the asymptotic limit of the ``time average''~\eqref{SomBir} under
the hypothesis of the $\ud$ $\pi$ of the sequence
$\left(\theta_n^\omega\right)_{n\ge 1}$ of Pr\"ufer angles.

\begin{lemma}\label{LRP}
Let $\left(R_n(\theta_0)\right)_{n\ge 1}$ be the sequence of the Pr\"ufer 
radii which satisfy the initial conditions $\mathbf{v}_0=(\cos\vartheta, 
\sin\vartheta)$. Suppose  there is a set $A\subset\mathbb R$  of null Lebesgue
measure so that
the sequence $\left(\theta_n^\omega\right)_{n\ge 1}$ of 
the Pr\"ufer angles is $\ud\pi$ for $\varphi\in[0,\pi)\backslash A$. Then,
\begin{eqnarray}
C_{N}^{-1}r^N \le \left(R_{N}(\theta_0)\right)^2\le C_{N} r^N \;,
\label{R2N1}
\end{eqnarray}
\noindent
where $C_{N}$ is a real number such that $C_{N}>1$ and $\lim_{N\rightarrow\infty}
C_{N}^{1/N}=\left(R_{0}\right)^2$, with $r$ given by~\eqref{erre}.
\end{lemma}
\begin{proof} The inequalities~\eqref{R2N1} follow from the
hypotheses of the lemma, by equations~\eqref{Rnj}, \eqref{e4.17MW} and
 the estimate 
\begin{eqnarray*}\label{Disc}
\left\vert\frac{1}{N}\sum_{n=1}^N\ln f\left(v,\theta_n^\omega\right)-\frac{1}{\pi}
\int_0^\pi \ln f(v,\theta)d\theta\right\vert \le C D^\ast_N\;,
\end{eqnarray*}
\noindent
where 
\begin{eqnarray*}
D^\ast_N(\theta)=D^\ast_N(\theta_1,\ldots,\theta_N)=\sup_{0<\vartheta\le\pi}
\left\vert\frac{\text{card}(\{k:\theta _{k}~\text{mod~}\pi
\in \lbrack 0,\theta ),1\leq k\leq N\})}{N}-\vartheta\right\vert
\end{eqnarray*}
is the so-called {\em discrepancy} of the sequence $\left(\theta_n\right)_{n\ge 1}$
(see~\cite{KN} for an ample discussion on discrepancy) and~$C$ is some positive
constant. For the second part of the lemma, we need the 

\begin{theorem}[Corollary~1.1 of Chapter~2 in~\cite{KN}]\label{C1.1KN}
A sequence $w$ is $\ud$ $\pi$ if, and only if, $\lim_{N\rightarrow
\infty}D^\ast_N(w)=0$.
\end{theorem}

It follows by the hypothesis of uniform distribution and Theorem~\ref{C1.1KN}
that 
\[
\lim_{N\rightarrow\infty}D^\ast_N(\theta^\omega)=0\;;
\] hence, $\lim_{N
\rightarrow\infty}C_{N}^{1/N}=\left(R_{0}(\theta_0)\right)^2$, and the proof of
Lemma~\ref{LRP} is complete\end{proof}
 
Now we finally deal with the uniform distribution of the sequence
$\left(\theta_n^\omega
\right)_{n\ge 1}$.

\begin{theorem}
\label{thethe} The sequence of Pr\"{u}fer angles $(\theta _{n}^\omega)_{n\geq 1}$
is $\ud$ $\pi $ for all $\varphi/\pi \in \lbrack 0,1]\backslash \mathbb{Q}$ 
and all $\omega\in \Lambda $, apart from a set with null $\nu $ measure.
\end{theorem}
\begin{proof} The proof is exactly the same of~Theorem~3.2 in
\cite{CMW1}.\end{proof}

\section{Hausdorff dimension of the spectral measure}
\label{EHD}

\setcounter{equation}{0} \setcounter{theorem}{0}

This section is devoted to the determination of the Hausdorff dimension of
the spectral measure of $H_{v,\phi}(m,c)$. At the end, the proofs of
Theorems~\ref{thethe1} and~\ref{tmarwre} are presented.

\subsection{Basic definitions and generalized subordinacy theory}

Firstly we recall some useful definitions. An almost complete description is 
found in~\cite{Last}. Given a Borel set $S\subset \mathbb{R}$ and $\alpha \in
\lbrack 0,1]$, consider the number 
\begin{equation}
Q_{\alpha ,\delta }(S)=\inf \left\{ \sum_{\nu =1}^{\infty }|b_{\nu
}|^{\alpha }:|b_{\nu }|<\delta ;S\subset \bigcup_{\nu =1}^{\infty }b_{\nu
}\right\} \;,  \label{Qhaus}
\end{equation}
with the infimum taken over all covers by intervals of size at most $\delta $.
The limit 
\begin{equation}
h^{\alpha }(S)=\lim_{\delta \downarrow 0}Q_{\alpha ,\delta }(S)\;,
\label{Mhaus}
\end{equation}
 is called $\alpha $-\textit{dimensional Hausdorff (outer)  measure}. The 
counting measure,
at $\alpha=0$, and the Lebesgue measure, at $\alpha =1$, are important particular
cases. 
It is clear by the definitions \eqref{Qhaus} and \eqref{Mhaus} that $h^{\alpha }(
S)$ is an outer measure on~$\mathbb{R}$ (see~\cite{Falconer}). For $\beta <\alpha
<\gamma $, 
\begin{equation*}
\delta ^{\alpha -\gamma }Q_{\gamma ,\delta }(S)\leq Q_{\alpha ,\delta
}(S)\leq \delta ^{\alpha -\beta }Q_{\beta ,\delta }(S)\;,
\end{equation*}
\noindent holds for any $\delta >0$ and $S\subset \mathbb{R}$. So, if $
h^{\alpha }(S)<\infty $, then $h^{\gamma }(S)=0$ for $\gamma >\alpha $; if $
h^{\alpha }(S)>0$, then $h^{\beta }(S)=\infty $ for $\beta <\alpha $. Thus,
for every Borel set $S$, there is a unique $\alpha _{S}$ such that $
h^{\alpha }(S)=0$ if $\alpha >\alpha _{S}$ and $h^{\alpha }(S)=\infty $ if $
\alpha _{S}<\alpha $. The number $\alpha _{S}$ is called the \textit{Hausdorff
dimension} of the set~$S$.

Another useful concept is the \textit{exact dimension} of a measure, taken
from~\cite{RodTay}.

\DEFI
\label{Exactdim} A Borel measure~$\mu$ in $\mathbb{R}$ is said to be of
exact dimension $\alpha $, $\alpha \in \lbrack 0,1]$, if  two
requirements hold: (1)~for every $\beta \in \lbrack 0,1]$ with $\beta
<\alpha $ and $S$ a set of dimension $\beta $, $\mu (S)=0$ (which means that 
$\mu (S)$ gives zero weight to any set $S$ with $h^{\alpha }(S)=0$); (2)~there is a
set $S_{0}$ of dimension $\alpha $ which supports $\mu $ in the
sense that $\mu (\mathbb{R}\backslash S_{0})=0$.
\DEFF



Finally, we recall the notions of continuity and singularity of a measure
with respect to the Hausdorff measure. Given $\alpha \in \lbrack 0,1]$, a
measure $\mu $ is called $\alpha $-continuous if $\mu (S)=0$ for every set $
S$ with $h^{\alpha }(S)=0$; it is called $\alpha $-singular if it is
supported on some set $S$ with $h^{\alpha }(S)=0$. 

\OBSI\label{DSC}
It is possible to reformulate
Definition~\ref{Exactdim} in this context: a measure $\mu $ is said to have
exact dimension $\alpha $ if, for every $\epsilon >0$, it is simultaneously $(\alpha
-\epsilon )$-continuous and $(\alpha +\epsilon )$-singular.
\OBSF



Jitomirskaya and Last \cite{JitLast} extended, to Hausdorff measures, the 
Gilbert-Pearson theory of subordinacy \cite{GP} for Lebesgue measures, which 
relates the spectral
property of $\rho $ to the rate of growth of the solutions to the Schr\"{o}dinger
equation. Now we describe the extension of these results to Dirac
operators~\eqref{MDir}. 

A solution $\Psi$ to~\eqref{eqdirre} is said to be subordinate if 
\begin{equation*}
\lim_{l\rightarrow \infty }\frac{\left\Vert \Psi\right\Vert _{l}}{\left\Vert
\Phi\right\Vert _{l}}=0  \label{sub}
\end{equation*}
\noindent holds for any linearly independent solution $\Phi$ to~\eqref{eqdirre},
where $\left\Vert \cdot \right\Vert _{l}$ denotes the $l^{2}(\mathbb{Z}_{+},
\mathbb{C}^2)$-norm truncated at the length $l\in \mathbb{R}$, i.e., 
\begin{equation*}
\left\Vert \Psi\right\Vert _{l}^{2}\equiv
\sum_{n=0}^{[l]}\left[|\psi_{1,n}|^{2}+|\psi_{2,n}|^{2}\right]+(l-[l])\left(|\psi_{1,
[l]+1}|^{2}+|\psi_{2,[l]+1}|^{2}\right)\;,
\end{equation*}
\noindent $\lbrack l]$ the integer part of $l$.

Following Jitomirskaya-Last \cite{JitLast},  for any given $
\epsilon >0$ introduce a length $l(\epsilon)\in (0,\infty )$ by the equality 
\begin{equation}
\left\Vert u^D\right\Vert _{l(\epsilon )}\left\Vert
u^N\right\Vert _{l(\epsilon )}=\frac{c}{2\epsilon }  \label{eps}
\end{equation}
\noindent (see equation (1.12) in~\cite{JitLast}), where $u^{D}$ and $u^{N}$ are the
solutions to~\eqref{eqdirre} which satisfy the initial conditions 
\eqref{DNboucon}.

At most one of the solutions $\left\{ u^D,u^N\right\} $ to~\eqref{eqdirre} belongs
to~$l^{2}(\mathbb{Z}_+,\mathbb{C}^2)$, thanks to the constancy of the Wronskian,
which follows by the Green's identity; that is,  for $n\ge -1$, 
\[
\sum_{n=0}^N\left(\Psi^
\ast_n(H_D\Phi)_n-(H_D\Psi)_n^\ast\Phi_n\right)=W[\Phi,\Psi](N)-W[\Phi,\Psi](-1)=0,
\]
 (where
$W[\Phi,\Psi](n)=c(\phi_{1,n+1}\psi^\ast_{2,n}-\phi_{2,n}\psi^\ast_{1,n+1})$; see
also
Chapter 9 in~\cite{CodLev}). Hence, the left-hand side
of~\eqref{eps} is a monotone increasing function of $l$ which vanishes at $
l=0$ and diverges as $l\rightarrow \infty $. On the other hand, the
right-hand side of~\eqref{eps} is a monotone decreasing function of $
\epsilon $ which diverges as $\epsilon \rightarrow 0$. It is then concluded that the
function $l(\epsilon )$ is a well-defined monotone decreasing and continuous
function of $\epsilon $ which diverges as $\epsilon \rightarrow 0$.

What follows is the version of Jitomirskaya-Last inequalities  for the discrete
Dirac operators.

\begin{theorem}\label{JLineq} Let $H_D(m,c)$ be the Dirac
operator~\eqref{MDir} that satisfies the boundary condition~\eqref{boucond}.
Given $\epsilon>0$, we obtain
\begin{equation*}\label{DM}
\frac{5-\sqrt{24}}{m(E +i\epsilon )}\leq \frac{\left\Vert
u^D\right\Vert _{l(\epsilon )}}{\left\Vert u^N\right\Vert
_{l(\epsilon )}}\leq \frac{5+\sqrt{24}}{m(E +i\epsilon )}\;.~
\end{equation*}
\end{theorem}

 In order to prove this theorem, we only need the following lemma, which is an
adaptation of Lemma~3.1 in~\cite{JitLast}; with such result at hand, the proof of
Theorem~\ref{JLineq} follows the same lines as the proof of Theorem~1.1
in~\cite{JitLast}. 

The function $\chi_n(z)$  in Lemma~\ref{LFVP}
is nothing but the unique $l^2(\mathbb{Z}_+,\mathbb{C}^2)$ solution of the Dirac 
equation \eqref{eqdir}, the one associated with the definition of~$m(z)$ (see
Section~\ref{SFN}).

\begin{lemma} [Variation of Parameters] \label{LFVP} For every $n\ge 0$, $\chi_n(z)$
satisfies the identity
\begin{eqnarray}\label{FVP}
\nonumber\left(\begin{array}{c}\chi_{1,n+1}(z)\\ \chi_{2,n}(z)\end{array}\right) 
&=&-\left(\begin{array}{c}u^N_{1,n+1}(E)\\ u^N_{2,n}(E)\end{array}\right)+
m(z)\left(\begin{array}{c}u^D_{1,n+1}(E)\\ u^D_{2,n}(E)\end{array}\right)\\
\nonumber &-&\frac{i\epsilon}{c}\left(\begin{array}{c}u^N_{1,n+1}(E)\\ u^N_{2,n}(
E)\end{array}\right)\sum_{k=1}^n\left\{u^D_{1,k+1}(E)\chi_{1,k}(z)+u^D_{2,k}(E)
\chi_{2,k}(z)\right\}\\
&+& \frac{i\epsilon}{c}\left(\begin{array}{c}u^D_{1,n+1}(E)\\ u^D_{2,n}(E)
\end{array}\right)\sum_{k=1}^n\left\{u^N_{1,k+1}(E)\chi_{1,k}(z)+u^N_{2,k}(E)
\chi_{2,k}(z)\right\}\;.
\end{eqnarray}
\end{lemma}
\begin{proof} We denote by $\binom{\tilde{v}_{1,n+1}(E)}{\tilde{v}_{2,
n}(E)}$ the right-hand side of~\eqref{FVP}, and let $\binom{\tilde{v}_{1,0}(E)}
{\tilde{v}_{2,-1}(E)}=\binom{m(z)}{-1}$. Regarding the Wronskian constancy,
 it is easy to verify that 
\begin{eqnarray*}
\binom{\tilde{v}_{1,n+1}(E)}{\tilde{v}_{2,n}(E)}=\binom{\tilde{v}_{1,n}(E)+\frac{
mc^2-V_n+E}{c}\tilde{v}_{2,n}(E)+\frac{i\epsilon}{c}\chi_{2,n}(z)}{\tilde{v}_{2,
n-1}(E)+\frac{mc^2-V_n-E}{c}\tilde{v}_{1,n}(E)-\frac{i\epsilon}{c}\chi_{1,n}(z)}
\end{eqnarray*}
holds for every $n\ge -1$. Since
\begin{eqnarray*}
\binom{\chi_{1,n+1}(z)}{\chi_{2,n}(z)}=\binom{\chi_{1,n}(z)+\frac{(mc^2-V_n+E+i
\epsilon)\chi_{2,n}(z)}{c}}{\chi_{2,n-1}(z)+\frac{(mc^2-V_n-E-i\epsilon)\chi_{1,n}(
z)}{c}}
\end{eqnarray*}
\noindent
and that $\binom{\tilde{v}_{1,0}(E)}{\tilde{v}_{2,-1}(E)}=\binom{m(z)}{-1}=
\binom{\chi_{1,0}(E)}{\chi_{2,-1}(E)}$, it follows by induction that $\tilde{
v}_n(z)=\chi_n(z)$ for every $n\ge 0$. 
\end{proof}



It is a direct consequence of Theorem~\ref{JLineq} that Theorem~1.2 
in~\cite{JitLast} and its corollaries also hold true: if $
\rho$ is defined by~\eqref{TBor}, then, with $b=\alpha /(2-\alpha )$,
\begin{equation*}
D_\rho^\alpha(E):=\limsup_{\epsilon \downarrow 0}\frac{\rho\left( (E -\epsilon,E
+\epsilon )\right)}
{(2\epsilon )^{\alpha }}=\infty  \label{sup}
\end{equation*}
if, and only if,
\begin{equation*}
\liminf_{l\rightarrow \infty }\frac{\left\Vert u^D\right\Vert _{l}}{
\left\Vert u^N\right\Vert _{l}^{b}}=0~,  \label{inf}
\end{equation*}
and $D_\rho^\alpha(E)$ is the so called \textit{Hausdorff upper derivative}
of~$\rho$ at~$E$ (see~\cite{Last} for a detailed discussion of this concept).
 
Before we proceed, we need some important results regarding the behavior of
the generalized eigenfunction $u^D$. Next we present a version of Theorem~3.10
in~\cite{LS}.

\begin{proposition}\label{PROPO} Fix $\delta>0$. For almost every
$E$ with respect to the measure $\rho$,
\begin{eqnarray}\label{eT3.10LS}
\frac{1}{l}\sum_{n=0}^l \left(\left\vert u_{1,n}^D(E)\right\vert^2+\left\vert 
u_{2,n}^D(E)\right\vert^2\right)\le C_E\,(\ln l)^{1+\delta}\;, \quad l\ge 2\;.
\end{eqnarray}
In particular, by taking $\delta=1$,
\begin{equation}\label{e1.13JL}
\limsup_{l\rightarrow\infty}\frac{\Vert u^D(E)\Vert}{l^{1/2}\ln l}<\infty\;.
\end{equation}
\end{proposition}
\begin{proof} Define the function $g_k(E)=2^{-k}\sum_{n=0}^{2^k}
\left(\vert u_{1,n}^D(E)\vert^2+\vert u_{2,n}^D(E)\vert^2\right)$. It follows from
Proposition 3.3 in~\cite{LS} (in fact, from a straightforward adaptation of this
result) that $\int g_k(E)d\rho_{\textrm{ac}}(E)\le C<\infty$; thus, $\sum_{k=0
}^\infty k^{-1-\delta}g_k(E)\in {\LL}^1(\mathbb{R},d\rho)$. This implies the
inequality $g_k(E)\le \widetilde{C}_E k^{1+\delta}$ for almost every $E$
with respect to $\rho$. Let $2^{k-1}\le l\le 2^k$. Then
\begin{eqnarray*}
\frac{1}{l}\sum_{n=0}^l\left(\left\vert u_{1,n}^D(E)\right\vert^2+\left\vert 
u_{2,n}^D(E)\right\vert^2\right)\le 2\widetilde{C}_E k^{1+\delta}\le C_E(\ln l)^{1+
\delta}\;,
\end{eqnarray*}
with $C_E=2\left(1+\frac{1}{\ln 2}\right)\widetilde{C}_E$. In particular, by picking
the
particular value $\delta=1$, relation~\eqref{e1.13JL} follows from~\eqref{eT3.10LS}.
This completes the proof of the proposition. \end{proof}

\begin{lemma} Let $u^D(E)$ and $u^N(E)$ be the solutions to~\eqref{eqdirre} that
satisfy the initial conditions~\eqref{DNboucon}. Then
\begin{eqnarray*}
\Vert u^D(E)\Vert_{l}\Vert u^N(E)\Vert_{l} \ge c\,l\;.
\end{eqnarray*}
\label{LCW}
\end{lemma}
\noindent \textit{Proof}. Since the Wronskian is constant, it follows that
\begin{eqnarray*}
W[u^D,(u^N)^\ast](n)=c\left(u^D_{1,n+1}u^N_{2,n}-u^N_{1,n+1}u^D_{2,n}\right)=W[u^D,
(u^N)^\ast](-1)=c
\label{conswro}
\end{eqnarray*}
\noindent
for every $n\ge 0$. Thus,
\begin{eqnarray*}
\nonumber \sum_{n=0}^{l-1}c &=&\sum_{n=0}^{l-1}W[u^D,(u^N)^\ast](n)\le\left\vert\left
\langle
\binom{(u_{1,n+1}^D(E))^\ast}{(u_{2,n}^D(E))^\ast},\binom{u_{2,n}^N(E)}{-u_{1,n+1}^N
(E)}\right\rangle_{l-1}\right\vert \\
&\le &\left\Vert\binom{u_{1,n+1}^D(E)}{u_{2,n}^D(
E)}\right\Vert_{l-1} \left\Vert\binom{u_{1,n+1}^N(E)}{u_{2,n}^N(E)}\right\Vert_{l-
1}\;, 
\end{eqnarray*}
\noindent
where we have used Cauchy-Schwarz inequality in the last step. Hence,
\begin{eqnarray*}
\Vert u^D(E)\Vert_{l}\Vert u^N(E)\Vert_{l} \ge c\,l \;,
\end{eqnarray*}
\noindent
which concludes the proof of the lemma. \hfill $\Box$


By taking into account the above adaptations to the Dirac operator, the proofs of
Corollary~\ref{C4.4}(a) and~(b)  follow the same lines of the proofs of
Corollaries~4.4 and~4.5 in~\cite{JitLast}, respectively.

\begin{corollary}\label{C4.4} 
(a)~Suppose that for some $\alpha \in \lbrack 0,1)$ and every $E$ in some Borel set 
$F$, every solution~$\Psi$ to the Dirac eigenvalue equation~\eqref{eqdirre} obeys
\begin{equation*}
\limsup_{l\rightarrow \infty }\frac{\left\Vert \Psi\right\Vert _{l}^{2}}{
l^{2-\alpha }}<\infty ~.
\end{equation*}
Then, the restriction $\rho(F\cap \cdot )$ is $\alpha $-continuous.

(b)~Suppose that 
\begin{equation}
\liminf_{l\rightarrow \infty }\frac{\left\Vert u^D(E)\right\Vert _{l}^{2}}{
l^{\alpha }}=0  \label{c4.5a}
\end{equation}
\noindent is satisfied for every $E$ in some Borel set $F$. Then the 
restriction $\rho(F\cap \cdot )$ is $\alpha $-singular.
\end{corollary}



Corollary~\ref{C4.4} can be rewritten in terms of the one-dimensional $
2\times 2$ transfer matrices $T(n;E)$ defined by~\eqref{Tn}. This approach, based on
Corollary~3.7 in~\cite{CMW}, is
of particular importance in our problem, since we have obtained in Section 
\ref{TMPV} the exact behavior of these matrices for a sparse potential like~\eqref{Ve}.

\begin{corollary}\label{C4.4a}
Suppose that for some $\alpha \in \lbrack 0,1)$ and every $E$ in some Borel set
 $A\subset \mathbb R$, 
\begin{equation}
\limsup_{l\rightarrow \infty }\frac{1}{l^{2-\alpha }}\sum_{n=0}^{l}\left
\Vert T(n;E)\right\Vert ^{2}<\infty \;,  \label{dnor}
\end{equation}
\noindent  with $\left\Vert \cdot\right\Vert $ some matrix norm. Then the 
restriction $\rho(A\cap \cdot)$ is $\alpha $-continuous.
\end{corollary}
\begin{proof} By choosing $\theta_1=0$ and $\theta_2=\pi/2$, it follows
 from a straightforward adaptation of Theorem~2.3 in~\cite{KLS} that 
there exists a 
constant $D$ such that  
\begin{equation*}
\left\Vert T(n-1;E)\right\Vert \ge C\max \left\{
R_{n}(0),R_{n}(\pi/2)\right\} \;,  \label{lr}
\end{equation*}
\noindent 
where $R_n(\theta)$ is the Pr\"{u}fer radius at $n$ starting from the initial 
condition $\mathbf{v}_{\theta }=\dbinom{\cos \theta }{\sin \theta }$; explicitly,
$C=\sqrt{\frac{(E+mc^2)(2c+E-mc^2)}{(E-mc^2)(2c-E-mc^2)}}$. Since
\begin{eqnarray*}
R_{n}^{2}(\theta _{1(2)}) &=& \frac{4c^2(E-mc^2)}{(E+mc^2)(4c^2+m^2c^4-E^2)}\left
\vert u_{1,n}^{D(N)}\right\vert^2+\frac{4c^2}{(4c^2+m^2c^4-E^2)}\left\vert u_{2,n
-1}^{D(N)}\right\vert^2 \\
&-& \frac{4c(E-mc^2)}{4c^2+m^2c^4-E^2}\Re\left\{u_{1,n}^{D(N)}(u_{2,n-1}^{D(N)})^
\ast\right\}
\end{eqnarray*}
(see~\eqref{EFGPT}), we obtain the inequality
\begin{equation*}
D\left[\left\vert u_{1,n}^{D(N)}\right\vert^2+\left\vert u_{2,n-1}^{D(N)}\right
\vert^2\right] \leq R_{n}^{2}(\theta _{1(2)})\;,
\end{equation*}
with $D=\frac{2c(E-mc^2)(2c-E-mc^2)}{(E+mc^2)(4c^2+m^2c^4-E^2)}$. Thus,
\begin{equation}
\sum_{n=0}^{l}\left\Vert T(n;E)\right\Vert ^{2}\geq C_1\max
\left\{\left\Vert u^D(E)\right\Vert _{l+1}^{2},\left\Vert u^N(E)\right\Vert
_{l+1}^{2}\right\}\;,  \label{c4.4a}
\end{equation}
\noindent
$C_1=C\times D$. Hypothesis~\eqref{dnor}, together
with~\eqref{c4.4a}, imply Corollary~\ref{C4.4a}.\end{proof}

Note that the growth of the norm of the transfer matrix gives exactly the 
growth of the increasing solution. This fact will be of great importance later on.

\subsection{Hausdorff dimension and spectral transition}
\label{HDST}

The last step in the determination of the Hausdorff dimension of the
measure $\rho$ is the following extension of Proposition~3.9 in~\cite{CMW}:

\begin{proposition}
\label{l2.1} Let $\mathcal{A=}\left( a_{n}\right) _{n\geq 1}$ be given
by~\eqref{spacon}, $E\in \mathbb{R}$ and assume that the sequence $\left( 
\theta _{n}^\omega\right) _{n\geq1}$ of Pr\"{u}fer angles~\eqref{PA} is $\ud$ $\pi $
for every $\theta _{0}\in \lbrack 0,\pi )$, 
almost every $\varphi\in\lbrack 0,\pi )$ (w.r.t. Lebesgue measure) and almost
every $\omega\in\Xi$. Then, there 
is a generalized eigenfunction $\Psi$ (i.e., $\Psi$ satisfies $H_{v,\phi}(m,c)
\Psi=E\Psi$ and the phase boundary condition~\eqref{bouconphi}) such that 
\begin{equation}
C_{n}^{-1}r^{n/2}\leq R_n(\phi )\leq C_{n}r^{n/2}
\label{lru}
\end{equation}
holds with~$r$ given by~\eqref{erre} and $C_{n}^{1/n}\searrow R_0(\phi)$
 as $n\rightarrow \infty $. In addition, there exists a subordinate solution 
$\Phi$ (with $\alpha ^{\ast }$-phase
boundary condition) for energy $E$ such that, for all sufficiently
large $n$, the Pr\"{u}fer radius associated with $\Phi$ satisfies 
\begin{equation}
\left\vert R_n(\alpha ^{\ast })\right\vert \leq \tilde{C}
_{n}r_{j}^{-n/2}\;  \label{dj}
\end{equation}
with $\tilde{C}_{n}^{1/n}\searrow R_0(\alpha^\ast)$ as $n\rightarrow \infty $.
\end{proposition}
\begin{proof} The proof of the proposition follows the same steps
of the proof of Proposition~3.9 in~\cite{CMW}. We, nevertheless, trace them out. 

The first part of the proposition is simply Lemma~\ref{LRP}. The second part
follows the same steps of Lemma 3.6 in~\cite{CMW1}, which adapts the results
presented in Theorem~8.1 in~\cite{LS} (the latter gives a sufficient condition
for the existence of a subordinate solution to the Dirac equation 
\eqref{eqdirre}).

The inequalities~\eqref{lru} and the results of Theorem~2.3 in~\cite{KLS} imply
that
\begin{equation}
C_{n}^{-1}r^{n/2}\leq t_{n}\leq C_{n}r^{n/2}~,  \label{rtr}
\end{equation}
\noindent
with $t_n:=\Vert T(a_n+1;E)\Vert$ and $r$ given by~\eqref{erre}. From now on,
the proof of the existence of a subordinate solution, as well as the 
determination of its asymptotic behavior, follow from the proof of Lemma 3.6
in~\cite{CMW1}. \end{proof}

\OBSI
The corresponding modifications of the known results for Schr\"o\-ding\-er operators,
employed  in the proof of Proposition~\ref{l2.1} to the Dirac
setting studied in this work, are straightforward, being therefore omitted.
\OBSF

Finally, we are able to prove our main result.



\noindent\begin{proof}(Theorem~\ref{thethe1}) We base the arguments on the proof of
Theorem~3.11 in~\cite{CMW}. First, introduce the notation
\begin{equation}\label{defAeq}
A=-\sqrt{m^2c^4+2c^2
(1-\cos(\mathbb{Q}\pi))} \cup \sqrt{m^2c^4+2c^2(1-\cos(\mathbb{Q}\pi))},
\end{equation} which will also be used in other occasions ahead. 

Theorem~\ref{thethe} implies that the sequence 
$\left(\theta_{n}^{\omega}\right) _{n\geq 0}$ of Pr\"{u}fer angles is $\ud$ $\pi$ 
for every $\theta _{0}\in \lbrack 0,\pi ]$, every $E\in {\LL}^\prime\equiv L\setminus
A$ and almost every $\omega\in\Xi$. We obtain from~\eqref{rtr} the estimates
\begin{equation*}
\left\Vert T(k;E)\right\Vert \leq C_{n}r^{n/2}\leq
C_{n}^{\prime }a_{n}^{\gamma/2}\leq C_{n}^{\prime \prime }k^{\gamma/2}\;,
\end{equation*}
\noindent which hold for every $E\in {\LL}^\prime$ and every $a_n^\omega\le k
<a_{n+1}^\omega$, with $\gamma\equiv \ln r/\ln \beta $, $C_n^{\prime\prime}>0$ and 
$\lim_{n\rightarrow \infty }\left( C_{n}^{\prime \prime }\right) ^{1/n}<\infty$.

It follows by the constancy of $\left\Vert T(k;E)\right\Vert $
on $[a_{n}+1,a_{n+1}]$ (since the free matrix $T_0$ is equivalent to a rotation; 
see Section~\ref{TMPV}) that
\begin{equation}
\sum_{k=0}^{l}\left\Vert T(k;E)\right\Vert ^{2}\leq c\,l^{1+\gamma}  \label{somatra}
\end{equation}
\noindent 
 holds for some $c>0$ and every $E\in {\LL}^\prime$.

The application of Proposition~\ref{l2.1} together with~\eqref{INTI}
guarantees, for these values of $E$, the existence of a subordinate solution
 $\Phi^{\mathrm{sub}}$  such that its sequence of Pr\"ufer radii satisfies the
estimate 
\begin{equation*}
\left\vert R_n\left(\theta_{\alpha^\ast}\right)\right\vert\le C_n^{\prime\prime
\prime}\,a_n^{-\gamma/2}~,
\end{equation*}
for some  constant $C_n^{\prime\prime\prime}>0$.

Since every solution to~\eqref{eqdirre} has constant modulus
on the interval $[a_{n}+1,a_{n+1}]$, we have 
\begin{equation}
\left\Vert \Phi^{\mathrm{sub}}\right\Vert _{l}^{2}\leq c^{\prime
}\,l^{1-\gamma}\;,  \label{modusub}
\end{equation}
\noindent for some $c^{\prime }>0$.




Since the restriction of the measure $\rho$  to~$I$  is supported on the set of
those~$E$ for which $\Phi^{\mathrm{sub}}$ satisfies the
boundary condition~$\phi $ (thanks to the fact that $\rho$ has no absolutely 
continuous part; see Theorem~1 in~\cite{GP}), we have $u^D=\Phi^{\mathrm{sub}}$
 for a.e.\ $E\in I$ with respect to $\rho$ and for a.e.\ boundary condition 
$\phi $ (by the theory of rank-one 
perturbation; see Theorem~1.3 in~\cite{JitLast}).



Thus, by~\eqref{somatra} and~\eqref{modusub}, on the intervals~\eqref{SUPO} one has 
\begin{equation*}
\limsup_{l\rightarrow \infty }\frac{1}{l^{2-\alpha }}\sum_{k=0}^{l}\left
\Vert T(k;E)\right\Vert ^{2}<\infty  \label{c4.4c}
\end{equation*}
\noindent and 
\begin{equation*}
\liminf_{l\rightarrow \infty }\frac{\left\Vert u^D(E)\right\Vert
_{l}^{2}}{l^{\alpha ^{\prime }}}=0\;,  \label{c4.5b}
\end{equation*} 
\noindent provided $2-\alpha=1+\gamma+\varepsilon$ and $\alpha^{\prime }=1-\gamma
+\varepsilon$, respectively, where $\varepsilon$ is an arbitrary positive 
number.

It follows, by Corollary~\ref{C4.4a}, that the spectral 
measure~$\rho$ is simultaneously $(1-\gamma-\varepsilon)$-continuous and 
$(1-\gamma+\varepsilon)$-singular. Since $\varepsilon$ is arbitrary, we have
 from Remark~\ref{DSC} that the restriction $\rho(I^\prime\cap\cdot)$ has exact
Hausdorff dimension given by~\eqref{HDim}, where $I^\prime\subseteq {\LL}^\prime$.

Finally, from the theory of rank one perturbations (more specifically, Theorem~8.1
of \cite{Si}), we know that  $\rho\left(A\right)=0$ (see \eqref{defAeq}) holds for
almost
 every~$\phi $; therefore, for almost every~$\phi $, the restriction $\rho(I\cap
\cdot )$ has~\eqref{HDim} as its Hausdorff dimension. This concludes the proof 
of the theorem.
\end{proof}



\OBSI
We have checked that it is possible
to extend the results of~\cite{GP} and~\cite{KP} to the Dirac setting, in the same
way as exposed
here for the results of generalized subordinacy  discussed in~\cite{JitLast}.
\OBSF
Despite  the obtained exact Hausdorff dimension of the spectral measure
 $\rho$, what can we infer from the spectral nature of the operator $H_{v,\phi}
(m,c)$ on the interval~\eqref{SUPO}? This is a tricky question, since we are
induced to think that for $\alpha_\rho=0$, the spectrum is simply dense pure 
point and for $\alpha_\rho>0$, the spectrum is singular continuous. There are,
however, some examples in the literature \cite{DRJLS,CMW1,Zla} where the
spectrum is singular continuous with null Hausdorff dimension.
Note that Theorem~\ref{tmarwre}  settles  the problem. Now we present its proof.


\noindent \begin{proof} (Theorem~\ref{tmarwre}) We must prove, for fixed $v$ and
$\beta$, the inclusion
\begin{equation*}
I_1\setminus A_1\subseteq\left\{E\in\mathbb{R}:\sum_{n=0}^
\infty\Vert T(n;E)\Vert^{-2}=\infty\right\}
\end{equation*}
\noindent
($A_1$ is a set with zero Lebesgue measure), since (a)~follows directly from
an adaptation of Theorem~2.1 in~\cite{SS}. Now, if 
\begin{equation*}
[I_1\setminus A_1]^C\subseteq\left\{E\in\mathbb{R}: 
\sum_{n=1}^{\infty }\beta ^{n}(t_{n})^{2}\left( \sum_{m=n}^{\infty
}(t_{m}^j)^{-2}\right) ^{2}<\infty \right\}\;,
\end{equation*}
\noindent
then (b)~follows from an adaptation of Lemma~4.1 in~\cite{CMW1} and by 
Proposition~4.2 in~\cite{MWGA}.

We pick~$n$ such that $a_N^\omega \le n <a_{N+1}^\omega$ is valid for some $N\in
\mathbb{N}$. We obtain from~\eqref{spacon} and~\eqref{R}, 
\begin{eqnarray}
\!\!\!\!\!\!\!\!\! \sum_{m\le n}\Vert T(m;E)\Vert_{U}^{-2} = 
\sum_{k\le N+1}\Vert T(a_k^\omega;E)\Vert_{U}^{-2}\beta^{k+1} + \Vert 
T(a_N^\omega;E)\Vert_{U}^{-2}\sum_{a_{N+1}^\omega\le m < n}1 \,,
\label{MTM}
\end{eqnarray}
\noindent with $\Vert \cdot \Vert_U:=\Vert U\cdot U^{-1}\Vert$ (see Section 4
of \cite{MWGA}). 

We have by Theorem~2.3 in~\cite{KLS} the inequalities
\begin{eqnarray}
\Vert T(m;E)\Vert^{-2}\le C^2\Vert T(a_k^\omega;E)
\Vert^{-2}_{U} \le C^2\left(\max_{i\in\{1,2\}}R_{k}(\theta_{i})
\right)^{-2}
\label{MTM1}
\end{eqnarray}
\noindent
and
\begin{eqnarray}
\Vert T(m;E)\Vert^{-2}\ge C^{-2}\Vert T(a_k^\omega;E)\Vert^{-2}_{U} \ge 
\tilde C^{-2}\left(\max_{i\in\{1,2\}}R_{k}(\theta_{i})\right)^{-2}\;,
\label{MTM2}
\end{eqnarray}
\noindent
for every $a_k^\omega\le m<a_{k+1}^\omega$, with $C$ as in the proof of Corollary
\ref{C4.4a} and 
\[
\tilde C=C/|\sin(\theta_{1}-\theta_{2})/2|
\] ($\theta_{1
}$, $\theta_{2} \in [0,\pi)$ represent a pair of initial Pr\"ufer angles
such that $0<|\theta_{1}-\theta_{2}|<\pi/2$; see Lemma 2.2 of \cite{KLS}). 

Introduce 
\begin{eqnarray}
S_{N,M}^{\pm}(\beta,\varphi)\equiv \sum_{k=N}^MC_{k}^{\pm}\left(\beta^{k+1} 
\pm 2k\right)r^{-k} \;;
\label{SNM}
\end{eqnarray}
\noindent
(the term $2k$ is due to the uncertainty associated with the position of the 
$k$-th barrier: $\beta^{k+1}-2k\le a_k^\omega \le \beta^{k+1}+2k-1$ for all 
$\omega\in\Xi$), with $r$  given by~\eqref{erre}. 
It follows by Lemma~\ref{LRP}, and equations~\eqref{MTM}, \eqref{MTM2}
and~\eqref{SNM}, 
\begin{eqnarray}
\tilde{C}^{-2}S^-_{N+1,M}\le\sum_{m=n}^{a_M+1} \Vert T(m;E) \Vert^{-2}\le 
C^{2}S_{N,M}^{+} 
\;, 
\label{theinq}
\end{eqnarray}
for every $M\ge N+1$ and $E\in B$, with 
\begin{eqnarray}\label{Be}
B\equiv -\sqrt{m^2c^4+2c^2(1-\cos([0,\pi]\setminus\mathbb{Q}\pi))} \cup \sqrt{
m^2c^4+2c^2(1-\cos([0,\pi]\setminus\mathbb{Q}\pi))}\;,
\end{eqnarray}
and the set $2\cos(\mathbb{Q}\pi\cap[0,\pi))$ represents the  ``energies'' where the
sequence $(\theta^{\omega}_n)_{n\ge 0}$ is not $\ud$ $\pi$; see the proof of
Theorem~\ref{thethe}.

It is clear by Lemma~\ref{LRP}, and equations~\eqref{erre}, \eqref{theinq},
and~\eqref{Be} 
that
\begin{eqnarray*}
B \subseteq \left\{ E: \lim_{n\to\infty}\Vert
T(n;E) \Vert = \infty \right\} \;,
\end{eqnarray*}
\noindent
i.e., $B$ belongs to the complement of the essential support of the absolutely 
continuous part of the spectrum of $H_{v,\phi}(m,c)$, $\Sigma_{\mathrm{ac}}$, 
according to Theorem 1.1 of \cite{LS}. Hence, except for the set $A_1 = 
(\mathbb{Q}\pi\cap[0,\pi)) \cup A^*$ of Lebesgue zero measure ($A^*$ is some 
set of Lebesgue zero measure presented in the definition of~$\Sigma_{\mathrm{ac}}$),
the set
\begin{eqnarray}
B_1\equiv -\sqrt{m^2c^4+2c^2(1-\cos([0,\pi]\setminus A_1))} \cup \sqrt{
m^2c^4+2c^2(1-\cos([0,\pi]\setminus A_1))}%\;  
\label{Be1}
\end{eqnarray}
\noindent
belongs to the singular spectrum. It follows from~\eqref{SNM} and the left-hand
member of~\eqref{theinq}, for $M\rightarrow\infty$, that 
\begin{eqnarray*}
\sum_{n=0}^{\infty} \Vert
T(n;E) \Vert^{-2} = \infty
\end{eqnarray*}
\noindent
if $\beta>r$. Therefore, we conclude from~\eqref{Be1} and Theorem 2.2 of 
\cite{KLS} that the essential spectrum of $H_{v,\phi}(m,c)$ is purely singular
continuous when restricted to $I\backslash A_1$, and this proves part~(a) of the
theorem.

We prove part~(b) by assuming that~$\beta<r$. By introducing 
\begin{eqnarray}
S = \sum_{m=n}C_{m}r^{-m}\;,
\label{ESSE}
\end{eqnarray} 
\noindent
it follows by Lemma~\ref{LRP} and Eqs.~\eqref{MTM},~\eqref{MTM1}, that
\begin{eqnarray}
\sum_{m=n}^\infty \Vert T(a_m+1;E) \Vert^{-2} \le C^2 S \;.
\label{ESSE1}
\end{eqnarray}

Finally, we have from Lemma~\ref{LRP} and
Eqs.~\eqref{MTM1},~\eqref{MTM2},~\eqref{ESSE},~\eqref{ESSE1} that 
\begin{eqnarray}
\nonumber \sum_{n=1}^\infty \left(\beta^n+\omega_n-\omega_{n-1}\right) \Vert T(a_n+
1;E) \Vert^2\left(\sum_{m=n}^\infty \Vert T(a_m+1;E) \Vert^{-2}\right)^2 &  & \\
\nonumber \le C^\prime\sum_{n=1}^\infty \left(\beta^n+2n\right)C_{n}r^nS^2  & & \\
 \le C^{\prime\prime}\sum_{n=1}^\infty \left(\beta^n+2n\right)(C_{n})^3r^{-n} & &
\label{Epp}
\end{eqnarray}
\noindent
converges for every $E\in I^C$. It follows by Lemma~3.6 in~\cite{CMW1}
and~\eqref{Epp} above, that if $E\in I^c\setminus A_1$, then the Dirac equation
\eqref{eqdirre} have a $l^2(\mathbb{Z}_+,\mathbb{C}^2)$ solution. Since, by
Theorem~\ref{tee}, $I^c\in\sigma_{\mathrm{ess}}(H_{v,\phi}(m,c))$ for every $\phi\in
[0,\pi]$, the hypothesis of Proposition 4.2 of \cite{MWGA} is fulfilled and
consequently $H_{v,\phi}(m,c)$ have only dense pure point spectrum in $I^C$ for
almost every $\phi\in[0,\pi]$. This concludes the proof of part~(b) of the
theorem.\end{proof}




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\end{document}
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