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%%     journal="Proc. Amer. Math. Soc. 126, 1685-1695 (1998)",
%%     doi="10.1090/S0002-9939-98-04310-X",
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\copyrightinfo{1997}{by G. Teschl}
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\commby{Palle E. T. Jorgensen}
\date{October 28, 1996}

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\begin{document}

\title{Renormalized Oscillation Theory for Dirac Operators}


\author{Gerald Teschl}
\address{Institut f\"ur Reine und Angewandte Mathematik\\
RWTH Aachen\\ 52056 Aachen\\ Germany}
\curraddr{Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ 1090 Wien\\ Austria}
\email{gerald.teschl@univie.ac.at}
\urladdr{http://www.mat.univie.ac.at/\string~gerald/}

\keywords{Oscillation theory, Dirac operators, spectral theory}
\subjclass{Primary 36C10, 39L40; Secondary 34B24, 34L15}

\begin{abstract}
Oscillation theory for one-dimensional Dirac operators with separated boundary
conditions is investigated. Our main theorem reads: If $\lambda_{0,1}\in \mathbb R$
and if $u,v$ solve the Dirac equation $H u= \lambda_0 u$, $H v= \lambda_1 v$ (in
the weak sense) and respectively satisfy the boundary condition on the left/right,
then the dimension of the spectral projection $P_{(\lambda_0, \lambda_1)}(H)$
equals the number of zeros of the Wronskian of $u$ and $v$. As an application we
establish finiteness of the number of eigenvalues in essential spectral gaps of
perturbed periodic Dirac operators.
\end{abstract}

\maketitle



\section{Introduction}




In \cite{st} Sturm originated oscillation theory for second-order differential
equations one hundred and fifty years ago. Since then numerous extensions have been
made (see, e.g., \cite{co},\cite{kr},\cite{re},\cite{sw}, and the references
therein). In \cite{wd2} Weidmann extended results for Sturm--Liouville operators
from Hartman
\cite{ha1}, \cite{ha2}, Hartman and Putnam \cite{hp}, and himself \cite{wd1} to the
case of Dirac operators. In particular, he proves Sturm-type comparison theorems
and applies the results to investigate the essential spectrum of Dirac operators.
With the present paper we want to complement \cite{wd2} in the sense that we will
use oscillation theory to investigate the discrete spectrum.

Using standard oscillation theory would mean to count zeros of components of
solutions of the Dirac equation. Unfortunately this approach soon leads into severe
troubles:

(i). Components of solutions might vanish identically on some
intervals.

(ii). Zeros of components of solutions are not monotone with respect to the
spectral parameter. Hence solutions can pick up or lose zeros as the
spectral parameter increases which, in general, destroys the connection
between zeros and number of eigenvalues (cf.\
Remark~\ref{rmzofuj}).

The natural remedy is to look at zeros of the Wronskian instead, that is, use a
renormalized version of oscillation theory developed in \cite{gst} for the case of
Sturm--Liouville operators (see \cite{tosc} in the case of Jacobi operators). In
addition, this approach avoids technical difficulties arising from the fact that
Dirac operators, in contradistinction to Sturm--Liouville operators, are not bounded
from below.

To set the stage, let $I=(a,b) \subseteq \R$ (with $-\infty \le
a < b \le \infty$) be an arbitrary interval and consider the
Dirac differential expression
\begin{equation}
\tau = \frac{1}{\I} \sig_2 \frac{d}{dx} + \phi(x).
\end{equation}
Here
\begin{equation}
\phi(x) =  \phi_{\rm el}(x)\id  + \phi_{\rm am}(x)\sig_1 +
(m+ \phi_{\rm sc}(x)) \sig_3,
\end{equation}
$\sig_1$, $\sig_2$, $\sig_3$ denote the Pauli matrices
\begin{equation}
\sig_1=\left(\ba{cc} 0 & 1 \\ 1 & 0\ea\right), \quad 
\sig_2=\left(\ba{cc} 0 & -\I \\ \I & 0\ea\right), \quad 
\sig_3=\left(\ba{cc} 1 & 0 \\ 0 & -1\ea\right),
\end{equation}
and $m$, $\phi_{\rm sc}$, $\phi_{\rm el}$, and $\phi_{\rm am}$ are interpreted as
mass, scalar potential, electrostatic potential, and anomalous
magnetic moment, respectively (see \cite{th}, Chapter~4). 
As usual we require $m\in[0,\infty)$ and $\phi_{\rm sc}, \phi_{\rm el}, \phi_{\rm
am}
\in L^1_{loc}(I,\R)$ real-valued. We don't include a
magnetic moment $\hat{\tau} = \tau +\sig_2 \phi_{\rm mg}(x)$
since it can be easily eliminated by a simple gauge transformation
$\tau = U \hat{\tau} U^{-1}$, $U =\exp(\I\int^x \phi_{\rm mg}(t) dt)$
(there is also a gauge transformation which gets rid of $\phi_{am}$ or
$\phi_{\rm el}$ (see \cite{ls}, Section~7.1.1)).
Explicitly we have
\begin{equation}
\tau f = \left(\ba{cc} \phi_{11} & -\frac{d}{dx} + \phi_{12}\\
\frac{d}{dx} + \phi_{12}& \phi_{22}\ea\right)
\left(\!\!\ba{c} f_1 \\ f_2\ea\!\!\right)
= \left(\ba{c} -f_2' + \phi_{12}f_2 + \phi_{11} f_1 \\ f_1'
+\phi_{12}f_1 + \phi_{22} f_2 \ea\right),
\end{equation}
$f\in AC_{loc}(I,\C^2)$, where primes denote derivatives with respect to $x$ and
$\phi_{11} = \phi_{\rm el} + m + \phi_{\rm sc}$, $\phi_{12} = \phi_{21} =
\phi_{\rm am}$, $\phi_{22} = \phi_{\rm el} - m - \phi_{\rm sc}$.

If $\tau$ is limit point at both $a$ and $b$, then $\tau$ gives rise to a unique
self-adjoint operator $H$ when defined maximally (cf., e.g., \cite{ls},
\cite{wdl}, \cite{wd2}). Otherwise, we fix a boundary
condition at each endpoint where $\tau$ is limit circle.

By $u_\pm(z,x)$ we will denote (non identically vanishing) solutions of the
differential equation $\tau u = z u$, $z\in\C$, which satisfy the following
requirements (whenever such solutions exist).

\begin{list}{(\roman{me}).}{\usecounter{me}}
\item $u_\pm(z,.) \in AC_{loc}(I,\C^2)$ and  $\tau u_\pm(z)
= z u_\pm(z)$.
\item $u_+(z,.)$ (resp.\ $u_-(z,.)$) is square integrable near $b$
(resp.\ $a$) and  fulfills the boundary condition of $H$ at $b$
(resp.\ $a$) if any (i.e$.$, if $\tau$ is limit circle at $b$
(resp.\ $a$)).
\end{list}

Explicitly, $H$ is given by
\begin{equation}
H: \ba[t]{lcl} \db(H) &\to& L^2(I,\C^2) \\ f &\mapsto& \tau f \ea ,
\end{equation}
where
\begin{equation} \label{domH}
\db(H) = \{ f \in L^2(I,\C^2) | \ba[t]{l} f \in AC_{loc}(I,\C^2), \, \tau f \in
L^2(I,\C^2),\\ W_a(u_-(\lam_0),f) = W_b(u_+(\lam_0),f) =0 \} \ea
\end{equation}
with
\begin{equation}
W_x(f,g) = f_1(x) g_2(x) - f_2(x) g_1(x)
\end{equation}
the usual Wronskian (we remark that the limit $W_{a,b}(.,..) = \lim_{x \to
a,b} W_x(.,..)$ exists for functions as in (\ref{domH})).
The resolvent of $H$ can be expressed in terms of $u_\pm(z)$ as follows
\begin{equation} \label{resolv}
(H-z)^{-1} f(x)= \int_a^b G(z,x,y) f(y) dy,
\end{equation}
where
\begin{equation}
G(z,x,y)= \frac{u_\pm(z,x) \otimes u_\mp(z,y)}{W(u_+(z),u_-(z))},
\quad \pm(x-y)>0.
\end{equation}
Recall that $W_x(u_+(z),u_-(z))$ is independent of $x$
(cf.\ (\ref{wprime})). In addition, we set $G(z,x,x) =
\lim_{\eps\to 0} (G(z,x+\eps,x) + G(z,x-\eps,x))/2$.

Denote by $H_{x,-}$ (resp.\ $H_{x,+}$), $x\in I$ self-adjoint operators associated
with $\tau$ on $L^2((a,x),\C^2)$ (resp.\  $L^2((x,b),\C^2)$) obtained from $H$ by
imposing the additional boundary condition $f_1(x)=0$. Then $H_{x,-}\oplus H_{x,+}$
is a rank one resolvent perturbation of $H$ and hence $\sig_{ess}(H) =
\sig_{ess}(H_{x,-}) \cup \sig_{ess}(H_{x,+})$ (cf.\ \cite{wd3}, Korollar~6.2). Here
$\sig_{ess}(.)$ denotes the essential spectrum. If $G_{x,\pm}(z,.,..)$ denotes the
resolvent kernel of $H_{x,\pm}$ we define the Weyl $m$-functions $m_{x,\pm}(z)$
(w.r.t.\ the base point $x$) by
\begin{equation}
G_{x,\pm}(z,x,x) = \left(\ba{cc} 0 & \pm\frac{1}{2} \\ \pm\frac{1}{2} &
m_{x,\pm}(z) \ea\right).
\end{equation}
The first resolvent identity shows that $m_{x,\pm}(z)$ are Herglotz functions
(cf., e.g., \cite{ti}).

\begin{lemma}
The solutions $u_\pm(z,x)$ exist for $z\in\C\bs\sig_{ess}(H_{x_0,\pm})$.
They can be assumed real analytic with respect to $z\in\C\bs
\sig(H_{x_0,\pm})$. In addition,  we can include a finite number of isolated
eigenvalues in the domain of holomorphy of $u_\pm(z,x)$ by removing the
corresponding poles.
\end{lemma}

\begin{proof}
If $U(z,x,x_0)$, $z\in\C$ is a fundamental matrix solution for $\tau u = z u$
(i.e., $U(z,x_0,x_0) = \id$, $x_0 \in I$) and
$m_{x_0,\pm}(z)$ are the Weyl $m$-functions with respect to the base point
$x_0$. Then  we can choose
\begin{equation}
u_\pm(z,x) =  U(z,x,x_0) \left(\!\!\ba{c} 1 \\ \pm
m_{x_0,\pm}(z)\ea\!\!\right).
\end{equation}
By removing the corresponding poles of $m_{x_0,\pm}(z)$ we can include a finite
number of isolated eigenvalues in the domain of holomorphy of $u_\pm(z,x)$.
\end{proof}

A finite end point is called regular if $\phi_{11}, \phi_{12},
\phi_{22}$ are integrable near this end point.  In this case boundary values for
all functions exist at this end point. In particular, $\tau$ is called regular if
both end points $a,b$ are regular, that is, $a,b \in \R$ and $\phi_{11},
\phi_{12}, \phi_{22} \in L^1(I,\R)$. In the regular case the resolvent of
$H$ is Hilbert-Schmidt and hence the spectrum is purely discrete (i.e.,
$\sig_{ess}(H)=\emptyset$).



\section{Wronskians}



In this section we want to investigate the Wronskian of two solutions $u,v$.
A straightforward calculation gives
\begin{equation} \label{wprime}
W_x'(u,v) = (\lam_0-\lam_1) u(x) v(x)
\end{equation}
if $\tau u = \lam_0 u$ and $\tau v = \lam_1 v$. Note that (in contradistinction
to the Sturm--Liouville case) the Wronskian of two solutions can only
have simple zeros (unless $\lam_0=\lam_1,\: u=v$ or $u\equiv 0$ (resp.\ $v\equiv 0$)
of course). Moreover, $W_x(u,v)=0$ if $u(x),v(x)$ are parallel and $W_x'(u,v)=0$ if
$u(x),v(x)$ are orthogonal.

Clearly this implies

\begin{lemma} \label{wvanish}
Let $\tau u = \lam_0 u$ and $\tau v = \lam_1 v$ for some $\lam_1\ne\lam_0$.
If $u,v \in L^2((c,d),\C^2)$ and $W_c(u,v) = W_d(u,v)$ for some $a \le c < d \le
b$ then $u,v$ are orthogonal on $(c,d)$, that is, $\int_c^d u(t) v(t) dt =0$.
\end{lemma}

\begin{proof}
Integrating (\ref{wprime}) we obtain
$W_d(u,v)-W_c(u,v)=(\lam_1-\lam_0) \int^d_c u(t) v(t) \, dt$,
$c,d \in I$, and hence the result is immediate (take limits if $c=a$ or $d=b$).
\end{proof}

\begin{lemma}
Let $\lam \in \R\bs\sig_{ess}(H)$. Then
\begin{equation} \label{wrpsidot}
W_x(u_\pm(\lam), \dot{u}_\pm(\lam)) = \left\{
\ba{l} -\int_x^b u_+(\lam,t)^2 dt \\ \int_a^x
u_-(\lam,t)^2 dt \ea \right. ,
\end{equation}
where the dot denotes a derivative with respect to $\lam$.
\end{lemma}

\begin{proof}
{}From Lemma \ref{wvanish} we know
\begin{equation}
W_x(u_\pm(\lam), u_\pm(\ti{\lam})) = (\ti{\lam}-\lam) \left\{
\ba{l} -\int_x^b u_+(\lam,t) u_+(\ti{\lam},t) dt \\ \int_a^x
u_-(\lam,t) u_-(\ti{\lam},t) dt \ea \right. .
\end{equation}
Now use this to evaluate the limit
$\lim_{\ti{\lam} \to \lam}W_x(u_\pm(\lam),
(u_\pm(\lam) - u_\pm(\ti{\lam}))/(\lam-\ti{\lam}))$.
\end{proof}




\section{Oscillation theory}




We first introduce Pr\"ufer variables for $u\in C(I,\R)$ defined by
\begin{equation}
u_1(x)=\rho_u(x)\sin(\theta_u(x)) \qquad u_2(x)=\rho_u(x)
\cos(\theta_u(x)).
\end{equation}
If $u$ is never $(0,0)$ and $u$ is continuous, then $\rho_u$ is
positive and $\theta_u$ is uniquely determined once a value of
$\theta_{u}(x_0)$, $x_0\in I$ is chosen by the requirement $\theta_u \in C(I,\R)$.

Clearly
\begin{equation} \label{wpruefer}
W_x(u,v)=\rho_u(x)\rho_{v}(x)\sin(\theta_u(x)-\theta_v(x)).
\end{equation}

An important role is played by the following observation.

\begin{lemma} \label{delincx}
Let $\lam_0<\lam_1$, let $u,v$ solve $\tau
u = \lam_0 u$, $\tau v= \lam_1 v$, and introduce
\begin{equation}
\Delta_{u,v}(x)= \theta_u(x) - \theta_v(x).
\end{equation}
Then, if $\Delta_{u,v}(x) \equiv 0 \mod \pi$,
\begin{equation}
\lim_{x \to x_0} \frac{\Delta_{u,v}(x) -
\Delta_{u,v}(x_0)}{x-x_0}= (\lam_1-\lam_0)>0.
\end{equation}
\end{lemma}

\begin{proof}
If $\Delta_{u,v}(x_0)\equiv 0 \mod \pi$, then (from (\ref{wpruefer}))
\begin{equation}
\lim_{x \to x_0} \frac{\rho_u(x)\rho_v(x) \sin(\Delta_{u,v}(x))}{x-x_0} =
W'_{x_0}(u,v)>0
\end{equation}
implies the assertion using (\ref{wprime}).
\end{proof}

Or, put differently, the last proposition implies that the integer part of
$\Delta_{u,v}(x)/ \pi$ is increasing.

\begin{lemma} \label{lemnbzer}
Let $\lam_0 < \lam_1$ and let $u,v$ solve $\tau u = \lam_0 u$, $\tau v= \lam_1 v$.
Denote by $\#(u,v)$ the number of zeros of $W(u,v)$ inside the interval $(a,b)$. 
Then
\begin{equation} \label{nprva}
\#(u,v) = \lim_{x \uparrow b} \intp{\Delta_{u,v}(x) / \pi} -
\lim_{x \downarrow a} \intp{\Delta_{u,v}(x) / \pi},
\end{equation}
where $\intp{x}$ denotes the integer part of a real number $x$, that is,
$\intp{x} = \sup\{n \in \Z | n \le x \}$.
\end{lemma}

\begin{proof}
We start with an interval $[x_0,x_1]$ containing no zeros of
$W(u,v)$. Hence $\intp{ \Delta_{u,v}(x) / \pi} = \intp{
\Delta_{u,v}(x) / \pi}$. Now let $x_0 \downarrow a$, $x_1 \uparrow b$ and
use Lemma~\ref{delincx}.
\end{proof}

If $\lam \in\R\bs\sig_{ess}(H)$ holds, then
equation (\ref{wrpsidot}) clearly implies 
\begin{equation} \label{phidot}
\dot{\theta}_+(\lam,x) = \frac{\int_x^b u_+(\lam,t)^2 dt}{\rho_+(\lam,x)^2}
> 0, \qquad \dot{\theta}_-(\lam,x) = -\frac{\int_a^x u_-(\lam,t)^2
dt}{\rho_-(\lam,x)^2} < 0,
\end{equation}
where we have abbreviated $\rho_\pm(\lam,x) = \rho_{u_\pm(\lam)}(x)$ and
$\theta_\pm(\lam,x) = \theta_{u_\pm(\lam)}(x)$.

\begin{remark} \label{rmzofuj}
We remark that linking zeros of $u_j$ to the rotation number $\theta_u$ is not
possible since (unlike in the Sturm--Liouville case) the integer part of
$\theta_u$ does not count zeros of $u_j$. Indeed, (assuming $\phi$
 continuous for a moment) shows that
$u_1(x)=0$ implies $\theta_u'(x_0) = \phi_{22}(x_0) - \lam_0$
which is not necessarily positive. Hence the integer part of
$\theta_u/ \pi$ can increase or decrease (or stay the same) at zeros of
$u_1$ (cf.\ the discussion at the end of Section~2 in \cite{wd2}).
In addition, this implies that zeros of $u_{\pm,1}(\lam,.)$, are not monotone with
respect to $\lam$. Hence solutions can pick up or lose zeros as $\lam$ increases.
This, in general, destroys the connection between zeros and number of eigenvalues.
Moreover, if $\phi_{22}(x) - \lam_0$, vanishes on a subinterval of $I$,
then $u_1(x)$ can vanish on the same interval (without $u$ being identically zero).

However, if $\phi_{22}$, is bounded from above (resp.\ below), we can apply
standard oscillation theory for values of $\lam$ with $\phi_{22}(x) -\lam <0$
(resp.\ $\phi_{22}(x) -\lam >0$) for all $x\in I$ (cf.\ Remark~\ref{remsusy}~(ii)).
Similar for $u_2$. 

To further illustrate these problems we consider the following example
with 
\begin{equation}
\phi = \left(\ba{cc} \theta' & 0 \\ 0 & \theta' \ea\right).
\end{equation}
We will normalize $\theta(x_0)=0$ for some $x_0\in I$. The
solution $u$ of $\tau u = \lam_0 u$ satisfying the initial condition $u(x_0)=
\rho_0 (\sin\theta_0, \cos\theta_0)$ is given by
\begin{equation}
u(x) = \rho_0 \left(\ba{c} \sin(\theta_0 - \lam_0 (x-x_0) + \theta(x) ) \\
\cos(\theta_0 - \lam_0 (x-x_0) + \theta(x) )\ea\right).
\end{equation}
Clearly, if $\theta'(x)=\lam_0$ for $x\in(x_0,x_0+\eps)$ and $\theta_0=0$, then
$u(x) = (0, \rho_0)$ for $x\in(x_0,x_0+\eps)$.

To get more specific, let $I=(0,1)$, $\theta(x) = 4x(x-1)$, $x_0=0$, and impose
the boundary conditions $f_1(0)=f_1(1)=0$. We easily obtain $\sig(H) = \pi\Z$ and
\begin{equation}
\theta_-(\lam,x)= \theta(x) - \lam x, \quad \theta_+(\lam,x)= \theta(x) - \lam
(x-1).
\end{equation}
This implies the following for the zeros of $u_{-,1}(\lam,.)$ as $\lam$ increases.
At $\lam=0\in\sig(H)$ there are no zeros. At $\lam=4(\sqrt{\pi}-1) \not\in\sig(H)$
we pick up two zeros one of which gets lost again at $\lam=\pi\in\sig(H)$. As soon
as $\lam>4$ we have $\theta'(x)-\lam>0$ for all $x\in I$ and from now on
$u_{-,1}(\lam,.)$ picks up precisely one zero whenever $\lam$ hits an eigenvalue
(and no zeros get lost).

To end this remark we compute $\Delta_{u_-(\lam_0), u_+(\lam_1)}(x)=
\lam_1 (x-1) - \lam_0 x$, where all unpleasant factors cancel.
\end{remark}





\section{Number of eigenvalues and zeros of Wronskians}



The objective of this section is to establish the connection between zeros of the
Wronskian and spectra of Dirac operators. As a warm up we considers the regular
case.

\begin{theorem} \label{thmreg}
Suppose $\tau$ is regular. Denote by $P_\Omega(H)$ the
family of spectral projections for $H$. Then we have for $\lam_0 < \lam_1$
\begin{equation}
\dim\Ran \, P_{(\lam_0,\lam_1)}(H) = \#(u_-(\lam_0),u_+(\lam_1)) =
\#(u_+(\lam_0),u_-(\lam_1)),
\end{equation}
where $\#(u,v)$ is the number of zeros of $W(u,v)$ inside $(a,b)$.
\end{theorem}

\begin{proof}
We only carry out the proof for the $\#(u_-(\lam_0),u_+(\lam_1))$ case.
Defining $\#(u_-(\lam_0),u_+(\lam_1))$ as in
(\ref{nprva}) shows that our claim is true for $\lam_1$ close to $\lam_0$.
Abbreviate $\Delta(\lam,x) = \Delta_{u_-(\lam_0),u_+(\lam)}(x)$.
Since $\Delta(\lam,b)$ is independent of $\lam$ it
suffices to look at $\Delta(\lam,a)$. As $\lam$ increases from $\lam_0$ to
$\lam_1$, $-\Delta(\lam,a)$ increases by (\ref{phidot}) and is $0 \mod
\pi$ if and only if $\lam$ is an eigenvalue of $H$ (Lemma~\ref{lemnbzer},
equation (\ref{domH})) completing the proof.
\end{proof}

Next, we want to prove Theorem \ref{thmreg} in the general case. This will be
done in two parts.

\begin{theorem}
Let $\lam_0 < \lam_1$ and $\sig_0,\sig_1 \in \{\pm \}$. Suppose
$u_{\sig_j}(\lam_j,.)$, $j=0,1$ exist. Then
\begin{equation}
\dim\Ran \, P_{(\lam_0,\lam_1)}(H)  \ge
\#(u_{\sig_0}(\lam_0),u_{\sig_1}(\lam_1)).
\end{equation}
\end{theorem}

\begin{proof}
Again the proof is only done for $\sig_0=-$. Abbreviate
$u=u_-(\lam_0)$ and $v=u_+(\lam_1)$ and $n=\#(u,v)$.
Suppose $n$ finite, otherwise the following argument works for arbitrary large
$n$. Let $x_1,\dots, x_n$ be the zeros of $W_x(u,v)$.
Since $W_{x_j}(u,v)=0$ there exists constants $\gam_j$ such that
\begin{equation}
\eta_j(x) = \left\{ \ba{c@{\quad}l} u(x), & x \le x_j \\
\gam_j v(x), & x > x_j \ea \right. , 1 \le j \le n,
\end{equation}
is in the domain of $H$ (i.e., $u(x_j) = \gam_j v(x_j)$). Furthermore, set
\begin{equation}
\ti{\eta}_j(x) = \left\{ \ba{c@{\quad}l} -u(x), & x \le x_j \\
\gam_j v(x), & x > x_j \ea \right. , 1 \le j \le n.
\end{equation}
If $\lam_1$ is an eigenvalue of $H$ we define in
addition $\eta_0 = v = \ti{\eta}_0$, $x_0=a$ and if
$\lam_0$ is an eigenvalue of $H$, $\eta_{n+1} = u = -\ti{\eta}_{n+1}$,
$x_{n+1}=b$. Lemma~\ref{wvanish} implies $\int^{x_k}_{x_j}
uv\,dx =0$ and hence $\int^a_b \eta_j\eta_k\,dx = \int^a_b 
\ti{\eta}_j\ti{\eta}_k\,dx$ for all $j,k$. Using
\begin{equation}
(H-\frac{\lam_1+\lam_0}{2})\eta_j = \frac{\lam_1-\lam_0}{2}\ti{\eta}_j
\end{equation}
we obtain
\begin{equation}
\| (H-\frac{\lam_1+\lam_0}{2}) \eta \| = \frac{\lam_1-\lam_0}{2}\, \|\eta\|
\end{equation}
for any $\eta$ in the span of the $\eta_j$'s. Thus, $\dim\Ran \,P_{[\lam_0,
\lam_1]}(H) \ge \dim(\text{span}\{ \eta_j \})$. But $u$ and $v$ are
independent on each interval (since their Wronskian is non-constant) and so the
$\eta_j$ are linearly independent. This proves the theorem in the
$u=u_-(\lam_0)$, $v=u_+(\lam_1)$ case.

The case $u=u_-(\lam_0)$, $v=u_-(\lam_1)$ is similar. We define
\begin{equation}
\eta_j(x) = \left\{ \ba{c@{\quad}l} u(x) + \gam_j v(x), & x \le x_j \\
0, & x > x_j \ea \right. , 1 \le j \le n
\end{equation}
(with $\eta_j \in \db(H)$), and
\begin{equation}
\ti{\eta}_j(x) = \left\{ \ba{c@{\quad}l} -u(x) + \gam_j v(x), & x \le x_j \\
0, & x > x_j \ea \right. , 1 \le j \le n.
\end{equation}
If $\lam_1$ is an eigenvalue of $H$ we define in addition $\eta_0 = v =
\ti{\eta}_0$, $x_0=b$ and if $\lam_0$ is an eigenvalue of $H$, $\eta_{n+1}
= u = -\ti{\eta}_{n+1}$, $x_{n+1}=b$. Again, $\eta_j$'s are linearly independent
by considering their supports. And since $\int^{x_j}_{a} uv\,dx =0$,
$1\le j \le n$ we can proceed as before.
\end{proof}

Fix functions $u,v$. Pick $a_m\downarrow a$,
$b_m\uparrow b$ and set $I_m=(a_m,b_m)$. Define $\ti{H}_m:
\db(\ti{H}_m) \to L^2(I_m,\C^2)$, $f \mapsto \tau f$ 
with
\begin{equation}
\db(\tilde{H}_m) = \{ f \in L^2(I_m,\C^2) | \ba[t]{l} f \in
AC(I_m,\C^2), \, \tau f \in L^2(I_m,\C^2),\\ W_{a_m}(u,f) =
W_{b_m}(v,f) =0 \}.
\ea
\end{equation}
Consider $H_m=\alpha \id\oplus \ti{H}_m \oplus\alpha\id$ on
$L^2(I,\C^2)=L^2((a,a_m),\C^2)\oplus L^2(I_m,\C^2)\oplus L^2((b_m,
b),\C^2)$, where $\alpha$ is a fixed real constant. Then we have
the following standard result (\cite{wdl}, Chapter~16, \cite{wd2},
Section~1, and \cite{gst}, Section~5). 

\begin{lemma} \label{strconh} 
Suppose that either $H$ is limit point at $a$ or
that $u = u_-(\lam_0)$ for some $\lam_0$ and similarly, that either
$H$ is limit point at $b$ or $v=u_+(\lam_1)$ for some $\lam_1$.
Then $H_m$ converges to $H$ in strong resolvent sense as $m\to\infty$
and hence
\begin{equation} \label{strecon}
\dim\Ran\,P_{(\lam_0,\lam_1)}(H) \le \liminf \,\dim\Ran\,
P_{(\lam_0,\lam_1)}(H_m).
\end{equation}
\end{lemma}

Now we are ready to prove

\begin{theorem}
If $u=u_\mp(\lam_0)$ and $v=u_\pm (\lam_1)$, then
\begin{equation} \label{ngedim}
\dim\Ran\, P_{(\lam_0,\lam_1)}(H) \le \#(u,v).
\end{equation}
If $H$ is limit point at $b$ (resp. $a$) we can replace $u_-(\lam_j)$
(resp. $u_+(\lam_j)$) by an arbitrary solution of $\tau u= \lam_j u$.
\end{theorem}

\begin{proof}
We can assume $\#(u,v)<\infty$ (otherwise there is nothing to prove).
Pick $a_m\downarrow a, b_m\uparrow b$. Let $H_m$ be given as in Lemma
\ref{strconh} with $\alpha\notin [\lam_0, \lam_1]$. If $m$ is so large, that
all zeros of $W(u,v)$ are in $(a_m,b_m)$, Theorem \ref{thmreg} implies
$\#(u,v) = \dim\Ran\, P_{(\lam_0,\lam_1)}(\ti{H}_m) =
\dim\Ran\, P_{(\lam_0,\lam_1)}(H_m)$ since $\alpha\notin [\lam_0,\lam_1]$.
Thus by Lemma~\ref{strconh}, (\ref{ngedim}) holds as was to be proven.
\end{proof}

Combining the last two theorems we get:

\begin{theorem} \label{mainthm}
Let $\lam_0<\lam_1$, then
\begin{equation}
\dim\Ran\, P_{(\lam_0,\lam_1)}(H) = \#(u_-(\lam_0),u_+(\lam_1)) =
\#(u_+(\lam_0),u_-(\lam_1)),
\end{equation}
where $\#(u,v)$ denotes the number of zeros of $W(u,v)$ inside $(a,b)$.
The result still holds for  $u=u_-(\lam_0)$, $v=u_-(\lam_1)$ (resp. 
$u=u_+(\lam_0)$, $v=u_+(\lam_1)$) if $H$ is limit point at $b$
(resp. $a$).
\end{theorem}

\begin{remark}
The limit point assumption in the case $u=u_\mp(\lam_0)$,
$v=u_\mp(\lam_1)$ is clearly crucial, since the Wronskian contains no
information about the boundary condition at $a$ respectively $b$ in this case.
\end{remark}

Finally we state

\begin{theorem}
Let $\lam_0 \neq \lam_1$. Let $\tau u =\lam_0 u$, $\tau v =
\lam_1 v$, and $\tau \tilde{v} = \lam_1 \tilde{v}$ with $v$ independent of
$\tilde{v}$. Then the zeros of $W(u,v)$ interlace the zeros of $W(u,\tilde{v})$ (in
the sense that there is exactly one zero of one function in between two zeros of
the other). In particular, $|\#(u,v)-\#(u,\tilde{v})| \leq 1$.
\end{theorem}

\begin{proof}
The result is immediate from $0< \Delta_{v,\tilde{v}}(x) < \pi$ (for a suitable
normalization of $\Delta_{v,\tilde{v}}(x)$) which follows from constancy of
$W(v,\tilde{v})$.
\end{proof}

By applying this theorem twice, we conclude

\begin{theorem} \label{thmaesol}
Let $\lam_0 \neq \lam_1$. Let $u, \tilde{u}$ and $v, \tilde{v}$ be the linearly
independent solutions of $\tau u = \lam_0 u$ and $\tau v= \lam_1 v$, respectively.
Then
\begin{equation}
|\#(u, v)-\#(\tilde{u},\tilde{v})|\leq 2.
\end{equation}
\end{theorem}

Moreover, we infer the following useful result.

\begin{corollary}\label{coresssp}
Let $u,v$ satisfy $\tau u = \lam_0 u$, $\tau v = \lam_1 v$. Then
\begin{equation}
\#(u,v) < \infty \quad\Leftrightarrow\quad \dim\Ran\,
P_{(\lam_0,\lam_1)}(H) < \infty.
\end{equation}
\end{corollary}

\begin{proof}
Using the split up $H_{x_0,-}\oplus H_{x_0,+}$ reduces the problems to the case
with one regular endpoint. Thus the solutions $u_\pm(\lam)$ exist at least at one
end point. Using first Theorem~\ref{mainthm} and then Theorem~\ref{thmaesol}
finishes the proof.
\end{proof}

\begin{remark} \label{remsusy}
(i). We remark that all results obtained thus far also hold for
the more general system
\begin{equation}
\tau = k(x)^{-1} \Big( \frac{1}{\I} \sig_2 (p(x) \frac{d}{dx} + \frac{d}{dx} p(x))
+ \phi(x) \Big),
\end{equation}
where $p \in AC_{loc}(I,(0,\infty))$ and $k$ is a symmetric positive definite matrix
with coefficients $k_{ij}\in L^1_{loc}(I,\R)$. The necessary modifications
are straightforward (see also \cite{wd2}, Section~5).\\
(ii).In the case of supersymmetric Dirac operators (i.e.,
$\phi_{11}=\phi_{22}=0$)
\begin{equation}
H= \left(\ba{cc} 0 & A^* \\ A & 0 \ea\right), \quad A=\frac{d}{dx} +
\phi_{12}(x)
\end{equation}
(note that $H$ and $-H$ are unitarily equivalent) we have
\begin{equation}
H^2 = \left(\ba{cc} H_1 & 0 \\ 0 & H_2 \ea\right), \quad H_1= A^*A, \: H_2 = A A^*.
\end{equation}
Moreover, $\tau u = \lam u$ implies $\tau_j u_j = -u_j'' + (\phi_{12}^2 - (-1)^j
\phi_{12}') u_j= \lam u_j$, $j=1,2$, where $\tau_j$ is the differential expression
corresponding to $H_j$. This says that all oscillation theoretic results for
supersymmetric Dirac operators follow immediately from the corresponding results
for (semi-bounded) one-dimensional Schr\"odinger operators.
\end{remark}


\section{Applications}

\label{secapp}


In our final section we want to apply our results to investigate the spectra of
short-range perturbations of periodic Dirac operators. Our objective is to prove
the analog of the Theorem by Rofe-Beketov \cite{rof} about the finiteness of the
number of eigenvalues in essential spectral gaps of the perturbed Hill operator.
The reader might find some results for the special case of perturbed constant
operators in \cite{hmrs}, \cite{fr} and for the general case in \cite{hs1},
\cite{hs2}.

We first recall some basic facts from the theory of periodic Dirac operators
(cf., e.g., \cite{bgu}, \cite{un}, \cite{wdl}, Chapter~12). Let $H_p$ be a Dirac
operator associated with periodic potential $\phi_p$, that is, $\phi_p(x+1) =
\phi_p(x)$,
$x\in I=\R$. The spectrum of $H_p$ is purely absolutely
continuous and consists of a countable number of gaps, that is,
\begin{equation}
\sig(H_p) = \bigcup_{j\in\Z} [E_{2j},E_{2j+1}]
\end{equation}
with $\cdots E_{2j} < E_{2j+1} \le E_{2j+2} < E_{2j+3}\cdots$.
Moreover, Floquet theory implies the existence of solutions $u_{p,\pm}(z,.)$ of
$\tau_p u = z u$, $z\in\C$ ($\tau_p$ the differential expression corresponding to
$H_p$) satisfying
\begin{equation}
u_{p,\pm}(z,x) = p_\pm(z,x) m(z)^{\pm x}, \quad p_\pm(z,x+1) = p_\pm(z,x),
\end{equation}
where $m(z) \in \C$ is called Floquet multiplier. $m(z)$ satisfies
$m(z)^2 =1$ for $z \in \{ E_j \}_{j\in\Z}$,
$|m(z)|=1$ for $z \in \sig(H_p)$, and $|m(z)|<1$ for $z \in
\C\bs\sig(H_p)$. (This says in particular, that $u_{p,\pm}(z,.)$ are
bounded for $z\in\sig(H_p)$ and linearly independent for $z \in \C\bs\{ E_j
\}_{j\in\Z}$.)

As anticipated, we will study perturbations $H$ of $H_p$ associated with
potential satisfying $\phi(x) \to \phi_p(x)$ as $|x| \to
\infty$. Both $H$ and $H_p$ are limit point (cf.\ \cite{wd3}, Satz~5.1) and hence
give rise to a unique self adjoint operator when defined maximally. Using this
notation our theorem reads:

\begin{theorem} \label{thmappl}
Suppose $\phi_p$ is a given periodic potential and $H_p$ is the
corresponding Dirac operator. Let $H$ be a perturbation of $H_p$ such that
\begin{equation} \label{decay}
\int_\R (1+|x|) |\phi(x) - \phi_p(x)| dx < \infty.
\end{equation}
Then we have $\sig_{ess}(H)=\sig(H_p)$, the point spectrum of $H$ is confined to
the spectral gaps of $H_p$, that is, $\sig_p(H) \subset
\R\bs\sig(H_p)$ and finite in each gap. Furthermore, the essential spectrum of
$H_p$ is purely absolutely continuous.
\end{theorem}

\begin{proof}
Using (\ref{resolv}) plus $|u_{p,\pm}(z,x)| \le C_\pm |m(z)|^{\pm x}$ shows that
$H$ is relatively compact with respect to $H_p$, implying
$\sig_{ess}(H)=\sig_{ess}(H_p)$. To prove the remaining claims it suffices to show
the existence of solutions $u_\pm(\lam,.)$ of $\tau u = \lam u$ for $\lam \in
\sig(H_p)$ (continuous w.r.t. $\lam$) satisfying
\begin{equation} \label{asupm}
\lim_{x\to\pm\infty} |u_\pm(\lam,x) - u_{p,\pm}(\lam,x)| =0.
\end{equation}
In fact, for $\lam \in \sig(H_p)$ there exists at least one bounded solution
which is not square integrable and hence there are no eigenvalues in the essential
spectrum of $H$ (since the Wronskian of a bounded and a square integrable
solution must vanish). Invoking Theorem XIII.20 of \cite{rs4} shows that the essential
spectrum of $H$ is purely absolutely continuous. Moreover, since $W_x(u_{p,-}(E_{2j-1}),
u_{p,+}(E_{2j}))$ has no zeros, we infer that
$W_x(u_-(E_{2j-1}), u_+(E_{2j}))$ has only finitely many zeros. Thus by
Corollary~\ref{coresssp} there are only finitely many eigenvalues in each gap.
It remains to show (\ref{asupm}). Suppose $u_+(\lam,.)$, $\lam \in\sig(H_p)$
satisfies 
\begin{equation} \label{voltseq}
u_\pm(\lam,x) = u_{p,\pm}(\lam,x) - \I \sig_2
\int_{\pm\infty}^x U_p(\lam,x,y) (\phi(y)-\phi_p(y)) u_\pm(\lam,y) dy,
\end{equation}
where $U_p(\lam,.,y)$ is the fundamental matrix solution of of $\tau_p u = \lam u$
satisfying the initial conditions $U_p(\lam,y,y)=\id$.
Then $u_\pm(\lam,.)$ fulfills $\tau u = \lam u$ and (\ref{asupm}). Existence of a
solution of (\ref{voltseq}) follows upon applying a standard iteration argument
(compare also \cite{hs1} and \cite{ti} in the special case $\phi_p=0$)
using
\begin{equation}
|U_p(\lam,x,y)| \le C (1 + |x-y|), \quad \lam\in\sig(H_p),\, C>0.
\end{equation}
\end{proof}

Clearly, there are several other strategies to prove Theorem~\ref{thmappl}.
The proof given here has the advantage of being rather short and
transparent. In addition, the idea of proof applies to much general scattering
situations (where $H_p$ is not necessarily periodic) as long as sufficient
information about the spectrum of $H_p$ and the asymptotic behavior of (weak)
solutions of $H_p$ and $H$ is available.

\begin{remark}
The fact that the essential spectrum of $H$ is purely absolutely continuous
has first been proven by \cite{hs2} under the weaker assumption $\int_\R |\phi(x) -
\phi_p(x)| dx < \infty$. Since (\ref{decay}) is only needed to ensure existence
of $u_\pm(\lam,x)$ for $\lam$ at the boundary of $\sig(H_p)$
(for $\lam$ in the interior of $\sig(H_p)$ we have $|U_p(\lam,x,y)| \le C$)
the weaker assumption above suffices) our proof also covers this situation.
However, the following example
\begin{equation}
\phi(x) = \phi_p(x) + \left(\ba{cc} \frac{x^2-1}{(x^2+1)^2} & 0 \\ 0&0\ea\right),
\quad \phi_p(x) = \left(\ba{cc} 1 & 0 \\ 0 & -1\ea\right)
\end{equation}
shows that (\ref{decay}) cannot be replace by $\int_\R (1+|x|^\eps) |\phi(x) - \phi_p(x)|
dx < \infty$, $\eps<1$. Indeed, $H$ has an eigenvalue $1 \in \sig(H_p) =
(-\infty,-1]\cup[1,\infty)$ with corresponding eigenfunction
\begin{equation}
u(1,x) = \frac{1}{(x^2+1)^2} \left(\ba{c} x^2+1 \\ -x \ea\right).
\end{equation}
\end{remark}



\section*{Acknowledgments}

I thank S.~Timischl for discussions and F. Gesztesy, B.~Thaller, and K.~Unterkofler for
hints with respect to the literature.



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\end{document}