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\begin{document}
\title{Accumulation of Discrete Eigenvalues of the Radial Dirac Operator}
\author{Marcel Griesemer}
\address{Mathematik\\Universit\"at Regensburg\\D-93040 Regensburg}
\email{Marcel.Griesemer@mathematik.uni-regensburg.de}
\author{Joseph Lutgen}
\address{Mathematik\\Universit\"at Regensburg\\D-93040 Regensburg}
\email{Joseph.Lutgen@mathematik.uni-regensburg.de}
\begin{abstract}
For bounded potentials which behave like \(-cx^{-\gamma}\) at
infinity we investigate whether discrete eigenvalues of the radial
Dirac operator $H_{\kappa}$ accumulate at +1 or not. It is well
known that $\gamma=2$ is the critical exponent. We show that
\(c=1/8+\kappa(\kappa+1)/2\) is the critical coupling constant in
the case $\gamma=2$. Our approach is to transform the radial Dirac
equation into a Sturm-Liouville equation nonlinear in the spectral
parameter and to apply a new, general result on accumulation of
eigenvalues of such equations.
\end{abstract}
\maketitle
\section{Introduction}
The Dirac operator on $\R^3$ with (bounded) potential behaving like
\(-c|\mathbf{x}|^{-\gamma} \ (\gamma>0)\) at infinity has finitely
many discrete eigenvalues if \(\gamma>2\) and infinitely many for
\(\gamma<2\) \cite{Kurbenin1969}. In the case of a spherically
symmetric potential this is still true for the Dirac operator
restricted to a subspace of definite angular momentum and definite
parity (radial Dirac operator) \cite{Kurbenin1969}. We rederive this
result for the radial Dirac operator by a new method which allows us
to determine the critical coupling constant in the case \(\gamma=2\).
Our strategy is to transform the radial Dirac eigenvalue equations
(see below) into a Sturm-Liouville equation nonlinear in the spectral
parameter to which we then apply a new, general theorem
(Theorem~\ref{maintheorem}, Lutgen) on accumulation of eigenvalues of
such equations. First and foremost our result is a nice application of
this general theorem and a demonstration of its strength.
The eigenvalue equation for the Dirac operator with spherically
symmetric potential $V$ is equivalent to the system of first
order differential equations
\begin{align}\label{d1}
-F'+\frac{\kappa}{x}F+(1+V-\lambda)G &= 0\\ \label{d2}
G'+\frac{\kappa}{x}G+(-1+V-\lambda)F&= 0
\end{align}
on $\R_{+}$, to be solved for \(F,G\in L^2(\R_{+}),\ \lambda\in \R\)
and \(\kappa\in\Z\backslash\{0\}\) with boundary conditions
\(F(0)=0=G(0)\). $\kappa$ parameterizes both the total angular
momentum and the parity. Let us assume the potential $V$ is
bounded, nonpositive, and \(-V(x)x^{\gamma}\rightarrow c\) as $x=|\mathbf{x}|$
goes to infinity, the coupling constant $c$ being positive. Under
further assumptions on $V'$ to be discussed below, we show that
accumulation of eigenvalues at $+1\ (=mc^2)$ from below occurs whenever
\begin{equation*}
\gamma <2,\makebox[5em]{or}\gamma=2\makebox[4em]{and}c>\frac{1}{8}+
\frac{\kappa(\kappa+1)}{2},
\end{equation*}
while for
\begin{equation*}
\gamma >2,\makebox[5em]{or}\gamma=2\makebox[4em]{and}c\leq\frac{1}{8}+
\frac{\kappa(\kappa+1)}{2}
\end{equation*}
there is no accumulation. An analogous theorem (without conditions on
$V'$) with the same critical exponent and the same critical coupling
constant holds true for the radial Schr\"odinger equation which
emerges in the nonrelativistic (particle) limit of the Dirac
equations (\ref{d1}) and (\ref{d2}). This should not come as a
surprise, for if the potential is bounded, all but finitely many of
the bound states of the Dirac operator are localized far away from
zero where the potential is small by assumption; hence, the kinetic
energy will also be small which means that relativity is of no
importance for the question of whether eigenvalues accumulate at $1$.
This is true as long as relativistic corrections are not large enough
to generate infinitely many bound states localized near the origin, a
requirement which explains our condition on $V'$ for finite $x$: at
least for large values of $|\kappa|$ it says that the spin-orbit
interaction, which is one of the leading relativistic corrections to
the Schr\"odinger equation, is in some sense smaller than the kinetic
energy due to the orbital angular momentum.
The general theorem from which we derive our result is a statement
about accumulation of eigenvalues for a Sturm-Liouville eigenvalue
problem nonlinear in the eigenvalue parameter. This is exactly the
form of our problem once we have eliminated $F$ by means of equation
(\ref{d2}). Roughly speaking, the theorem says that the family of
Sturm-Liouville equations we get has infinitely many $L^2$-solutions
if and only if the equation for $\lambda=1$ is oscillatory (i.e.,
every solution has infinitely many zeros). It thus generalizes folk
wisdom from the theory of classical (i.e., linear in $\lambda$)
Sturm-Liouville eigenvalue problems. To determine whether the equation
for $\lambda=1$ is oscillatory a standard result (Kneser's criterion)
is applied.
Infinitude of the discrete spectrum for $\gamma<2$ and finiteness for
$\gamma>2$ was proved by Kurbenin \cite{Kurbenin1969}. For the
critical case $\gamma=2$ and the special potential
\(V(x)=-\lambda(1+x^2)^{-1}\) Klaus showed finiteness for
\(\lambda<1/8\) and infinitude for \(\lambda>1/8\) \cite{Klaus1980}.
This is the only previous result we are aware of which identifies a
critical coupling constant. There is a variety of results in the case
of finite discrete spectrum usually assuming \(\gamma<2\) or \(V\in
L^3\cap L^{3/2}\), which excludes \(\gamma=2\)
\cite{Tamura1976,Klaus1980,BirmanLaptev1994,Cancelieretal1996}. Most
interesting in the present context is one of Birman and Laptev, which
compares the number of discrete eigenvalues of the Pauli and Dirac
operators for large coupling constants \cite{BirmanLaptev1994}. It is
found that the asymptotic numbers coincide, which nicely complements
the picture given in the present paper (see
\cite[Theorem 4]{BirmanLaptev1994}, the factor $2^{q+1}$ there is absent
when the comparison operator in \cite[eqn. 12]{BirmanLaptev1994} is
chosen to describe the nonrelativistic particle limit).\\
This paper is organized as follows. In Section 2 we recall the angular
momentum decomposition of the Dirac operator and in Section 3 we
eliminate $F$ from equation~(\ref{d2}) to obtain a SL-equation
equivalent to the system~(\ref{d1},\ref{d2}). The general theorem of
Lutgen (Theorem~\ref{maintheorem}) and our main result
(Theorem~\ref{mainresult}) are contained in Section 4. In this section
we also speculate about a more general relationship between the
spectra of the radial Dirac and the Schr\"odinger operator. In the appendix we
give a second proof of the accumulation result under different
hypotheses.
\section{The Radial Dirac Operator}
This section serves to recall, without proofs, the angular momentum
decomposition of the Dirac operator. More detailed rigorous
expositions can be found in \cite{Thaller1992,Rejto1971,Weidmann1987}.
Consider a Dirac-electron subject to an external spherically symmetric
potential $V$. Due to the $O(3)$-symmetry of this system, the total
angular momentum and the parity of an electronic state are conserved
quantities. This means that the Dirac operator $H$ is reduced by the
subspaces of definite angular momentum and definite parity. On such a
subspace it is unitarily equivalent to a 2 by 2 matrix operator
$H_{\kappa}$ on \(L^2(\R_+)\otimes\C^2\). In the following this is
spelled out in more mathematical terms.
The three-dimensional Dirac operator with
potential \(V:\R^3\rightarrow\R\) reads
\begin{equation*}
H = -i\boldsymbol{\alpha}\cdot\boldsymbol{\nabla}+\beta +
V\hspace{2em}\hbox{on}\ \H=L^2(\R^3)\otimes\C^4
\end{equation*}
where
\(\boldsymbol{\alpha}\cdot\boldsymbol{\nabla} = \sum_{i=1}^{3}\alpha_i
\frac{\partial}{\partial x_i}\)
and in the standard representation
\begin{equation*}
\alpha_i = \begin{pmatrix}0&\sigma_i\\\sigma_i&0\end{pmatrix},\hspace{2em}
\beta =
\begin{pmatrix}1&0\\0&-1 \end{pmatrix},
\end{equation*}
with the Pauli matrices $\sigma_i$. In our units \(\hbar=m=c=1\). Assume
for simplicity that $V$ is bounded (and measurable). The Hamiltonian $H$ is thus
self-adjoint on \(H^1(\R^3)\otimes\C^4\) \cite{Thaller1992}.
The total angular momentum is represented in $\H$ by the operator
(actually the triple of operators) \(\mathbf{J}=(J_1,J_2,J_3)\) which
is the sum
\(\mathbf{J}=\mathbf{L}\otimes\mathbf{1}+\mathbf{1}\otimes\mathbf{S}\)
of the operators representing the orbital angular momentum
\(\mathbf{L}=-i\mathbf{x}\times\boldsymbol{\nabla}\) and the spin
\(\mathbf{S}=\frac{1}{2} \bigl(\begin{smallmatrix}\boldsymbol{\sigma}&0\\
0&\boldsymbol{\sigma}\end{smallmatrix}\bigr)\). The space reflection
\(\mathbf{x}\rightarrow-\mathbf{x}\) is represented by the parity
operator $P$ defined by
\(P\psi(\mathbf{x})=\beta\psi(-\mathbf{x})\). Since
\(\mathbf{J}^2=J_1^2+J_2^2+J_3^2,\ J_3\) and $P$ are mutually
commuting and have pure point spectrum there is a complete family of
simultaneous eigenspaces \(\H=\bigoplus \H_{j,m,\pi}\). The
eigenvalues on $\H_{j,m,\pi}$ are given by
\begin{align*}
\mathbf{J}^2 &= j(j+1)&
j\in&\left\{1/2,3/2,5/2,\ldots\right\},\\
J_3 &=m& m\in&\{-j,\ldots,j\},\\
P &= \pi& \pi\in&\{\pm 1\}.
\end{align*}
To achieve a economic notation we now introduce a new parameter \(\kappa\in\Z\backslash
\{0\}\) which replaces $j$ and $\pi$. This is usually done in such a way
that \(j=|\kappa|-\frac{1}{2}\) and \(\pi=\pm(-1)^{\kappa}\)
if \(\pm\kappa>0\). With \(\H_{\kappa,m}=\H_{j,m,\pi}\) one then has
\begin{equation*}
\H = \bigoplus_{\kappa\in\Z\backslash\{0\}}\,
\bigoplus_{m=-(|\kappa|-1/2)}^{|\kappa|-1/2}\H_{\kappa,m}
\end{equation*}
and \(\H_{\kappa,m}\) is of the form
\begin{equation}\label{rad1}
\H_{\kappa,m} =
\left\{\left.\begin{pmatrix}\frac{iG}{x}\Omega_{\kappa,m}\\
\frac{F}{x}\Omega_{-\kappa,m}\end{pmatrix}
\right|G,F\in L^{2}(\R_{+})\right\}
\end{equation}
where \(x=|\mathbf{x}|\) and \(\Omega_{\kappa,m}\) are $\C^2$-valued functions
on the unit sphere whose components are proportional to spherical harmonics
\(Y_{l,m}\), their index $l$ being connected with $j$ and $\kappa$ by
\begin{equation}\label{rad2}
\kappa = \pm\left(j+\frac{1}{2}\right)\quad \mbox{if} \quad
l=j\pm\frac{1}{2}.
\end{equation}
This relation is usually taken as the definition of $\kappa$ (see
\cite{BjorkenDrell1964}); it says that the sign of $\kappa$ indicates
whether spin and orbital angular momentum of the upper component are
``parallel'' or ``anti-parallel''.
Now let us assume that $V$ is spherically symmetric. Then $H$ commutes
as well with \(\mathbf{J}^2,\ J_3\) and $P$, and is thus reduced by
the spaces \(\H_{\kappa,m}\). Its effect on the functions $G$ and $F$
in (\ref{rad1}) is described by the radial Dirac operator
\begin{equation*}
H_{\kappa}:= \begin{pmatrix}1+V & -\frac{d}{dx}+\frac{\kappa}{x}\\
\frac{d}{dx}+\frac{\kappa}{x} & -1+V\end{pmatrix},
\hspace{2em} D(H_{\kappa})=H_0^1(\R_{+})\otimes\C^2
\end{equation*}
in \(L^2(\R_{+})\otimes\C^2\), more precisely, the operator
\(H\restricted \H_{\kappa,m}\) is unitarily equivalent to
\(H_{\kappa}\), the unitary map being \((G,F)\in
L^2(\R_{+},\C^2)\rightarrow
(\frac{iG}{x}\Omega_{\kappa,m},\frac{F}{x}\Omega_{-\kappa,m})
\in\H_{\kappa,m}\).
The domain of $H_{\kappa}$ is the inverse image of
\(D(H)\cap\H_{\kappa,m}\) under this map. It is equal to
\(H_0^1(\R_{+})\otimes\C^2\)
because \(fx^{-1}Y_{l,m}\in H^1(\R^3)\) if and only if $f$ belongs to
\(H_0^1(\R_{+})\). The spectrum of $H_{\kappa}$ is qualitatively the
same as the spectrum of $H$: if \(V\equiv 0\) then
\(\sigma(H_{\kappa})=(-\infty,-1]\cup[1,\infty)=\sigma(H)\) and otherwise
\begin{equation*}
\sigma_{\mathrm ess}(H_{\kappa})=(-\infty,-1]
\cup[1,\infty)=\sigma_{\mathrm ess}(H)
\end{equation*}
if at least \(V(x)\rightarrow 0\) as \(x\rightarrow\infty\)
\cite{Weidmann1987}. If in addition \(V\leq 0\), then $-1$ is not an
accumulation point of discrete eigenvalues \cite[Thm.
10.37]{Weidmann1980}.
\section{A Sturm-Liouville Equation equivalent to the Dirac Equation}
The purpose of this section is to prove the equivalence of the radial
Dirac equations $(H_\kappa - \lambda)\psi=0$ to the
$\lambda$-dependent Sturm-Liouville equation obtained by eliminating
$F$. Formal equivalence is easy to see. The hard part is to show that
the solutions belong to the prescribed spaces.
To begin with we recall that \(f\in H_0^1(\R_{+})\) if and only if $f$
is absolutely continuous on $[0,\infty)$, \(f,f'\in L^2(\R_{+})\), and
$f(0)=0$ (we define absolute continuity locally). Suppose \(\eps :=
2-\sup_{x>0} V(x)>0\), $V$ is absolutely continuous and
\(\lambda\in(1-\eps,1)\). By the definition of $H_{\kappa}$, $\lambda$ is
an eigenvalue if and only if the equations (\ref{d1},\ref{d2}) are
satisfied for some \(F,G\in H_0^1(\R_{+})\), which may be assumed
real-valued. By~(\ref{d2}), $G'$, and hence \((\lambda+1-V)^{-1}G'\),
is absolutely continuous. Eliminating $F$ with the help of~(\ref{d2})
we see that $G$ solves the boundary value problem
\begin{equation}\label{d3}
-(r(x,\lambda)G'(x))'+p(x,\lambda)G(x)=0, \quad G(0)=0,
\end{equation}
where
\begin{align*}
r(x,\lambda) &= (1+\lambda-V(x))^{-1}\\
p(x,\lambda) &= (1+\lambda-V(x))^{-1}\frac{\kappa(\kappa+1)}{x^2}+(1+V(x)
-\lambda)
-(1+\lambda-V(x))^{-2}V'(x)\frac{\kappa}{x}.
\end{align*}
Under some additional assumptions on $V$ the converse is true as well:
\begin{theorem}\label{equivalent}
Suppose \(V\in L^{\infty}(\R^3;\R)\) is absolutely continuous and
$\eps := 2-\sup_{x>0} V(x)$ is positive. Furthermore, suppose
\(xV'(x)\rightarrow 0\) as \(x\rightarrow 0\), $V'(x)/x \rightarrow
0$ as \(x\rightarrow \infty\), $\liminf_{x \rightarrow
\infty}V(x)\geq 0$ and \(\kappa\neq -1\). Then
\(\lambda\in(1-\eps,1)\) is an eigenvalue of $H_{\kappa}$ if and
only if~(\ref{d3}) has an $L^2$-solution $G$ which belongs to
\(C^1(\R_{+})\) and has absolutely continuous derivative. In this
case \((G,F)\) with
\begin{equation*}
F := (\lambda+1-V)^{-1}\left(G'+\frac{\kappa}{x}G\right)
\end{equation*}
is an eigenvector of $H_{\kappa}$ belonging to $\lambda$.
\end{theorem}
\noindent
{\em Remark.} Analogously, if $V(x)\geq -2+\eps$ and $\kappa\neq 1$, then
$\lambda\in (-1,-1+\eps)$ is an eigenvalue of $H_{\kappa}$ if and only
if there exists a solution $F\in L^2$ of the Sturm-Liouville equation
obtained by eliminating $G$.
\begin{proof}
Suppose $G$ is an $L^2$-solution of~(\ref{d3}), \(G\in C^1(\R_{+})\)
and $G'$ is absolutely continuous. If
\((\lambda+1-V)^{-1}\left(G'+\frac{\kappa}{x}G\right)\) is substituted for
$F$ in the Dirac equations (\ref{d1},\ref{d2}), then equation~(\ref{d2}) is
trivially satisfied and~(\ref{d1}) follows from the equation
in~(\ref{d3}). It remains to prove that $G$ and $F$ belong to
\(H_0^1(\R_{+})\). To do this we show that
\[\begin{array}{rl}
(i) & G'\ \mbox{is square integrable at}\ +\infty.\\
(ii) & \mbox{There exists a $k>1$ such that}\ |G(x)|\leq \const\
x^{k+1/2}\ \mbox{for small $x$}.\\
(iii) & G'(x)/x\ \mbox{is square integrable at 0 and}\
G'(x)\rightarrow 0\ \mbox{as}\ x\rightarrow 0.
\end{array}\]
It then follows that \(G', G/x\in L^2(\R_{+})\) and hence that
\(F\in L^2(\R_{+})\), that \(F(0)=0\), and that \(F'\in L^2(\R_{+})\)
because \(G'/x, G/x^2\in L^2(\R_{+})\) and $F$ solves (\ref{d1}). Thus
\(F,G\in H_0^1(\R_{+})\).
Henceforth the second argument of $r(x,\lambda)$ and
$p(x,\lambda)$ will be fixed, and we therefore drop it. The conditions on
$V$ ensure that there exist constants \(x_0,x_1,r_0,R\in
\R_{+}\), depending on $\lambda$, such that
\begin{align}\label{preq1}
0 < r_0 &\leq r(x) \leq R& x&\in\R\\ \label{preq2}
p(x)& \geq 0& x&\geq x_1\\ \label{preq3}
p(x)& \geq r(x)\frac{k^2}{x^2}& x&\leq x_0
\end{align}
where \(k^2=\kappa(\kappa+1)-1/2>1\). This is the only place in this
proof where we need \(\kappa\neq -1\). In the following the function
\(H(x)=rG'G(x)\) will be of importance. Its derivative is by (\ref{d3})
\begin{equation}\label{preq4}
H'(x) = pG^2(x) + r(G')^2(x).
\end{equation}
\noindent
{\em Proof of (i).} If $x\geq x_1$, then $H'(x)\geq0$, i.e., $H(x)$ is
monotonically increasing. There are thus two cases: Either $H(x_2)>0$
for some $x_2\geq x_1$ (case 1), or $H(x)\leq0$ for all $x\geq x_1$
(case 2). In case 1
\begin{equation*}
0< H(x_2)\leq H(x)=\frac{1}{2}r(G^2)'(x)\leq
\frac{1}{2}R(G^2)'(x),\hspace{2em}x\geq x_2
\end{equation*}
which implies
\begin{equation*}
\frac{R}{2}[G^2(x)-G^2(x_2)]\geq(x-x_2)H(x_2)\rightarrow
\infty\hspace{2em} (x\rightarrow\infty)
\end{equation*}
in contradiction to \(G\in L^2(\R_{+})\). Thus we must have case 2; hence,
\begin{equation*}
r_0\int_{x_1}^{x}(G')^2dy \leq \int_{x_1}^{x}r(G')^2dy \leq
\int_{x_1}^{x}H'(y)dy\leq-H(x_1)
\end{equation*}
for all $x>x_1$. This proves (i).
\noindent
{\em Proof of (ii).} By (\ref{preq3}) and (\ref{preq4}) we have for \(x\leq
x_0\)
\begin{equation}\label{preq5}
H'(x)\geq r\left(\frac{k}{x}\right)^2G^2+ r(G')^2\geq 2r\frac{k}{x}|GG'|
=2\frac{k}{x}|H(x)|.
\end{equation}
This implies
\begin{equation}\label{preq6}
\frac{d}{dx}\left(H(x)x^{\pm 2k}\right)\geq 0,\hspace{2em}x\leq x_0,
\end{equation}
and \(H'(x)\geq 0\) for \(x\leq x_0\). There are again two possible
cases: either $H(x_2)<0$ for some $x_2\in (0,x_0]$ (case 1), or
$H(x)\geq 0$ for all $x\in (0,x_0]$ (case 2). In case 1
\begin{equation*}
0> H(x_2)\geq H(x)=\frac{1}{2}r(G^2)'(x)\geq
\frac{1}{2}R(G^2)'(x)\hspace{2em}x\leq x_2
\end{equation*}
which, using $G(0)=0$, leads to the contradiction
\begin{equation*}
\frac{R}{2}G^2(x_2)=\frac{R}{2}\int_0^{x_2}(G^2)'dy\leq x_2H(x_2) <0.
\end{equation*}
Therefore, case 2 is realized and \(H(0)=\lim_{x\rightarrow
0}H(x)\geq 0\). By (\ref{preq6}), \(H(x)x^{-2k}\leq
H(x_0)x_0^{-2k}=:c_0\), that is
\begin{equation}\label{preq7}
H(x) \leq c_0 x^{2k},\hspace{2em}x\leq x_0,
\end{equation}
and by (\ref{preq5})
\begin{equation*}
\left|\frac{d}{dx}G^2(x)\right|=\frac{2}{r}|H(x)|\leq \frac{x}{kr}H'(x).
\end{equation*}
This with $G(0)=0$ leads to
\begin{multline*}
G^2(x) \leq \int_0^x\left|\frac{d}{dy}G^2(y)\right|dy \leq
\frac{1}{r_0}\int_0^x\frac{y}{k}H'(y)dy\\
\leq \frac{x}{kr_0}[H(x)-H(0)]\leq \frac{c_0}{kr_0}x^{2k+1}\hspace{2em}x
\leq x_0
\end{multline*}
which proves (ii).
\noindent
{\em Proof of (iii).} Recall from above that $H(x)\geq 0$
for all $x\in(0,x_0]$. By (\ref{preq3}) and (\ref{preq4}),
\(rG'(x)^2/x^2\leq H'(x)/x^2\) for $x\in(0,x_0]$. Now integrate by
parts and use (\ref{preq7}) to see that
\begin{equation*}
\int_{\eps}^{x_0}\frac{H'(y)}{y^2}dy\leq
\frac{H(x_0)}{x_0^2}+2c_0\int_{\eps}^{x_0}x^{2k-3}dx\leq\const
\end{equation*}
where the constant is independent of $\eps>0$ because
\(2k-3>-1\). This shows that \(G'(x)/x\) is square integrable near 0.
Finally, we prove that \(G'(x)\rightarrow 0\) as \(x\rightarrow 0\). For
\(x\leq x_0\) the function $G^2(x)$ is monotonically increasing
because \(G'(x)G(x)=H(x)/r(x)\geq 0\). We may assume $G(x)>0$
for \(x\in (0,x_0]\) (otherwise $-G(x)>0$ near $0$ or there is nothing
to prove). Then \((rG')'=pG\geq 0\) for \(x\leq x_0\), i.e., $rG'$ is
monotonically increasing near $0$. Furthermore, \(rG'(x)\geq 0\) near
$0$ because $G(x)>0$ and \(rG'G(x)=H(x)\geq 0\) near $0$. Hence, the
limit \(c_1=\lim_{x\rightarrow 0}rG'(x)\geq 0\) exists and \(G'(x)\geq
c_1/R\) near $0$. If $c_1>0$ this is not compatible with \(|G(x)|\leq
\const\ x^{k+1/2}\). Therefore, $c_1=0$ and \(G'(x)\rightarrow 0\) as
$x\rightarrow 0$.
\end{proof}
\section{The Main Result}
The main theorem, which is a special, simplified case of the results
in \cite{Lutgen1997,Lutgen1998}, treats accumulation/nonaccumulation of
eigenvalues for the problem
\begin{eqnarray}\label{SLE}
-(r(x;\lambda)f'(x))' + p(x;\lambda)f(x) &=& 0 \quad x \in
[a,\infty),\\
\label{SLBC}
\alpha (\lambda)f(a) + \beta (\lambda) f'(a) &=&0
\end{eqnarray}
\noindent where the spectral parameter $\lambda$ varies in an
interval
$\Lambda =(\mu,\nu]$ with $\nu$ finite (an eigenvalue is a value $\lambda_0$ such that
for $\lambda = \lambda_0$ eqn. (\ref{SLE}) has an $L^2$-solution
satisfying (\ref{SLBC})). The following conditions are assumed to
hold:\\
\noindent {\bf Assumptions}\\
\indent {\bf i)} $r,p:[a,\infty) \times \Lambda \rightarrow \R$ are
continuous and $r$ is positive.\\
\indent {\bf ii)} $\alpha, \beta:\Lambda \rightarrow \R$ are
continuous,
$\alpha(\lambda)^2 + \beta(\lambda)^2 \not = 0$ for all
$\lambda \in \Lambda$, and
$\beta$ is either never zero or is identically zero.\\
\indent {\bf iii)} There exist functions $\chi:(\mu,\nu) \rightarrow
[a,\infty)$ and $\eta:(\mu,\nu) \rightarrow (0,\infty)$, the first one
being
continuous, such that $p(x ;\lambda) \geq \eta(\lambda)$ for all
$(x;\lambda) \in [\chi(\lambda), \infty) \times (\mu,\nu)$.
\begin{theorem}\label{maintheorem} (Main Theorem)\\
\indent {\bf i)} If equation (\ref{SLE}) with $\lambda = \nu$ is
oscillatory on $[a,\infty)$, then $\nu$ is an
accumulation point of eigenvalues of (\ref{SLE}, \ref{SLBC})
from the left.\\
\indent {\bf ii)} If $r(a;\lambda)\alpha(\lambda)/\beta(\lambda)$ is
increasing, $r(x;\lambda)$ is decreasing, and $p(x;\lambda)$ is
strictly decreasing in $\lambda$ for each $x$, and if equation
(\ref{SLE}) with $\lambda = \nu$ is nonoscillatory on $[a,\infty)$,
then $\nu$ is not an accumulation point of eigenvalues of
(\ref{SLE}, \ref{SLBC}) from the left.
\end{theorem}
The question whether (\ref{SLE}) is oscillatory for $\lambda = \nu$
can be
settled by applying a generalization \cite{Lutgen1997,Lutgen1998} of Kneser's Criterion
\cite[Cor.XIII.7.37]{DS2}. Specifically, if for some $\varrho \in
\R$,
\[
\limsup_{x\rightarrow \infty}\frac{r(x;\nu)}{x^\varrho}< \infty,
\quad
\limsup_{x \rightarrow \infty}\frac{p(x;\nu)}{x^{\varrho -2}} <
-\frac{(1-\varrho)^2}{4}\limsup_{x \rightarrow \infty}\frac{r(x;\nu)}
{x^\varrho},
\]
then we have oscillation, whereas if
\[
\liminf_{x\rightarrow \infty}\frac{r(x;\nu)}{x^\varrho} >0, \quad
\liminf_{x \rightarrow \infty}\frac{p(x;\nu)}{x^{\varrho -2}} >
-\frac{(1-\varrho)^2}{4}\liminf_{x \rightarrow \infty}\frac{r(x;\nu)}
{x^\varrho},
\]
then we have nonoscillation.
\begin{theorem}\label{mainresult}(Main Result) Assume the
hypotheses in Theorem \ref{equivalent}, that \(V'\) is continuous, and that
\(V(x) \rightarrow 0\) as \(x\rightarrow \infty\).\\
\indent {\bf i)} If \(-\limsup_{x \rightarrow \infty}\left \{V(x)x^2
- \kappa V'(x)x/(2-V(x))^2 \right \} > 1/8 + \kappa(\kappa+1)/2,\) then
$+1$ is an accumulation point of eigenvalues of $H_\kappa$ in the
gap $(-1,1)$.\\
\indent {\bf ii)} Suppose there is a constant
$\delta <1$ such that $\kappa V'(x)x < \delta \left \{ \kappa(\kappa
+1) + 4x^2 \right \}$ for all $x>0$. If \(-\liminf_{x \rightarrow \infty}
\left \{V(x)x^2 - \kappa V'(x)x/(2-V(x))^2 \right \} < 1/8 +
\kappa(\kappa+1)/2,\) then $+1$ is not an accumulation point of eigenvalues
of $H_\kappa$ in the gap $(-1,1)$.
\end{theorem}
\indent
That a condition similar to the one in part i) of the Theorem still
implies accumulation when $\kappa = -1$ and $-20$. This occurs because
the ``only if'' part of Theorem 1 also holds for $\kappa = -1$, i.e.,
finiteness of the number of eigenvalues of (5) in the gap for $\kappa
= -1$ implies the same for the Dirac problem (1,2). Further, the
proof of Theorem 3 still works when $p(x;\lambda) \geq k/x$ for $x
\in (0,x_0], \lambda \in (0,1)$ with some constants $k,x_0 >0$ which
is indeed the case in (5) when $\kappa = -1$ and $V'(0)>0$.
Applying these results under the additional assumptions that
$-V(x)x^\gamma \rightarrow c$ and $V'(x)x \rightarrow 0$ as $x
\rightarrow \infty$ for some positive constants $c,\gamma$ proves the
comments in the introduction for $c$ not equal to the critical
coupling constant (the critical case is treated in the following remarks).\\
\noindent {\bf Remarks i)} It may happen that neither of the
inequalities
in Kneser's Criterion hold; nevertheless, it is still
possible to obtain results in such ``critical cases''. For simplicity
consider the critical case
\[
0< \lim_{x \rightarrow \infty}r(x;\nu) < \infty, \quad
\lim_{x \rightarrow \infty}p(x;\nu)x^2 = -\frac{1}{4}\lim_{x
\rightarrow
\infty}r(x;\nu).
\]
Employing a sort of ``refined'' Kneser's criterion in which we compare
the function $p(x;\nu)$ with one of the form $x^{-2} \{-1/8+\const
\cdot (\log x)^{-2}\}$ near $+\infty$ rather than with $\const \cdot
x^{-2}$ (in fact, there is a whole sequence of such refinements
\cite[pp.325,362]{Hartman1982}) we find that now
\[
\limsup_{x \rightarrow \infty} x^2(\log x)^2 \left \{ p(x;\nu) +
\frac{1}{8x^2} \right \} < -\frac{1}{4}\lim_{x \rightarrow
\infty}r(x;\nu)
\]
is sufficient for oscillation of (\ref{SLE}) with $\lambda = \nu$ and
\[
\liminf_{x \rightarrow \infty} x^2(\log x)^2 \left \{p(x;\nu) +
\frac{1}{8x^2} \right \} > -\frac{1}{4}\lim_{x \rightarrow
\infty}r(x;\nu)
\]
is sufficient for nonoscillation. These comments can be applied to
the
critical case in the introduction: If $x(\log x)^2 V'(x)\rightarrow
0$ and $(\log x)^2(c+V(x)x^2) \rightarrow 0$ as $x \rightarrow
\infty$ with $c$ equal to the critical coupling constant
$1/8+\kappa(\kappa+1)/2$ (which is valid, for example, when
$V(x)= -c/x^2$ near $+\infty$), then the limits on the left above both
equal $0$, and, consequently, $+1$ is not an accumulation point of
eigenvalues.\\
\indent {\bf ii)} The procedure above can easily be applied to the
radial Schr\"odinger equation
\[
\left \{ -\frac{1}{2}\frac{d^2}{dx^2} + \frac{\kappa(\kappa+1)}{2x^2} +V(x) -
\lambda
\right \} f = 0 \quad (x \in (0,\infty)), \quad f(0)=0
\]
as well. In particular, assuming $V$ is continuous, bounded, and
nonpositive on $(0,\infty)$, $-V(x)x^{\gamma} \rightarrow c$ as $x
\rightarrow \infty$ for some constants $c,\gamma >0$, and
$\kappa(\kappa +1) >0$, then $0$ is an accumulation point of negative
eigenvalues if $\gamma<2$ or if $\gamma =2$ and
$c>1/8+\kappa(\kappa+1)/2$, and $0$ is not an accumulation point if
$\gamma >2$ or if $\gamma =2$ and $c<1/8+\kappa(\kappa+1)/2$. In the
critical case $(c+V(x)x^2)(\log x)^2 \rightarrow 0$ as $x \rightarrow
\infty$ where $c=1/8+\kappa(\kappa+1)/2$ accumulation does not occur
at $0$. Here the monotonicity conditions in part ii) of the theorem
are trivial, whereas for the radial Dirac equations we had to make
additional assumptions on $V'$ to guarantee
that these hold.\\
\indent {\bf iii)} The radial Schr\"odinger equation above emerges as
a nonrelativistic limit of the Dirac equations (\ref{d1},\ref{d2})
(see \cite{BjorkenDrell1964,Hunziker1975}). This explains the
similarity of the results in Theorem \ref{mainresult} and remark ii)
(see also the introduction). The condition on $V'$ in
Theorem~\ref{mainresult} can be understood by considering the first
relativistic corrections to the (radial) Schr\"odinger equation. It is
due to the term \(-(1+\lambda-V(x))^{-2}V'(x)\kappa/x\) in equation
(\ref{d3}), which is close to \(-V'(x)\kappa/(4x)\) for $\lambda=1$
and $V(x)\simeq 0$. But this is exactly the term proportional to $V'$
due to the Darwin term and the spin-orbit interaction, the spin-orbit
interaction providing the larger part if $|\kappa|>1$. Our condition
on $V'$ compares \(-V'(x)\kappa/(4x)\) with
\(\kappa(\kappa+1)/(2x^2)\), i.e., the spin-orbit interaction with the
energy due to the orbital angular
momentum.\\
\bigskip
The similarity of the above results for the radial
Dirac equations and the radial Schr\"odinger equation and the
intuition discussed in the introduction concerning the reason for this
similarity suggest the following conjecture: \bigskip
\noindent
{\bf Conjecture.} {\em Suppose $V$ satisfies the hypotheses in
Theorem~\ref{equivalent} and in ii) of Theorem~\ref{mainresult},
\(V(x)\rightarrow 0\) and \(xV'(x)\rightarrow 0\) as \(x\rightarrow
\infty\) (i.e., \(V'(x)/x\) decays faster than $x^{-2}$). Then the
eigenvalues of the system (\ref{d1},\ref{d2}) accumulate at 1 from
below if and only if the eigenvalues of the radial Schr\"odinger
equation accumulate at 0 from below.} \bigskip
The ``if'' part of this conjecture holds true whenever the hypotheses
of Theorem~\ref{minimax} are satisfied. This follows from
equation~(\ref{sc1}) in the proof of this theorem.
\bigskip
{\it Sketch of the proof of the main theorem.} See \cite{Lutgen1997,Lutgen1998}
for the details. For each $\lambda \in \Lambda$ let $y_\infty$ be the
solution of (\ref{SLE}) determined by the initial conditions
$y_\infty(\chi(\lambda);\lambda) = 1=
r(\chi(\lambda);\lambda)y_\infty'(\chi(\lambda);\lambda)$. Then, as
functions of $x$, $y_\infty$ and $ry_\infty'$ are increasing on
$[\chi(\lambda),\infty)$, and a second solution is defined by
\[
w(x;\lambda):= c_\infty(\lambda)y_\infty(x;\lambda)
\int_{\textstyle x}^\infty
\frac{ds}{r(s;\lambda)y_\infty(s;\lambda)^2} \qquad
\chi(\lambda) \leq x < \infty,
\]
where $c_\infty(\lambda)$ is a constant such that
$w(\chi(\lambda);\lambda) =1$. Extended as solutions on $[a,\infty)$,
the $w(\cdot\,;\lambda)$ have the following properties:
$w(\cdot\,;\lambda)$ is in $L^2[a,\infty)$ and is positive and
decreasing on $[\chi(\lambda);\infty)$, $w$ and $\partial w/\partial
x$ are jointly continuous in $x$ and $\lambda$, and, if $r(x;\lambda)$
is decreasing and $p(x;\lambda)$ is strictly decreasing in $\lambda$
for fixed $x$, then $\lambda \mapsto
r(a;\lambda)w'(a;\lambda)/w(a;\lambda)$ is strictly increasing on
intervals where the denominator is nonzero. In view of assumption
iii), (\ref{SLE}) is in the limit point case at $\infty$ for each
$\lambda$, i.e., there is, up to constant multiple, at most one
$L^2$-solution; hence, $\lambda$ is an eigenvalue if and only if
$\alpha(\lambda)w(a;\lambda) + \beta(\lambda)w'(a;\lambda) = 0$.
Extending the coefficients $r,p$ and the solutions $w$ to the whole
real axis by setting $r(x;\lambda) := r(a;\lambda)$ and $p(x;\lambda)
:= p(a;\lambda) + (x-a)$ for $x \leq a$, it follows that
$w(\cdot\,;\lambda)$ has infinitely many zeros $\cdots
0$ such that $r(x;\lambda) \geq r_0$,
$p(x;\lambda)
\geq k/x^2$, and $\partial p/\partial \lambda <0$ for all $(x;\lambda)
\in
(0,x_0] \times (0,1]$. Note that $\partial r/ \partial \lambda <0$
holds
automatically for all $x$ and $\lambda$. For each $\lambda \in (0,1]$
let
$y_0(\cdot\,;\lambda)$ be the solution of the differential equation in
(\ref{d3}) determined
by the initial conditions $y_0(x_0;\lambda) = 1$, $r(x_0;\lambda)y_0'
(x_0;\lambda) = -1$. As functions of $x$, $y_0$ is decreasing and
$ry_0'$ is
increasing on $(0,x_0]$. For each $\lambda \in (0,1]$ define a new
solution by
\[
v(x;\lambda):=c_0(\lambda)y_0(x;\lambda) \int_0^{\textstyle x}
\frac{ds}
{r(s;\lambda)y_0(s;\lambda)^2} \qquad 0< x \leq x_0,
\]
where $c_0(\lambda)$ is a constant such that $v(x_0;\lambda)=1$. The
following properties hold (see \cite{Lutgen1997,Lutgen1998} for the details): $v$ is
positive and
increasing on $(0,x_0]$, $v(x;\lambda) \rightarrow 0$ ($x \rightarrow
0$),
$v$ and $\partial v/\partial x$ are jointly continuous in $x$ and
$\lambda$,
and $\lambda \mapsto r(x_0;\lambda)v'(x_0;\lambda)$ is decreasing on
$(0,1]$.
Since $\lim_{x \rightarrow 0}y_0(x;\lambda) >0$, a solution $G$ will
satisfy
the boundary condition in (\ref{d3}) if and only if it is a constant
multiple
of $v$, i.e., if and only if $v'(x_0;\lambda)G(x_0) - G'(x_0) =0$.
Thus, the
eigenvalue problem (\ref{d3}) is equivalent to one of the type
(\ref{SLE},
\ref{SLBC}) on $[x_0,\infty)$. The additional boundedness assumption
in ii)
is made to guarantee that $\partial p/\partial \lambda <0$ for all $x
\geq
x_0$ and $\lambda \in (0,1)$. Simply calculating the limits in the
comments before the Theorem with $\varrho =0$ and $\nu =1$ and applying
the Main Theorem gives the result.\qed\\
\section{Appendix}
Here we give a second proof of the accumulation result which works for
\(\kappa=-1\) as well. It is based on the minimax principle of
Siedentop and Griesemer \cite{GriesemerSiedentop1997} and the
accumulation result for the radial Schr\"odinger equation explained in
remark ii) after Theorem~\ref{mainresult}.
\begin{theorem}\label{minimax}
Suppose \(V\in L^{\infty}(\R_{+}),\ 0\geq V(x)>-2\) a.e. and
\(V(x)\rightarrow 0\) as \(x\rightarrow \infty\). Then discrete
eigenvalues of $H_{\kappa}$ accumulate at $+1$ whenever
\begin{equation}\label{sc}
-\limsup_{x\rightarrow\infty}x^2V(x) > \frac{1}{8}+
\frac{\kappa(\kappa+1)}{2}.
\end{equation}
\end{theorem}
\begin{proof}
Notice that $H_{\kappa}$ has the form of a perturbed supersymmetric
Dirac operator. In fact \(H_{\kappa}=Q+\beta + V\) where
\begin{equation*}
Q=\begin{pmatrix}0 & -\frac{d}{dx}+\frac{\kappa}{x}\\
\frac{d}{dx}+\frac{\kappa}{x} &
0\end{pmatrix} \quad\mbox{and}\quad \beta=\begin{pmatrix}1&0\\0&-1
\end{pmatrix}
\end{equation*}
with respect to the decomposition \(L^2(\R_{+})\otimes \C^2=
L^2(\R_{+})\oplus L^2(\R_{+})\), and \(Q\beta+\beta Q=0\) on
$D(Q)$. Hence by \cite[Theorem 8]{GriesemerSiedentop1997}
\begin{equation}\label{sc1}
\dim\ P_{(-1,1)}(\H_{\kappa})\H \geq \dim\ P_{(-\infty,0)}(h)L^2(\R_{+})
\end{equation}
where $h$ is the restriction of \(Q^2/2+V\) to the subspace of vectors
with vanishing lower component. A straight forward computation shows
that $h$ is the radial Schr\"odinger operator, i.e., on
$C_0^{\infty}(\R_{+})$, \(h=-1/2(d^2/dx^2) +
\frac{\kappa(\kappa+1)}{2x^2}+V\). Moreover, as we shall prove below,
any solution $f\in L^2(\R_{+})$ of
\begin{equation}\label{sc2}
-\frac{1}{2}f'' + \frac{\kappa(\kappa+1)}{2x^2}f +(V-\lambda)f=0,
\quad f(0)=0,
\end{equation}
\(f,f'\) being absolutely continuous on $[0,\infty)$, is an
eigenvector of $h$ with eigenvalue $\lambda$. But this equation has
infinitely many $L^2$-solutions with eigenvalues $\lambda$
accumulating at $0$ from below by (\ref{sc}) and remark ii) after
Theorem~\ref{mainresult}. Hence $h$ has infinitely many negative
eigenvalues. The theorem now follows from (\ref{sc1}) and the fact
that discrete eigenvalues of $H_{\kappa}$ cannot accumulate at $-1$.
Clearly the operator $Q$ is self-adjoint on \(D(H_{\kappa})=H_0^1(\R_{+})
\otimes\C^2\) which is
thus the form domain of $Q^2$. Hence \(H_0^1(\R_{+})\) is the form
domain of $h$ and (by Hardy's inequality \cite[Theorem 10.35]{Weidmann1980})
\(C_{0}(\R_{+})\) is a
form core of $h$. This and $h=h^*$ imply that $D(h) =
\{f\in H_0^1(\R_{+})|\; \exists f^* \in L^2 \mbox{ s.t. }\sprod{hg}{f}
=\sprod{g}{f^*}\; \forall g\in C_{0}(\R_{+})\}$. A solution $f$ of (\ref{sc2})
belongs to this set because it belongs
to \(H_0^1(\R_{+})\) (see the proof of Theorem~\ref{equivalent}) and because
\begin{equation}\label{sc3}
\sprod{hg}{f}=\sprod{g}{\lambda f}\quad\mbox{for all}\quad g\in C_{0}(\R_{+})
\end{equation}
(integrate twice by parts). But (\ref{sc3}) shows that
\(hf=\lambda f\) which completes the proof.
\end{proof}
\noindent
{\bf Acknowledgement.} We thank Heinz Siedentop and Timo Weidl for
many useful discussions. We also acknowledge the support of the
European Union through its Training, Research, and Mobility program,
grant FMRX-CT 96-0001.
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