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\vphantom{f}
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\noindent{\bf DIRAC PARTICLES IN MAGNETIC FIELDS}
%
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\leftskip=2.1truecm
\noindent B. THALLER
{\nineit{
\noindent University of Graz
\noindent Institute of Mathematics
\noindent Heinrichstra\ss e 36
\noindent A-8010 Graz
\noindent Austria
}
}
\leftskip=0pt
\vskip 1truecm
\noindent
{\ninerm{ABSTRACT. We give a review of spectral
and scattering theory for spin-1/2 particles in an external
magnetic field. The supersymmetric point of view is strongly
emphasized. Recent results on Foldy-Wouthuysen transformations,
properties of the resolvent, threshold eigenvalues and
scattering theory are presented. }}
\vskip 1truecm
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%-------------endtitle---------------------------------
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\noindent
{\bf 1. Magnetic fields}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip%
\noindent%
%
In any space dimension $\nu\ge 2$ the magnetic field strength $B$
is given by a 2-form%
%
$$
B(x) =
\sum_{\scriptstyle i,k=1\atop \scriptstyle i0$, and $n-1$, if $\epsilon=0$.}
\medskip%%%%%%%%%%%%%%%
\noindent {\it Proof:}
The Green function of $\triangle$ in two dimensions is ${1\over 2\pi} \ln |x|$,
therefore
%
$$
\phi(x) = {1\over 2\pi} \int_{{\bbbr}^2} \ln |x-y| B(y) d^2y
\eqno(17)$$
%
satisfies $\triangle\phi(x) = B(x)$, and
%
$$
\phi(x) - F\ln|x| = O\bigl( {1\over |x|} \bigr), \quad
\hbox{\rm as $|x|\to\infty$}.
\eqno(18)$$
%
We choose the vector potential $A = (-\partial_2 \phi,\partial_1 \phi)$, and look for a solution of
%
$$
c\vec\sigma\cdot (p-A)\psi = 0,
\quad \vec\sigma = (\sigma_1,\sigma_2)
\eqno(19)$$
Writing
$$
\omega = e^{\sigma_3 \phi}\psi
\eqno(20)$$
we find that (19) is equivalent to
$$
\vec\sigma\cdot p \omega = 0
\quad{\rm or}\quad
\eqalign{
\left({\partial \over \partial x_1} + i{\partial \over \partial x_2}
\right) \omega_1(x) &= 0,\cr
\left({\partial \over \partial x_1} - i{\partial \over \partial x_2}
\right) \omega_2(x) &= 0.\cr}
\eqno(21)$$
These equations are equivalent to the Cauchy Riemann equations.
Hence $\omega_1$ (resp. $\omega_2$) has to be an entire analytic
function in the variable $z=x_1+ix_2$ (resp. $\overline{z} = x_1-ix_2$).
For large $|z|=|x|$ these functions behave as
%
$$
\eqalignno{
\omega_1(x) &\approx e^{+F\ln|x|}\psi_1(x) = |x|^{+F}\psi_1(x),
&(22)\cr
\omega_2(x) &\approx e^{-F\ln|x|}\psi_2(x) = |x|^{-F}\psi_2(x).
&(23)\cr}$$
%
If $F>0$ then $\omega_2$ is square integrabel at infinity and hence zero,
because an analytic function cannot vanish in all directions, as $|z|\to\infty$.
This shows that $\psi_2=0$ and therefore only $+mc^2$ can be an eigenvalue
of $H(A)$. But for this we have to fulfill the condition
$$
\psi_1 = e^{-\phi}\omega_1 \in L^2({\bbbr}^2)
\eqno(24)$$
which requires that $\omega_1$ should not increase faster than $|x|^{F-1-\delta}$,
for some $\delta > 0$. Since $\omega_1$ is an entire function,
it must be a polynomial in $x_1+ix_2$ of degree $\le n-1$
(resp. $n-2$, if $\epsilon=0$). Hence there are $n$ linearly independent
solutions $\psi_1$ of $D\psi_1=0$, namely (for $\epsilon\ne 0$)
$$
e^{-\phi},\enskip e^{-\phi}(x_1+ix_2),\enskip e^{-\phi}(x_1+ix_2)^2,
\enskip \ldots\enskip,\enskip e^{-\phi}(x_1+ix_2)^{n-1}.
\eqno(25)$$
An analogous reasoning applies to the case $F<0$.\hfill{\it QED.}
\medskip
\noindent
The proof of the Aharonov-Casher Theorem shows that if one can find
a solution $\phi$ of $\triangle \phi(x) = B(x)$, such that
$e^{-\phi}$ (or $e^{+\phi}$) is a rapidly decreasing function
in ${\cal S}({\bbbr}^3)$, then the eigenvalue $+mc^2$ (or $-mc^2$)
is infinitely degenerate and hence in $\sigma_{ess}(H(A))$ (see also Ref. [11]). This is indeed the case, e.g., for a homogeneous magnetic field.
In case of a spherically symmetric magnetic field in two dimensions ($B(x) = B(r)$, $r=|x|$) a solution of $\triangle \phi(r)$ = $\bigl( \partial^2/\partial r^2 + (1/r)\partial/\partial r\bigr) \phi(r)$ = $B(r)$ is given by
$$
\phi(r)=\int_0^r ds {1\over s}\int_0^s dt B(t) t
\eqno(26)$$
Hence for a magnetic field with infinite flux like
$$B(r)\approx r^{\delta-2},\quad\hbox{for some $\delta>0$, $r$ large,}
\eqno(27)$$
we find
$$\phi(r)\approx {1\over\delta}r^\delta,\quad \hbox{for $r$ large,}
\eqno(28)$$
i.e., $e^{-\phi}$ decreases faster than any polynomial in $|x|$.
Therefore $+mc^2$ is infinitely degenerate in this case.
If even $B(r)\to \infty$, as $r\to\infty$, then the next Theorem shows,
that $+mc^2$ is the only possible element in the essential spectrum of the Dirac operator.
%
%
\medskip
\noindent%
%%%%%%%%%%%%%%%%%
{\bf Theorem. (Helffer-Nourrigat-Wang, [2]).}
{\sl If in two dimensions $B(x)\to\infty$ (resp. $B(x)\to-\infty$),
as $|x|\to\infty$, then $\lambda$ with $\lambda\ne+mc^2$ (resp. $\lambda\ne-mc^2$) is not in the essential spectrum of the Dirac operator.}
\medskip
\noindent%
%%%%%%%%%%%%%%%%%
{\it Proof:} We assume $B(x)\to +\infty$, the other case can be
treated analogously. We show that $\lambda$ with $\lambda\ne mc^2$
is not in $\sigma_{ess}(H(A))$, because for all
$\Psi=\bigl( {\psi_1\atop\psi_2} \bigr) \in{\cal C}_0^\infty({\bbbr}^2)^2$
with support outside a ball with sufficiently large radius $R$ there
is a constant $C(\lambda)$ such that
$$
\left\| \bigl( H(A)-\lambda\bigr) \Psi\right\| \ge C(\lambda)\|\Psi\|
\eqno(29)$$
In order to prove this, we choose $R$ so large, that
$$
B(x) \ge {1\over c^2} |\lambda - mc^2| (3+2|\lambda+mc^2|),
\quad\hbox{\rm for all $|x|\ge R$.}
\eqno(30)$$
Denoting $\Phi = \bigl( {\phi_1\atop\phi_2} \bigr) = \bigl( H(A)-\lambda \bigr) \Psi$, i.e.,
$$\eqalign{
\phi_1 &= cD^*\psi_2 - (\lambda-mc^2)\psi_1\cr
\phi_2 &= cD\psi_1 - (\lambda+mc^2)\psi_2\cr}
\eqno(31)
$$
we find
$$\eqalignno{
\|D^*\psi_2\|^2 &= (\psi_2, D^*D\psi_2) = \|(p-A)^2\psi_2\|^2 + (\psi_2,B(x)\psi_2)\cr
&\ge {1\over c^2} |\lambda - mc^2| (3+2|\lambda+mc^2|) \|\psi_2\|^2,&(32)\cr}
$$
provided supp$\psi_2$ is outside the ball with radius $R$. Hence
$$\eqalignno{
(3+2|\lambda &+ mc^2| )\|\psi_2\|^2
\le
{1\over |\lambda - mc^2|} \|\phi_1 + (\lambda-mc^2)\psi_1\|^2 \cr
&=
{1\over |\lambda - mc^2|} \|\phi_1\|^2 + 2 ({\rm sgn}\lambda) {\rm Re}(\phi_1,\psi_1) + |\lambda-mc^2| \|\psi_1\|^2.&(33)\cr}
$$
Since
$$\eqalignno{
({\rm sgn}\lambda){\rm Re}(\phi_1,\psi_1)
&=
({\rm sgn}\lambda){\rm Re}(cD^*\psi_2,\psi_1) - |\lambda-mc^2|\|\psi_1\|^2\cr
&\le
\|\psi_2\| \|cD\psi_1\| - |\lambda-mc^2|\|\psi_1\|^2\cr
&\le \|\psi_2\| \|\phi_2\| + |\lambda+mc^2|\|\psi_2\|^2- |\lambda-mc^2|\|\psi_1\|^2&(34)\cr}
$$
we find
$$\eqalignno{
&3 \|\psi_2\|^2 + |\lambda - mc^2| \|\psi_1\|^2\cr
&\le
{1\over |\lambda - mc^2|}\|\phi_1\|^2 + 2 \|\psi_2\| \|\phi_2\|&(35)\cr}
$$
Now we have either $\|\psi_2\| \le \|\phi_2\|$ or $\|\psi_2\| \ge \|\phi_2\|$. In each case
$$
\|\Psi\|^2 = \|\psi_1\|^2 + \|\psi_2\|^2 \le 2\max \left\{ 1 , {1\over (\lambda-mc^2)^2} \right\} \|\Phi\|^2,
\eqno(36)$$
which proves the Theorem. \hfill{\it QED.}
\medbreak
\noindent
%
%
{\bf 7. Three space dimensions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\nobreak%
\medskip%
\nobreak%
\noindent%
There is no analogue of the Aharonov-Casher result for $\nu=3$. Also the theorem of Helffer, Nourrigat, and Wang is very specific to two dimensions. Concerning eigenvalues at $\pm mc^2$ in three dimensions, so far only some examples are known.
\medskip\noindent%
%
%%%%%%%%%%%
{\bf Example. (Loss-Yau, [12]).} If, for some real valued $\lambda$,
we had a
solution of $$
\vec\sigma\cdot {\bf p} \Psi({\bf x}) = \lambda({\bf x})\Psi({\bf x}),
\eqno(37)$$
which satisfies
$$
\langle \Psi , \Psi \rangle ({\bf x}) \equiv
\sum_{i=1}^2 \overline{\psi_i({\bf x})}\psi_i({\bf x}) \ne 0,
\eqno(38)$$
then we can find a solution of $\vec\sigma\cdot ({\bf p}-{\bf A}) \Psi = 0$.
First note that
$$
\langle \Psi , \vec\sigma \Psi \rangle
\langle \Psi , \vec\sigma \Psi \rangle =
\langle \Psi , \Psi \rangle ^2,
\eqno(39)$$
implies
$$
\vec\sigma\cdot {\langle \Psi , \vec\sigma \Psi \rangle
\over \langle \Psi , \Psi \rangle} \Psi = \Psi,
\eqno(40)$$
and hence
$$
\vec\sigma\cdot {\bf A}({\bf {\bf x}}) \Psi({\bf x}) = \lambda({\bf x})\Psi({\bf x}),
\eqno(41)$$
if we choose
$$
{\bf A}({\bf x}) = \lambda({\bf x}){\langle \Psi , \vec\sigma \Psi \rangle
\over \langle \Psi , \Psi \rangle}.
\eqno(42)$$
But a solution of (37) is easy to find. Choose, for example,
$$
\Psi({\bf x}) = {1+i\vec\sigma\cdot{\bf x} \over (1+x^2)^{3/2}}\Phi_0,
\eqno(43)$$
where $\Phi_0\in {\bbbc}^2$, with $\langle \Phi_0,\Phi_0\rangle =1$. Note that
$$
0 \ne \langle \Psi , \vec\sigma \Psi \rangle ({\bf x}) =
{1\over (1+x^2)^3} \{ (1-x^2){\bf w} + 2({\bf w}\cdot{\bf x}){\bf x}
+ 2 {\bf w}\wedge{\bf x}\},
\eqno(44)$$
where $w=\langle \Phi_0 , \vec\sigma \Phi_0 \rangle$ is a unit vector in ${\bbbr}^3$. We obtain
$$
\vec\sigma\cdot {\bf p} \Psi({\bf x}) = {3\over 1+x^2}\Psi({\bf x}),
\eqno(45)$$
and finally
$$
{\bf A}({\bf x}) = 3(1+x^2)\langle \Psi , \vec\sigma \Psi \rangle,
\qquad {\bf B}({\bf x}) = 12\langle \Psi , \vec\sigma \Psi \rangle.
\eqno(46)$$
The vector field $A$ can be obtained by stereographic projection from a parallel
basis vector field on the three dimensional sphere. Hence the flow lines are
circles on the Hopf tori.
\medskip\noindent
A characterisation of the essential
spectrum in two or three dimensions is given by the next Theorem. This result is
not so typical for Dirac operators, because similar statements are true for the
nonrelativistic Schr\"odinger operator without spin
(see Leinfelder [6], Miller and Simon [13], [14]).
\medskip
\noindent
%%%%%%%%%%%%%%%
{\bf Theorem. (Leinfelder-Miller-Simon).} {\sl
In two or three dimensions, if $|B(x)|\to 0$, as $|x|\to\infty$, then
$$
\sigma_{ess}(H(A)) = (-\infty,-mc^2] \cup [mc^2,\infty).
\eqno(47)$$
If $B(x)$ is bounded, then the distance from an arbitrary
$\lambda\not\in (-mc^2,mc^2)$ to $\sigma_{ess}(H(A))$ is less than $2
\sqrt{12}\sup \sqrt{|B(x)|}$.}
\medskip
\noindent
%%%%%%%%%%%%%%%
{\it Proof:} It is sufficient to consider the essential
spectrum of the operator $$
D_\nu = c\sum_{i=1}^\nu\sigma_i(p_i-A_i) \equiv c\vec\sigma\cdot(p-A)
\eqno(48)$$
in dimensions $\nu = 2,3$, because $H(A)$ is unitarily equivalent to
$$
\sigma_3 \sqrt{(D_2)^2 + m^2c^4}\quad
\hbox{\rm for $\nu=2$,}
\eqno(49)$$
and
$$\pmatrix{\sqrt{(D_3)^2 + m^2c^4} & 0 \cr
0 & -\sqrt{(D_3)^2 + m^2c^4} \cr}\quad
\hbox{\rm for $\nu=3$.}
\eqno(50)$$
In order to prove $k\in\sigma_{ess}(D_\nu)$ it is sufficient to find an orthonormal sequence of vectors $\Psi^{(n)}$ in the domain of $D_\nu$, such that
$$
\lim_{n\to\infty} \|(D_\nu - k)\Psi^{(n)}\| = 0
\eqno(51)$$
(Weyl's criterion). Moreover, the distance between $k$ and $\sigma_{ess}(D_\nu)$ is less than $d$, if for a suitable orthonormal sequence $\Psi^{(n)}$
$$
\|(D_\nu - k)\Psi^{(n)}\| \le d.
\eqno(52)$$
We are going to construct suitable vectors $\Psi^{(n)}$ as follows. Let $B_n = B_{\rho_n}(x^{(n)})$ be a sequence of disjoint balls with centers $x^{(n)}$ and radii $\rho_n$. Any two $L^2$-functions with support in different balls are orthogonal. We use the gauge freedom to define within these balls vector potentials $A^{(n)}$ which are determined by the local properties of $B$ in that region (unlike the original $A$-field). For each $n$ we define
$$
A^{(n)}(x) = \int_0^1 B\bigl( x^{(n)} + (x-x^{(n)})s\bigr)
\wedge (x-x^{(n)})sds,
\eqno(53)$$
or, written in components
$$
A^{(n)}_i(x) = \int_0^1 \sum_{i=1}^\nu
F_{ki}\bigl( x^{(n)} + (x-x^{(n)})s\bigr) (x_k - x_k^{(n)})sds,
\quad i = 1,\ldots,\nu.
\eqno(54)$$
It is easy to see that
$$
\sup_{x\in B_n} |A^{(n)}(x)| \le \rho_n\sup_{x\in B_n} |B(x)|.
\eqno(55)$$
Furthermore, if $A$ is the vector potential we started with, then
$$
A - A^{(n)} = \nabla g^{(n)},
\quad\hbox{\rm with $g^{(n)}\in {\cal C}^\infty({\bbbr}^\nu)$}.
\eqno(56)$$
Finally, we choose
$$
\Psi^{(n)}(x) = {1\over \sqrt{2}}
\left( {1\atop 1} \right) {1\over\rho_n^{\nu/2}}
\enskip j\left( {x-x^{(n)}\over \rho_n}\right)
\exp \bigl( ig^{(n)}(x) - ikx_1 \bigr) ,
\eqno(57)$$
where j is a localization function with the following properties:
$$
j\in {\cal C}_0^\infty({\bbbr}^\nu), \quad
{\rm supp}j \subset \{ x\mid |x|\le 1\} ,\quad
\int |j(x)|^2d^\nu x = 1.
\eqno(58)$$
It is easily verified that ${\rm supp}\Psi^{(n)} \subset B_n$, and $\|\Psi^{(n)}\|=1$. A little calculation gives for all $k\in {\bbbr}$
$$\eqalignno{
&\{ \vec\sigma\cdot \bigl( p-A(x)\bigr) - k \} \Psi^{(n)}(x)\cr
&= -i{1\over \rho_n} \vec\sigma\cdot
(\nabla j)\left( {x-x^{(n)}\over \rho_n}\right) \enskip {1\over \sqrt{2}}
\left( {1\atop 1} \right) {1\over\rho_n^{\nu/2}}\exp \bigl( ig^{(n)}(x) - ikx_1 \bigr)\cr
&- \vec\sigma\cdot A^{(n)}(x)\Psi^{(n)}(x).&(59)\cr}
$$
Using (55) we obtain the estimate
$$\| \{ \vec\sigma\cdot \bigl( p-A \bigr) - k \} \Psi^{(n)}\|
\le {1\over \rho_n} \int |\nabla j (x)|^2 d^\nu x + \rho_n \sup_{x\in B_n}
|B(x)|
\eqno(60)$$
Assume first, that $|B(x)|\le M$ for all $x$.
Choosing $\rho_n=\rho$, all $n$, we find that (60) is bounded, i.e.,
$\sigma_{ess}(D_\nu)$ is not empty. Moreover, if we choose $j(x) = {\rm const.}
\cos^2(\pi |x|/2)$, then it is easy to see that $\int |\nabla j (x)|^2 d^3 x
\approx 11.62$. Setting $\rho = (12/M)^{1/2}$ we obtain the bound $2(12M)^{1/2}$
for (60). Hence the distance of an arbitrary $k\in {\bbbr}$ to
$\sigma_{ess}(D_\nu)$ is less than this constant. The distance from an arbitrary
$\lambda\not\in (-mc^2,mc^2)$ to the next point in $\sigma_{ess}(H(A))$ is
bounded by the same constant.
If $|B(x)|\to 0$, as $|x|\to\infty$, we can proceed as follows.
There is a sequence of disjoint balls $B_n$ with increasing radius $\rho_n$,
such that $$
\sup_{x\in B_n} |B(x)| \le {1\over \rho_n^2}
\eqno(61)$$
But then (60) is bounded by constant$/\rho_n \to 0$, as $n\to\infty$.
Hence any $k\in{\bbbr}^\nu$ is in the essential spectrum of $D_\nu$.
\hfill{\it QED.}
\medbreak\noindent
In case of electric or scalar potentials which decay at infinity, the essential spectrum mainly consists of a continuous spectrum associated to scattering states. This is not necessarily the case for magnetic fields. This can be seen most clearly by looking at spherically symmetric examples.
\medbreak
\noindent
{\bf 8. Cylindrical symmetry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nobreak%
\medskip%
\nobreak%
\noindent%
%%%%%%%%%%%%%%%%%%%%%%
In two dimensions, if the magnetic field strength is cylindrically symmetric we
can pass to coordinates
$r = |x|$, $\phi = \arctan(x_2/x_1)$. We denote the coordinate unit vectors by
${\rm e}_r = {1\over r} ( x_1,x_2)$,
${\rm e}_\phi = {1\over r} (-x_2,x_1)$,
write $B(x) \equiv B(r)$, and choose $A(x) = A_\phi(r){\rm e}_\phi$, where
$$
A_\phi(r) = {1\over r}\int_0^r B(s)s\, ds,\quad
B(r) = \left( {d\over dr} + {1\over r} \right) A_\phi(r) =
{1\over r}{d\over dr}\bigl( A_\phi(r)r \bigr).
\eqno(62)$$
In this notation the flux of $B$ is given by $F =
\lim_{r\to \infty} \bigl( A_\phi(r)r \bigr)$.
The Dirac operator in cylindrical coordinates can be written as
$$\eqalignno{
H(A) &\equiv c{\vec\sigma}\cdot(p-A) + \sigma_3 mc^2\cr
&=c(\vec\sigma\cdot{\rm e}_r)\,
{\rm e}_r\cdot(p-A) + c(\vec\sigma\cdot{\rm e}_\phi)\,
{\rm e}_\phi\cdot(p-A) + \sigma_3 mc^2\cr
&=c(\vec\sigma\cdot{\rm e}_r) \left\{
-i\left( {\partial \over \partial r} + {1\over 2r}\right) +
i{1\over r} \sigma_3 J_3 -i\sigma_3A_\phi(r)
\right\} + \sigma_3 mc^2.&(63)\cr}
$$
The angular momentum operator $J_3 = L_3 + \sigma_3/2$ commutes with
$H(A)$ and the spinors
$$
\chi_{m_j} = \left( {a\, e^{i(m_j-1/2)\phi} \atop b\, e^{i(m_j+1/2)\phi}}
\right),
\quad a,b\in{\bbbc},\quad m_j = \pm{1\over 2}, \pm{3\over 2}, \pm{5\over 2},
\ldots
\eqno(64)$$
form a complete set of orthogonal eigenvectors of $J_3$
in $L^2(S^1)^2$ with the properties
$$\eqalignno{
J_3 \chi_{m_j} &= m_j \chi_{m_j},&(65)\cr
(\vec\sigma\cdot{\rm e}_r)\chi_{m_j} &= \left( {b\, e^{i(m_j-1/2)\phi} \atop a\, e^{i(m_j+1/2)\phi}} \right).&(66)\cr}$$
Any function $\Psi(r,\phi)$ in $L^2({\bbbr}^2)^2$ can be written as a sum
$$
\Psi(r,\phi) = \sum_{m_j} \left(
{\displaystyle {1\over \sqrt{r}} f_{m_j}(r)e^{i(m_j - 1/2)\phi}
\atop
\displaystyle -i{1\over \sqrt{r}} g_{m_j}(r)e^{i(m_j + 1/2)\phi}}
\right),
\eqno(67)$$
with suitable functions $f_{m_j}$ and $g_{m_j}$ in
$L^2([0,\infty),dr)$. The action of $H(A)$ on $\Psi$ can be described on each
angular momentum subspace as the action of a ``radial Dirac operator'' $h_{m_j}$
defined in $L^2([0,\infty),dr)^2$ $$ h_{m_j}\left( f_{m_j}\atop g_{m_j}\right) =
\pmatrix{mc^2 & cD^* \cr
cD & -mc^2 \cr}\left( f_{m_j}\atop g_{m_j}\right)
\eqno(68)$$
with
$$
D = {d\over dr} - {m_j \over r} + A_\phi(r),
\eqno(69)$$
and $H(A)$ is unitarily equivalent to a direct sum of the operators $h_{m_j}$.
A little calculation shows
$$
\left. {D^*D \atop DD^*} \right\} =
- {d^2\over dr^2} + {\bigl( m_j \mp {1\over 2}\bigr)^2 - {1\over 4} \over r^2}
- 2{m_j \mp {1\over 2} \over r}A_\phi(r) + A_\phi^2(r) \mp B(r).
\eqno(70)$$
From (62) we see that if $B(r)r\to\infty$ then also $A_\phi(r)\to\infty$,
as $r\to\infty$. In this case the term $A_\phi^2$ dominates in (70) the interaction at
large values of $r$ and clearly the Schr\"odinger operator $D^*D$ (resp. $DD^*$)
has a pure point spectrum. By (11) the same is true for the Dirac operator on
each angular momentum subspace and hence for $H(A)$. Let us summarize these
observations in the following Theorem.
%
%
%
\medbreak\noindent
{\bf Theorem. (Miller-Simon [13], [14]).} {\sl In two dimensions, if B is spherically symmetric
and $B(r)r\to\infty$, as $r\to\infty$, then the Dirac operator $H(A)$ has a pure
point spectrum.}
\medskip\noindent
In addition, if $B(r)\to 0$, then $B(r)r\to \infty$ implies
that there is a complete set of orthonormal eigenvectors of $H(A)$ belonging to eigenvalues
which are dense in the the union of the
intervals $(-\infty,-mc^2]$ and $[mc^2,\infty)$.
%
%
\medbreak
\noindent
{\bf 9. A review of further results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nobreak%
\medskip%
\nobreak%
\noindent%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is immediately clear from the proof of the Theorem of
Leinfelder-Miller-Simon that the condition $|B(x)|\to 0$ can be weakened
considerably. It is sufficient to require that there is a sequence of balls
with increasing radius on which $B$ tends to zero. These balls can be widely
separated and it does not matter how $B$ behaves elsewhere
[7,12]. It is even sufficient to require a similar behaviour of the
derivatives of $B$
[2]. More, precisely, one defines functions
$$\epsilon_r(x) =
{\sum_{|\alpha|=r}|D^\alpha B| \over 1+\sum_{|\alpha| mc^2$.}
%
%
\medbreak
\noindent
%
%
{\bf 10. Scattering Theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nobreak%
\medskip%
\nobreak%
\noindent%
One of the basic problems in scattering theory is proving
asymptotic completeness of the wave operators
%
$$
\Omega_\pm(H,H_0) \equiv
\mathop{\hbox{\rm s-lim}}_{t\to\pm\infty}
e^{iHt}e^{-iH_0t}\,P_{cont}(H_0),
\eqno(72)$$
%
Supersymmetry implies a relation between the wave operators
$\Omega_\pm(H_P(A),H_P(0))$ of the nonrelativistic theory and the relativistic
wave operators $\Omega_\pm(H(A),H(0))$. Unfortunately, the Dirac operator is not
an ``admissible'' function of the Pauli operator, therefore we cannot apply
directly the invariance principle
in order to conclude existence of wave operators in the relativistic case from
the existence of the nonrelativistic wave operators. Nevertheless
we have the following theorem, where $F$
denotes the projection operator to the subspace belonging
to the indicated region of the spectrum of the
self-adjoint operator.
\medbreak\noindent
{\bf Theorem. (Thaller, [9, 17]).} {\sl Let $H = Q + m\tau$, $H_0 = Q_0 + m\tau$
be two Dirac operators with supersymmetry. Assume that for all $00) +
\Omega_\mp(Q^2,Q_0^2)\,F(H_0<0)
\eqno(74)$$
}%
\medskip\noindent
A proof of this theorem is given in Ref. [17]. Now we apply it to the case of magnetic fields. Note that $Q^2 = H_P(A)$ is just the nonrelativistic Pauli operator.
Assume
that the magnetic field strength $B$ decays at infinity, such that we have, for some
$\delta>0$,
%
$$
B(x) \le \rm{const.}(1+|x|)^{-3/2-\delta}
\eqno(75)$$
%
Choose the transversal (or Poincar\'e) gauge
%
$$
A(x) = \int_0^1 \! s\, B(xs)\wedge x \,ds .
\eqno(76)$$
%
$A$ is uniquely characterized by $A(x)\cdot x = 0$, and $A(x)$ decays like $|x|^{-1/2-\delta}$,
as $|x|\to\infty$. Hence the expressions div$A$, $A^2$, $\vec\Sigma\cdot B$
occurring in $Q^2-Q_0^2$ are all of short-range. The remaining long-range term is
$A(x)\cdot p$. It can be written as $A(x)\cdot p = G(x)\cdot (x\wedge p)$, where
%
$$
G(x) = \int_0^1 \! s\, B(xs) \,ds
\eqno(77)$$
%
satisfies,
%
$$
|G(x)| \le \rm{const.}(1+|x|)^{-3/2-\delta)}
\eqno(78)$$
%
and since the angular momentum $L = x\wedge p$ remains
constant under the nonrelativistic free time evolution $\exp(-iQ^2_0t)$, we easily
obtain by a stationary phase argument for $\Psi$ in a suitable dense set (see also [22])
%
$$
\|A(x)\cdot p\,e^{-ip^2 t}\,\Psi\| \le \rm{const.} (1+|t|)^{-3/2-\delta}.
\eqno(79)$$
%
The condition $(73)_1$ simply becomes
%
$$
\|A(x)\,e^{-ip^2t}\,\Psi\| \le \rm{const.}(1+|t|)^{-\delta}
\eqno(80)$$
%
and is trivially satisfied. Hence we have proven existence of the nonrelativistic and relativistic wave operators. But also asymptotic completeness is true.
\medbreak\noindent%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Theorem. (Loss-Thaller, [18]).} {\sl Let $H(A)$ and $H(0)$ be given as in
(2) and assume that the magnetic field strength $B$ satisfies
%
$$
D^\gamma B(x) \le \rm{const.}(1+|x|)^{-3/2-\delta-\gamma},
\eqno(81)
$$
%
for some $\delta>0$ and multiindices $\gamma$ with $|\gamma| = 0,1,2$. Then the nonrelativistic and
relativistic wave operators in the transversal gauge are asymptotically complete.}
\medskip\noindent%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Existence of wave operators is usually expected to hold for ``short-range
potentials'', where (each component of) the potential matrix $V$ satisfies
%
$$
|V(x)| \le \rm{const} (1+|t|)^{-1-\delta}.
\eqno(82)$$
%
A famous counter example is the electrostatic Coulomb potential, where
$|V(x)|$ decays like $|x|^{-1}$. In this case the wave operators do not
exist [19] and one has to introduce modifications of the asymptotic time
evolution.
For the magnetic fields in the theorem above the potential matrix $-\vec\alpha\cdot A$
has a much slower decay. Indeed, previous results in the literature
have been
obtained only by introducing modifications of the wave operators (see,
e.g., Ref. [20] and the references therein). Asymptotic completeness is due to
the transversality of $A$. In another gauge $A$ is not transversal and if
$\nabla g$ is long-range, then the unmodified wave operators (72) would not
exist. Instead, asymptotic completeness holds for $\Omega_\pm(H(A),H(\nabla
g))$. These remarks might be of importance, because physicists use almost
exclusively the Coulomb gauge instead of the transversal gauge, which is best
adapted to scattering theory. Note that although the wave and scattering
operators depend on the choosen gauge, the physically observable quantities like
scattering cross sections are gauge independent.
In situations like the Aharonov Bohm effect one has used
the free asymptotics (e.g., plane waves for the asymptotic description of
stationary scattering states), together with the Coulomb gauge, although the
vector potential is long-range. But in this case the calculations are justified,
because in two dimensions and for rotationally symmetric fields the Coulomb
gauge coincides with the transversal gauge (see also Ref. [21], for a
discussion).
Under weaker decay conditions on the magnetic field strength
the wave operators would not exist in that
form, because then the term $A^2$ occurring in $Q^2-Q_0^2$ would become long-range.
In this case one really needs modified wave operators, similar to the
Coulomb case.
The scattering problem is nontrivial even in classical mechanics. From special
examples we know that classical paths of particles in magnetic fields satisfying
our requirements do not have asymptotes. It is easy to see that the
velocity of the particles is asymptotically constant. But if we compare
the asymptotic motion of a particle in a magnetic field with a free motion,
one would have to add a correction which is transversal to the
asymptotic velocity and which increases for $\delta < 1/2$ like $|t|^{1/2-\delta}$.
Thus the situation seems to be
worse than in the Coulomb problem. There the interacting particles also cannot
be asymptotically approximated by free particles, but at least the classical
paths do have asymptotes. (The correction in the Coulomb problem increases
like $\ln |t|$ and is
parallel to the asymptotic velocity). A discussion of these effects in
the classical scattering theory with magnetic fields is given in Ref. [22].
%
%
\medbreak \noindent%
%
%
{\bf References}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nobreak%
\medskip%
\nobreak%
%
\leftskip=0.5 truecm%
\ninerm{
\item{[1]} Berthier, A. and Georgescu, V. On the point spectrum for Dirac
operators. J. Func. Anal. 71: 309-338 (1987).
\item{[2]} Helffer, B., Nourrigat, J. and Wang, X. P. Sur le spectre de
l'equation de Dirac avec champ magnetique. Preprint. (1989).
\item{[3]} Hunziker, W. On the nonrelativistic limit of the Dirac theory.
Commun. Math. Phys. 40: 215-222 (1975).
\item{[4]} Gesztesy, F., Grosse, H. and Thaller, B. A rigorous approach to
relativistic corrections of bound state energies for spin-1/2 particles. Ann.
Inst. H. Poincar\'e. 40: 159-174 (1984).
\item{[5]} Grigore, D. R., Nenciu, G. and Purice, R. On the nonrelativistic
limit of the Dirac Hamiltonian. Preprint, to appear in Ann. Inst. H.
Poincar\'e. (1989).
\item{[6]} Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.
Schr\"odinger operators with applications to quantum mechanics and global
geometry. Springer Verlag. Berlin, Heidelberg, New York, London, Paris, Tokyo
1987.
\item{[7]} Leinfelder, H. Gauge invariance of Schr\"odinger operators
and related spectral properties. J. Op. Theory. 9: 163-179 (1983).
\item{[8]} Gr\"umm, H. R. Quantum mechanics in a magnetic field.
Acta Phys. Austriaca. {\bf 53}: 113-131, (1981).
\item{[9]} Thaller, B. Normal forms of an abstract Dirac operator and
applications to scattering theory. J. Math. Phys. 29: 249-257 (1988).
\item{[10]} Aharonov, Y. and Casher, A. Ground state of a spin-1/2
charged particle in a two dimensional magnetic field. Phys. Rev. A19:
2461-2462 (1979).
\item{[11]} Avron, J. E. and Seiler, R. Paramagnetism for
nonrelativistic electrons and euclidean massless Dirac particles. Phys. Rev.
Lett. 42: 931-934 (1979).
\item{[12]} Loss, M. and Yau, H. T. Stability of Coulomb systems
with magnetic fields III. Zero energy bound states of the Pauli operator.
Commun. Math. Phys. 104: 283-290 (1986).
\item{[13]} Miller, K. C. Bound states of quantum mechanical particles
in magnetic fields. Dissertation, Princeton University. (1982).
\item{[14]} Miller, K. and Simon, B. Quantum magnetic Hamiltonians
with remarkable spectral properties. Phys. Rev. Lett. 44: 1706-1707
(1980).
\item{[15]} Helffer, B. and Mohamed, A. Caract\'erisation du spectre essentiel
de l'op\'erateur de Schr\"odinger avec champ magnetique. Ann. Inst. Fourier 38:
95-112 (1988).
\item{[16]} Kalf, H. Non-existence of eigenvalues of Dirac operators.
Proc. Roy. Soc. Edinburgh. A89: 307-317 (1981).
\item{[17]} Thaller, B. Scattering theory of a supersymmetric Dirac operator.
Preprint. (1989).
\item{[18]} Loss, M. and Thaller, B. Short-range scattering in
long-range magnetic fields: The relativistic case. J. Diff. Eq. 73:
225-236 (1988).
\item{[19]} Dollard, J. and Velo, G. Asymptotic behaviour of a
Dirac particle in a Coulomb field. Il Nuovo Cimento. 45: 801-812 (1966).
\goodbreak\vbox{
\item{[20]} Thaller, B. Relativistic scattering theory for long-range
potentials of the nonelectrostatic type. Lett. Math. Phys. 12: 15-19
(1986).
\item{[21]} Perry, P. Scattering Theory by the Enss Method.
Math. Rep. 1, Harwood academic publishers, New York 1983
\item{[22]} Loss, M. and Thaller, B. Scattering of particles by
long-range magnetic fields. Ann. Phys. 176: 159-180 (1987).}
}
\end