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\vphantom{f}

\vskip 1.3truecm

\noindent{\bf DIRAC PARTICLES IN MAGNETIC FIELDS}

%

\vskip 52truept

\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---------------------------------

%

\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 i<k}^\nu F_{ik}(x)

dx_i\wedge dx_k

\eqno(1)$$

%

satisfying $dB=0$ (exterior derivative). We can write $B=dA$ with a

magnetic ``vector'' potential (1-form) $A$ and obtain

%

$$

F_{ik}(x)={\partial A_i(x)\over \partial x_k} - {\partial

A_k(x)\over \partial x_i},\qquad

A(x) = \sum_{i=1}^\nu A_i(x) dx_i.

\eqno(2)$$

%

We want to stress that the vector potential is not directly

observable. Therefore, one should formulate all results under

assumptions on the field strengths. Eventually, we shall use the

gauge freedom (see below) to make the description as simple as

possible. Throughout this lecture we assume that each component of

$B$ is a smooth function in ${\cal C}^\infty({\bbbr}^\nu)$. We

refer to the cited literature for possible generalizations (e.g., [1]).

We are

only interested in results for dimensions $\nu = 2,3$. For $\nu=3$

we identify $B$ and $A$ with vector fields ${\bf B}({\bf

x})=\bigl( F_{23}({\bf x}), F_{31}({\bf x}),F_{12}({\bf x})\bigr)$

and ${\bf A}({\bf x})=\bigl( A_1({\bf x}), A_2({\bf x}),A_3({\bf

x})\bigr)$ satisfying ${\bf B}={\rm rot}{\bf A}$, i.e., ${\rm

div}{\bf B} = 0$. In two dimensions, $B = \partial_1A_2

-\partial_2A_1$ is simply a scalar field. This corresponds to the

three dimensional situation

${\bf B}({\bf x}) = \bigl(0,0,B(x_1,x_2)\bigr)$.

%

%

\medbreak \noindent%

%

%

{\bf 2. Dirac and Pauli operators}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\nobreak%

\medskip%

\nobreak%

%

%

%

\noindent

We recall the Dirac operator with a

magnetic field,

$$

H(A) \equiv \cases{

c\sum_{i=1}^3

\alpha_i (p_i-A_i) +  \beta   mc^2 & if $\nu=3$,\cr

&\cr

c\sum_{i=1}^2

\sigma_i (p_i-A_i) + \sigma_3 mc^2 & if $\nu=2$,\cr}

\eqno(3)$$

%

where $p_i = -i\partial/\partial x_i$, and where $\vec\alpha =

(\alpha_1, \alpha_2, \alpha_3)$ and $\beta$ are the famous

$4\times 4$ ``Dirac matrices'', which in the standard representation

are defined by

%

$$

\beta =

      \pmatrix{ {\bf 1}  &  {\bf 0}  \cr

		              {\bf 0}  & -{\bf 1}  \cr},\quad  

\alpha_i =

\left( \matrix{ {\bf 0}  &  \sigma_i \cr

		              \sigma_i &  {\bf 0}  \cr} \right) ,\quad i = 1, 2, 3.

\eqno(4)$$

%


Here $\sigma_i$ denote the $2\times 2$ ``Pauli matrices''

%

$$

\sigma_1 =

\left( \matrix{0 & 1 \cr 1 & 0\cr} \right) ,\quad

\sigma_2 =

\left( \matrix{0 &-i \cr i & 0\cr} \right) ,\quad

\sigma_3 =

\left( \matrix{1 & 0 \cr 0 &-1\cr} \right) .

\eqno(5)$$

%

The Dirac operator (3) is essentially self-adjoint on ${\cal

C}_0^\infty({\bbbr}^3)^4$, resp. ${\cal C}_0^\infty({\bbbr}^2)^2$,

even without restriction on the growth of $B$ or $A$ at infinity

[2].



In the standard representation $H(A)$ has the abstract

``supersymmetric'' form

%

$$

H(A) = \pmatrix{

 mc^2  &  cD^*\cr

 cD  &  -mc^2 \cr} = cQ + mc^2\tau,

\eqno(6)$$

%

where

%

$$

Q=\pmatrix{

 0  &  D^*\cr

 D  &  0  \cr},

\quad \tau = \pmatrix{

 1  &  0  \cr

 0  &  -1 \cr}

 \eqno(7)$$

%

and

%

$$

D \equiv \cases{

\sum_{i=1}^3 \sigma_i (p_i-A_i) & if $\nu=3$,\cr

&\cr

(p_1-A_1) + i(p_2-A_2) & if $\nu=2$,\cr}.

\eqno(8)$$

%

Note that $D \ne D^*$ in two dimensions. The Pauli operator

%

$$

H_P(A) \equiv {1 \over 2m}D^*D = \cases{

H_S({\bf A}) {\bf 1} - (1/2m)\vec\sigma\cdot{\bf B}, & if $\nu=3$,\cr

&\cr

H_S(A)  - (1/2m)B, & if $\nu=2$\cr}

\eqno(9)$$

%

can be obtained as the nonrelativistic limit of $H(A)$, see Refs.

[3], [4], [5].

Here, $H_S(A)={1

\over 2m}(-i\nabla - A)^2$ is the nonrelativistic Schr\"odinger

operator for a spinless particle. See Chapter 6 of

Ref. [6]

for a review of its properties and references.

\medbreak

\noindent%

%

%

{\bf 3. Vector potentials and gauge freedom}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

\nobreak%

\medskip%

\nobreak%

\noindent%

The 1-form $A$ is not uniquely determined by (1).

Replacing $A$ by $A + dg$, where $g \in {\cal

C}^\infty({\bbbr}^\nu)$ (see [7] for more general $g$'s),

does not change the magnetic field

strength and leads to an equivalent

mathematical description of this physical system (``gauge

invariance'').

Indeed, if $A^{(2)} = A^{(1)} + \nabla g$,

then the unitary transformation $\Psi({\bf x}) \to

e^{ig({\bf x})} \Psi({\bf x})$ transforms $H(A^{(1)})$ into

$H(A^{(2)})$

%

$$

e^{ig}\,H(A^{(1)})\,e^{-ig} =

H(A^{(2)})

\eqno(10)$$

%

(similar for $H_P$ and $H_S$).



The use of the vector potential in quantum mechanics is sometimes

counterintuitive, because the vector potential is usually nonzero in

regions where the magnetic field strength vanishes. Consider, e.g., a

magnetic field $B(x)$ in two dimensions with compact support and

nonvanishing flux $\int B(x)d^2x$. Clearly in this case we expect the

particle to move freely once it has left the support of B. But using

Stokes law $\oint Ads = \int Bd^2x$ we see that $A(x)$ cannot decay

faster than $|x|^{-1}$, as $|x|\to\infty$. Hence the vector potential

keeps influencing the wavefunction of the particle also at large

distances from the support of $B$. There have been attempts to formulate

quantum mechanics entirely in terms of $B$ [8].

\medbreak



\noindent

{\bf 4.Supersymmetry}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\nobreak%

\medskip%

\nobreak%

\noindent%

Now we turn to the proof of some

spectral properties of $H(A)$. Any operator H of the form (6)

will be called a ``Dirac operator with supersymmetry''.

\medbreak\noindent

{\bf Theorem. (Thaller, [9]).} {\sl Let $H = cQ+mc^2\tau$ be a Dirac operator with

supersymmetry. Then there is a unitary transformation $U$, which brings $H$ to

the diagonal form

%

$$

UHU^{-1} =

\pmatrix{ \sqrt{c^2D^*D + m^2c^4}  & 0            \cr

   0     &    -\sqrt{c^2DD^* + m^2c^4}   \cr} = \tau |H|.

\eqno(11)$$

%

The unitary transformation is given by

$$

U = a_+ + \tau({\rm sgn}Q)a_-,\quad

a_\pm = {1\over\sqrt{2}}\,\sqrt{ 1 \pm {mc^2 \over|H|}}.

\eqno(12)$$

}

\medskip\noindent{\it Proof:}  

For simplicity, we set $c=1$. It is easy to verify the following formulas for the

bounded operators $a_\pm$ %

$$

a_+^2 + a_-^2 = 1, \quad a_+^2 - a_-^2 = m|H|^{-1}, \quad 2a_+a_- = |Q|\,|H|^{-1}.

\eqno(13)$$

%

Furthermore we note that $|H|=(Q^2 + m^2)^{1/2}$ commutes with $\tau$ and $Q$,

and the following commutation relations hold on ${\cal D}(H) = {\cal D}(Q)$

%

$$

[H,a_\pm] = [Q,a_\pm],\quad H\tau\, (\rm{sgn}\,Q) = -\tau\,(\rm{sgn}\,Q)\,H.

\eqno(14)$$

%

Now we can verify Eq. (14) in the following way

%

$$\eqalignno{

UHU^{-1} &=

(a_+ + \tau\,(\rm{sgn}\,Q)\,a_-)H(a_+ - \tau\,(\rm{sgn}\,Q)\,a_-) =\cr

&= (a_+^2 + 2\tau\,(\rm{sgn}\,Q)\,a_+a_- - a_-^1)H\cr

&= (m|H|^{-1} + \tau\,(\rm{sgn}\,Q)\,|Q|\,|H|^{-1})H\cr

&= (m+\tau Q)|H|^{-1}H = \tau(m\tau + Q)|H|^{-1}H\cr

&= \tau H^2|H|^{-1} = \tau|H|

&(15)\cr}$$

%

The matrix form of $\tau|H|$ immediately follows from (12), (13) and

$|H|=\sqrt{H^2}$. \hfill{\it QED.}



\medbreak\noindent

A generalization of this result has been obtained recently in Ref. [5].

\medskip

\noindent

%%%%%%%%%%%%%%

{\bf Corollary.} {\sl The spectrum of $H(A)$ is symmetric

with respect to $0$ except possibly at $\pm mc^2$.

The open interval $(-mc^2, +mc^2)$ does not

belong to the spectrum.}

\medskip \noindent

%%%%%%%%%%%%%%

{\it Proof:}

We use (11) and the fact that

$\sigma(D^*D) = \sigma(DD^*)$ except at $0$. The Theorem now follows

immediately from the spectral mapping theorem.\hfill{\it QED.}

\medskip

\noindent

We have $H({\bf A})\Psi= mc^2\Psi$ if and only if

$\Psi = \left({\psi_1\atop 0}\right)$ with $D^*D\psi_1 = 0$, or

equivalently $D\psi_1 = 0$. On the other hand we see that $H({\bf

A})\Psi= -mc^2\Psi$ if and only if $\Psi = \left({0\atop \psi_2}\right)$

with $DD^*\psi_2 = 0$, or equivalently $D^*\psi_2 = 0$. Hence in three

dimensions, where $D^*=D$, the spectrum is always symmetric, even at 0.

The situation is completely different in two dimensions, as we learn from the following theorem.

\medbreak

\noindent

{\bf 6. Eigenvectors belonging to $\pm mc^2$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\nobreak%

\medskip%

\nobreak%

\noindent%

%%%%%%%%%%%%%%

{\bf Theorem. (Aharonov-Casher, [10]).}

{\sl In two dimensions, let $B(x)$  be a magnetic field with compact support, and denote

%

$$

F={1\over 2\pi}\int_{{\bbbr}^2} B(x) d^2x

\eqno(16)$$



a) If $F=n+\epsilon$ (where n is a positive integer, and $0\le\epsilon<1$),

then $+mc^2$ (but not $-mc^2$) is an eigenvalue of the Dirac operator

$H(A)$ defined in (3).



b) If $F=-n-\epsilon$, then $-mc^2$ (but not $+mc^2$) is an eigenvalue of $H(A)$.



\noindent In both cases the multiplicity of the eigenvalue is $n$, if $\epsilon>0$, 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|<r}|D^\alpha B|},\quad

\hbox{\rm if $r\ge 1$, and $\epsilon_0(x) = |B(x)|$.}

\eqno(71)$$

and introduces the assumption

\smallskip\noindent

($A_r$): There exist a sequence of disjoint balls $B_n$ of radii $r_n$

with $r_n\to\infty$ such that the function $\epsilon_r(x)$ restricted to the

union of these balls tends to zero at infinity.

\smallskip\noindent

It can be seen that if ($A_r$) holds for some $r\ge 2$, then one of the

assumptions ($A_0$) or ($A_1$) is true

[2].

If the components of $B$ are polynomials of degree $r$, then ($A_{r+1}$) holds.

The following Theorem applies to this case.

\medbreak

\noindent%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\bf Theorem. (Helffer-Nourrigat-Wang, [2]).} {\sl In three dimensions, if ($A_r$) holds for some $r\ge

0$, then

$\sigma_{ess}(H(A)) = (-\infty,-mc^2] \cup [mc^2,\infty)$.}

\medskip%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent

This result is remarkable, because magnetic fields increasing like

polynomials are known to yield Schr\"odinger operators with compact resolvent

[15].

It is clear from Section 6 this result is very specific to three dimensions.

Since the essential spectrum may contain embedded eigenvalues it is

interesting to know criteria for the absence of eigenvalues as in the

next Theorem.

\medbreak

\noindent

{\bf Theorem. (Kalf-Berthier-Georgescu, [1,16]).} {\sl If $B(x)\to 0$ and

$x\wedge B(x)\to 0$, as $|x|\to\infty$ , then the Dirac operator has no

eigenvalues $\lambda$ with $|\lambda| > mc^2$.}

%

%

\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 $0<a<b<\infty$

and for $\Psi$ in some dense subset of $F(a<Q_0^2<b){\cal H}_{a.c.}(Q_0^2)$ the

following condition is satisfied with $k=1$,$2$

%

$$

\|(Q^k - Q_0^k)\,e^{-iQ_0^2t}\,\Psi\| \le \rm{const.}(1+|t|)^{1-k-\delta}.

\eqno(73)_k$$

%

Then the wave operators $\Omega_\pm(Q^2,Q_0^2)$ and $\Omega_\pm(H,H_0)$ exist, and

%

$$

\Omega_\pm(H,H_0) = \Omega_\pm(Q^2,Q_0^2)\,F(H_0>0) +

\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

