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Resonances, Dirac
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\documentstyle[11pt]{article}
\title{Resonances of the Dirac Hamiltonian in the non relativistic 
limit}
\author{L. Amour
\and R. Brummelhuis 
\and J. Nourrigat}
\date{D\'epartement de Math\'ematiques, ESA 6056 CNRS,\\
Universit\'e de Reims, Moulin de la Housse, B.P. 1039,\\
51687 Reims Cedex 2, France.}
\begin{document}
\maketitle  
\newcommand{\fp}{\hfill $\Box$}
\def\C{{\rm I}\!\!\!{\rm C}}
\def\R{{\rm I}\!{\rm R}}
\def\Q{{\rm I}\!\!\!{\rm Q}}
\def\Z{{\rm Z}\!\!{\rm Z}}
\def\N{{\rm I}\!{\rm N}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\eeqa}{\end{eqnarray}}
\newtheorem{th}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{prop} {Proposition}
\newcommand{\bl}{\begin{lem}}
\newcommand{\el}{\end{lem}}
\parindent=0cm
\parskip 10pt plus 1pt
\baselineskip 15pt
\begin{abstract}
 For a Dirac operator in $\R ^3$ with an electric potential 
 behaving at infinity like a power of $ | x | $, we prove  the 
existence of 
 resonances and we study, when $c\rightarrow +  \infty $,  the 
asymptotic
 expansion of their real part, and an estimation of their 
 imaginary part, generalizing an old result of Titchmarsh.   
\end{abstract}
\section{Introduction}\label{s1}
We are interested in the following Dirac operator $D(c)$ in 
 $\R^3$, 
depending on a parameter  $c>1$
\beq
D(c)=
\left(
\begin{array}{cc}
V( x)&c\sigma \cdot D_x\cr
c\sigma\cdot D_x& V( x)-2c^2
\end{array}\right),
\eeq
Here $\sigma\cdot D_x$ denotes $\sigma _1 D_1 + \sigma _2 D_2 
+ \sigma _3 D_3$, where the $\sigma _j $ are the Pauli matrices, and 
$V $ is a $C^{\infty }$ real-valued function, satisfying the 
following hypotheses. 

(H1) We assume that $V$ can be extended in an holomorphic function 
in the following open set of $ \C^3$, for some positive 
constants $a$ and $r$ 
\beq 
\Omega = S_a\cup B(0, r) 
\eeq
where  $S_a$  is the complex sector $\{z\in\C^3,\ \vert {\rm Arg } 
z_j\vert<a,
\ \forall\,j=1,2,3\}$,  and $B(0, r)$ be the open complex ball
 with center $0$ and radius $r$. We assume also that for some 
positive constants 
  $k$, $m_0$ and $R$, we have 
\beq
\vert V(z)\vert\leq m_0(1+\vert z\vert^k)\hskip 1cm 
 \forall\, z\in S_a
\eeq
(H2) We have also, if $x\in \R^3$ and $ | x | \geq R$
\beq 
\vert x\vert^k \leq m_0  V(x) 
\eeq
(H3)We have also, if $x\in \R^3$ and $ | x | \geq R$
\beq 
\vert x\vert^k \leq m_0 x. \frac {\partial V }{\partial x} 
\eeq 
We see easily that $D(c)$ is essentially self-adjoint, and 
Titchmarsh proved, when $V$ is radial, that $D(c)$ has the whole real 
line as a purely absolutely continuous spectrum (see Thaller [14]). 
Let $H_{\infty }$  be  the corresponding Schr\"odinger operator 
\beq 
H_{\infty } = - \frac {1} {2} \Delta + V(x)
\eeq  
The spectrum of $H_{ \infty }$ is discrete. 
We shall prove that, when $c$ is large enough, $D(c)$ has resonances 
near  the eigenvalues of $H_{\infty }$ and we shall study 
their asymptotic behaviour when $c\rightarrow  + \infty $. 
Recall that, in the semiclassical limit, 
the asymptotic behaviour of the resonances is studied in 
Parisse [9] (see also Balslev-Helffer [2]). For the Dirac operator 
in one dimension, with 
potential $V(x)=  | x | $, Titchmarsh [15] gave an explicit 
computation 
of the resonances (see also Veselic [16] and Thaller [14]).  

For the definition 
of resonances, we need the analytic dilations (see 
Aguilar-Combes [1]).  
For each $\theta \in \C $ such that $ |\Im   \theta  |  < a$, we 
denote 
by $H( \theta , c )$ the following Hamiltonian
\beq \label{a1} 
H(\theta , c)=
\left(
\begin{array}{cc}
V(e^{\theta} x)&e^{-\theta}c\sigma \cdot D_x\cr
e^{-\theta}c\sigma\cdot D_x& V(e^{\theta} x)-2c^2
\end{array}\right),
\eeq
with domain 
\beq 
B^1(\R ^3, \C ^4)  = \{ u\in H^1(\R^3, \C^4),\,  | x | ^k u\in 
L^2(\R^3,\C^4) \}
\eeq 
We shall prove in section 2 the following theorem. 
\begin{th}\label{t1}
 $H(c,\theta)$ has  pure point spectrum for small
 positive $\Im\,\theta$. Each eigenvalue $\lambda_j(\theta, c)$
is isolated and of finite even  multiplicity, and  
 does not depend on $\theta$. 
 \end{th}
 The eigenvalues of $H(\theta , c)$, denoted by $E _j(c)$ 
 since they do not depend on $\theta $, will be called 
 resonances. We shall prove in section 3 the following theorem.  
\begin{th}\label{t2}
If $\Im \, \theta $ is small enough, we have the following properties.

(i) Let $K$ be a compact set of $\, \C $ containing no eigenvalue of 
$H_{\infty}$. 
Then, if $c$ is large enough, $K$ contains no resonance. 

(ii) Let  $D$ be a compact disc centered at an eigenvalue $E_0$ of 
$H_{\infty }$, of multiplicity $\mu $, and containing no other 
eigenvalue. 
Then, if $c$ is large enough, $D$ contains a finite number of 
resonances, and the sum of their multiplicities is $2\mu $. 
 \end{th}
\begin{th}\label{t3} 
If $\Im \, \theta $ is small enough, we have the following property.
 If $D$ is a disc as in theorem 2,  if $E_0$ is a simple  
eigenvalue of $H_{\infty }$, then $D$ contains, for 
$c$ large enough, one resonance $\lambda (c)$ of 
multiplicity  $2$, and there exists a $C^{\infty}$  function $f$ 
in a neighborhood of $0$ such that $f(0)= E_0$ and, for 
$c$ large enough 
\beq 
 \lambda (c) = f( \frac {1} {c^2})
\eeq 
 \end{th}
This theorem is proved in Section 4. 
Recall that, when $V(x) = {\cal  O}(<x>^{-s})$ $(s>0)$, if $E_0$ is 
an isolated simple eigenvalue of $H_{ \infty }$, 
Grigore-Nenciu-Purice [3] proved that for $c$ large enough, 
$D(c)$ has a double eigenvalue $\lambda (c)$ defined by 
 an equality like (9), but where 
$f$ is analytic. If $V$ is a polynomial, we may think that the 
function $f$ in (9) belongs perhaps in some Gevrey class 
related to the degree of $V$. 

Now, we can study the imaginary 
part of the resonances. 
We consider the following Agmon 
metric $ds^2_c$ in $\R ^3$, depending on $c$ 
(see Wang [17])
\beq 
ds^2_c \ = \frac {1} {c^2} V(x)_+ \, (2c^2 - V(x))_+ \, dx^2
\eeq 
where $x_+=sup(x, 0)$. For each $\varepsilon >0$, 
we condider the ''sea''
\beq 
M(c, \varepsilon ) = 
\{ x\in \R^3, \, V(x)\geq (2-\varepsilon )c^2 \} 
\eeq 
We denote by $S(c, \varepsilon)$ the distance, for the metric 
$ds^2_c$, of the origin to $M(c, \varepsilon)$.   
\begin{th}\label{t4}
Under the hypothesis of Theorem 2 (point ii), for each $\varepsilon 
>0$, there exists $C_{\varepsilon }>0$ such that the resonances 
$E _j(c)$ contained in $D$ satisfy 
\beq 
 | \Im \, E _j(c)  |  \leq C_{\varepsilon }  e^{-(2-\varepsilon)
 S(c, \varepsilon)}
 \eeq
 \end{th}
 

 We are very grateful to X.P. Wang for useful discussions about the 
 exterior scaling, used in section 5. 
 
 \section{Proof of theorem 1.}\label{s2}
 
 We remark first that $D(c)$ is essentially self-adjoint, since we 
 have easily the following implication :
 \beq 
 u\in L^2(\R ^3, \C ^4), 
 \ \ \ \ \ 
 \Im \ z <0, 
 \ \ \ \ \ \ 
 (D(c)-z)u=0
 \Rightarrow 
 u=0
 \eeq 

 Now  $c$ is fixed. 
It can be seen using  Cauchy's estimate that (H1) implies
\beq \label{hv3}
\vert \partial _z^\alpha V(z)\vert
\leq C_{ \alpha }(1+ \vert z\vert)^{k- | \alpha  | }
\hskip 1cm 
\forall  z\in S_\frac{a}{2}
\eeq 

>From the calculus adapted to the harmonic oscillator
straightforward modifications are easily made to obtain a calculus
for global elliptic pseudo-differential operators adapted to first
order systems with a potential behaving like $\vert x\vert^k$. 
Therefore we briefly
give the main aspects. See Shubin[12] for more considerations.

For each  $m\in \R$ let $\Gamma ^m $ be the space of 
$d\in C^\infty(\R^6, M_4(\C))$ such that  
for all $\alpha $ and $\beta\in\N^3$, there exists 
$C_{\alpha \beta }$ such that, for all $(x,\xi)\in\R^6$, 
$$
\vert \partial _x^\alpha \partial _{\xi }^{\beta } d(x,\xi)\vert
\leq C_{\alpha \beta } 
(1+\vert x\vert^k+\vert\xi\vert)^{m-\frac { |\alpha |}{k}  -  |  
\beta  |  }.$$
For each $d\in \Gamma ^m$, let  $Op(d)$ be the corresponding 
operator, associated to $d$ by the standard calculus 
\[ 
(Op (d) \varphi )(x) = 
(2\pi)^{-3}\int e^{i\langle x-y,\xi\rangle}d(y,\xi)\varphi (y)\, 
dyd\xi
\hskip 1cm \forall \varphi \in {\cal S}(\R ^3; \C^4)
\]  

The operator $Op(d)$ ($d\in \Gamma ^m$) 
is said globally elliptic if,  for some
positive real number  $C$ 
\[
(\vert x\vert^k+\vert \xi\vert)^m \leq C (1 + 
\vert {\rm Det } d(x,\xi)\vert )^{1/4}
\]
 for all $( x , \xi) \in \R ^6 $.  

The notation $\langle \cdot,\cdot\rangle$  stands for the 
inner scalar product of
$L^2(\R^3;\C^4)$ and $\Vert\cdot\Vert$ denotes the corresponding 
norm. For $j\in\N$ let
\[
B^j(\R^3;\C^4)=\left\{\phi\in L^2(\R^3;\C^4), \ \ \ \ 
 x^\alpha D_x^\beta\phi\in L^2(\R^3;\C^4),\mbox{ for } 
\frac{\vert\alpha\vert}{k}+\vert\beta\vert\leq j\right\}\] 
In particular, for 
$d\in \Gamma ^m$, $Op(d)$ maps $B^{s-m}(\R^3;\C^4)$ into 
$B^s(\R^3;\C^4)$ for any
$s\in\N$.



It is seen in lemma \ref{l3} 
that for 
small {\it positive} $\Im\, \theta$ the family $H(c,\theta)$ is Kato 
analytic. The resonances are
defined as the eigenvalues of $H(c,\theta)$ for 
small positive $\Im\, \theta$.


\bl\label{l1} There exists $\tau_0>0$ such that if 
$0<\Im\theta<\tau_0$ then
$H(c,\theta)$ is globally elliptic.\el

{\it Proof: }  The symbol of
$H(c,\theta)$ satisfies
\begin{equation}\label{s20}
{\rm Det }\ h(x,\xi,c,\theta)=\left(
V_\theta(x)\left( V_\theta(x)-2c^2\right)-
c^2e^{-2\theta}\vert\xi\vert^2\right)^2.
\end{equation}
 where $V_\theta(x)=V(e^{\theta} x)$.
 We write $\theta=\sigma+i\tau$, $\sigma,\tau\in\R$ and
$K,C,\tau_0$ denotes three positives real numbers
independant of $x$ and $\tau$. The real numbers $K,C$ (resp. 
$\tau_0$) may
increase
(resp. decreases).

Following the analyticity of $V$, there exists $\tau_0>0$ such that 
for
$0<\Im\theta<\tau_0$,
for all $x\in\R^3$,
\[
V_\theta(x)=V(xe^{\sigma})+i\tau e^{\sigma}\sum_{j=1}^3x_j
\frac{\partial V}{\partial x_j}(xe^{\sigma})+
\tau^2M(x,\theta).
\]
There exists $K,C,\tau_0>0$ such that
\begin{equation}\label{s21}
\forall\,\theta\in\C\ {\rm with}\ 0<\Im\theta<\tau_0,\ \forall\,\vert
x\vert\geq C,
\
\vert M(x,\theta)\vert\leq K\vert x\vert^k.
\end{equation}
Then, for some $K,C,\tau_0>0$,
if $0<\Im\theta<\tau_0$,
if $\vert x\vert\geq C$ then
\begin{eqnarray}
\label{s22} K^{-1}\tau\leq {\rm Arg} V_\theta(x),\, {\rm Arg}
(V_\theta(x)-2c^2)\leq
K\tau,\cr\cr
\label{s23} \vert V_\theta(x)\vert,\, \vert V_\theta(x)-2c^2\vert
\geq K^{-1}\vert x\vert^k.
\end{eqnarray}
>From (\ref{s22}), there exist  $K,C,\tau_0>0$ such that for all 
$\theta $ and $x$ such that $0<\Im\theta<\tau_0$ and $\vert 
x\vert\geq C,$
\begin{equation}\label{s24}
K^{-1}\tau\leq {\rm Arg} (V_\theta(x)(V_\theta(x)-2c^2))\leq
K\tau.
\end{equation}
Then (\ref{s24}) shows that for some $K,C,\tau_0>0$ ($\tau _0<\pi/2$),
 if  $0<\Im\theta<\tau_0$,
if $\vert x\vert\geq C$ then
$\vert
V_\theta(x)\left( V_\theta(x)-2c^2\right)-
c^2e^{-2\theta}\vert\xi\vert^2\vert$
\begin{equation}\label{s25}
\geq
\sin(K^{-1}\tau)\vert V_\theta(x)(V_\theta(x)-2c^2)\vert
+c^2\sin(2\tau)\vert\xi\vert^2.
\end{equation}
The proof of Lemma 1 follows from 
(\ref{s20}),(\ref{s23}),(\ref{s25}).\fp
  
 
Theorem 1 will  follow from the two  lemma below.


\bl\label{l2} There exists $\tau_0>0$ such that if 
$0<\Im\theta<\tau_0$ then
the resolvant set of $H(c,\theta)$ is not empty.\el

{\it Proof: } For $m\in\N$,  let $\widetilde \Gamma^m$ 
be the space of 
$a(,\cdot,\cdot,\rho)\in C^\infty(\R^6,M_4(\C))$, depending on a 
parameter $\rho \geq 1$, such  that, fo rall $\alpha $ 
and $\beta \in \N^3$, there exists
$C_{\alpha,\beta}$, independent on $\rho $,  such that, 
for all $(x,\xi,\rho)\in\R^6\times [1, + \infty[$,
$$\vert \partial_x^\alpha\partial_\xi^\beta a(x,\xi,\rho)\vert
\leq C_{\alpha,\beta}
(1+\vert \xi\vert+\vert x\vert^k+\rho)^{m-\frac{\vert
\alpha\vert}{k}-\vert\beta\vert}.$$

The operator  $Op(a(\rho ))$ ($a\in \widetilde \Gamma^m$), is said 
globally elliptic with parameter $\rho$ if
there exists $C>0$ such that, for all $(x,\xi,\rho)\in\R^6\times 
[1, + \infty [$,
$$(\vert\xi\vert+\vert x\vert^k+\rho)^m
\leq
 C(1+\vert{\rm Det }\ a(x,\xi,\rho)\vert)^{1/4}.
$$
As in the proof of Lemma \ref{l1}, $\theta=\sigma+i\tau$, 
$\sigma,\tau\in\R$
and $K,C,\tau_0$ are three positives real numbers
independant of $x$ and $\tau$ which may change. Let $\rho>0$,
$\alpha\in [0,2\pi)$ and set $P=H(c,\theta)+\rho e^{i\alpha}$. The 
symbol
$p(x,\xi,\rho)$ of $P$ (associated with the standard calculus) belongs
to $\widetilde \Gamma^1$. Take $K,C,\tau_0$
such that (\ref{s22}) holds and set $\alpha=K\tau$. There exists
$K,C,\tau_0$ (possibly different) such that
if $0<\Im\theta<\tau_0$,
if $\vert x\vert\geq C$ then
\begin{eqnarray}
\label{s31} K^{-1}\tau\leq {\rm Arg}( V_\theta(x)+\rho 
e^{i\alpha}),\, {\rm
Arg}
(V_\theta(x)+\rho e^{i\alpha}-2c^2)\leq
K\tau,\cr
\label{s32} \vert V_\theta(x)+\rho e^{i\alpha}\vert
\geq\cos(K\tau)(\vert V_\theta(x)\vert+\rho)
\geq K^{-1}(\vert x\vert^k+\rho)\cr
\label{s33} \vert V_\theta(x)+\rho e^{i\alpha}-2c^2\vert
\geq\cos(K\tau)(\vert V_\theta(x)-2c^2\vert+\rho)
\geq K^{-1}(\vert x\vert^k+\rho).
\end{eqnarray}
Then (\ref{s31}) shows that for some $K,C,\tau_0>0$ ($\tau _0<\pi/2$),
 if  $0<\Im\theta<\tau_0$,
if $\vert x\vert\geq C$ then
$
\vert(V_\theta(x)+\rho e^{i\alpha})( V_\theta(x)+\rho 
e^{i\alpha}-2c^2)-
c^2e^{-2\theta}\vert\xi\vert^2\vert
$
\begin{equation}\label{s34}
\geq\sin(K^{-1}\tau)\vert (V_\theta(x)+\rho e^{i\alpha})
(V_\theta(x)+\rho e^{i\alpha}-2c^2)\vert
+c^2\sin(2\tau)\vert\xi\vert^2.
\end{equation}
Following (\ref{s32}),(\ref{s34}), $P$ is globally
elliptic with parameter $\rho$. Then, there are $q(\rho)$
and $r(\rho )$ in $\widetilde \Gamma^{-1}$ such that
\begin{equation}\label{s35}
(H(c,\theta)+\rho e^{i\alpha})Op(q(\rho ))=I+Op(r(\rho )).
\end{equation}
Moreover, $\sup_{\rho \geq 1}\rho \Vert Op(r)\Vert_{{\cal 
L}(L^2(\R^3))}<\infty$.
Thus, the r.h.s. of (\ref{s35}) is invertible for a sufficiently
large $\rho$. This proves Lemma \ref{l2}.\fp


\bl\label{l3} There exists $\tau_0>0$ such that the family of
operators $\{H(c,\theta),\, 0<\Im \theta<\tau_0\}$ is analytic in the
sense of Kato. \el

Let $\tau_0$ be as in Lemma \ref{l1} and set $\theta\in\C$ with
$0<\Im\theta<\tau_0$. The existence of parametrixes
for the global elliptic operator $H(c,\theta)$ shows that
\[\exists\, C>0,\ \forall\,\phi\in B^1,\
\Vert \phi\Vert_{B^1}\leq C\left(\Vert H(c,\theta)\phi\Vert
+\Vert \phi\Vert\right).
\]
It implies that for all $\theta\in\C$ with $0<\Im \theta<\tau_0$,
$H(c,\theta)$ is
closed on $B^1$.

There exists $\tau_0>0$ (with another $\tau_0$), $K>0$ such that
for all $\theta,h\in\C$ satisfying $0<\Im z,\Im
\theta<\tau_0$ and for all $x\in\R^3$,
\begin{eqnarray}
\label{s36}(V(xe^{\theta+h}-V(xe^\theta))/h=e^\theta\sum_{j=1}^3x_j
\frac{\partial V}{\partial x_j}(xe^\theta)+hN(x,\theta,h)\cr
\label{s37}
\vert e^\theta\sum_{j=1}^3x_j
\frac{\partial V}{\partial x_j}(xe^\theta) \vert,
\
\vert N(x,\theta,h) \vert
\leq K\langle x\rangle^k.
\end{eqnarray}
Fix $u,v\in L^2(\R^3,\langle x\rangle^{2k}dx)$ and let $F$ be the map:
$\theta\mapsto \langle V_\theta u,v\rangle_{L^2(\R^3;\C)}$.
>From (\ref{s36}), if $0<\Im\theta<\tau_0$ then
 $(F(\theta+h)-F(\theta))/h$ has a limit as $h\rightarrow 0$
 ($0<\Im h<\tau_0)$. For  each $u\in L^2(\R^3,\langle 
x\rangle^{2k}dx)$,
  $\theta\mapsto V_\theta u$
is a (weakly) analytic vector valued function.
Then, for each $\phi\in B^1$, $H(c,\theta)\phi$ is a vector valued
analytic function of $\theta\in\{z\in\C,\ 0<\Im z<\tau_0\}$.

The above closure and analyticity results, added to Lemma \ref{l2},
imply that $\{H(c,\theta),\, 0<\Im \theta<\tau_0\}$
is an analytic family of type $(A)$ [7, VII.2].\fp






{\it Proof of theorem \ref{t1}: }Using  lemma \ref{l2} there exists
$z\in\C$ such that $(H(c,\theta)-z)^{-1}$ maps $L^2(\R^3;\C^4)$
into $B^1(\R^3;\C^4)$, hence 
is a compact operator of $L^2(\R^3;\C^4)$. Therefore, the spectrum of 
$H(c,\theta)$  is a sequence of  eigenvalues $\lambda _j(c , \theta)$ 
 of finite multiplicity.
 It is clear that $H(c,\theta)
=U(\Re\,\theta)H(c,\Im\,\theta) U(\Re\theta)^{-1}$, that is to say,
$H(c,\theta)$ is unitarily equivalent to $H(c,\Im\,\theta) $. 
Therefore each $\lambda_j(c,\theta)$ does
not depend on $\Re\,\theta$. In addition, lemma \ref{l3} implies that
each $\lambda_j(c,\theta)$ depends analytically on $\theta$ with at 
most,
algebraic singularities.  As [11, pf of th1(i)] it can be proved 
using 
Puiseux series, that each $\lambda_j(c,\theta)$ is a constant function
of $\theta$.  The multiplicity of each of these eigenvalues 
$\lambda _j(c, \theta)$ is even. This can be proved like in 
Parisse [9]. This 
completes the proof.\fp

 
 
 
  \section{Proof of theorem 2.}\label{s3}

By arguments similar to that of section 2, 
the spectrum of the following Schr\"odinger operator 
\beq      
H_{\theta } = - \frac {1}{2} e^{-2\theta } \Delta 
+ V(x e^{\theta })
\eeq 
is discrete, and the eigenvalues  are the same as $H_{\infty }$, 
with the same multiplicities. (The only difference with section 2 
is that the sign of $\Im \ \theta $ plays no role, and there is 
$\tau >0$ such that the family $(H_{\theta })_{ | \Im \theta  |  < 
\tau }$ 
is analytic in the sense of Kato).   If $z$ is not in this spectrum, 
we set 
\beq 
R_{z \theta \infty }= 
\left(\begin{array}{rrrr}
(H_{\theta } -z)^{-1} I_2 &0\cr
0&0\cr
\end{array}\right)
\eeq 
We set also 
\[
B^+ ( \theta _0)= \{ \theta \in \C , \, \, \, 
 |  \theta  |  < 1, \, \, \, \, 0<\Im \theta < \theta _0 \} 
\]
and 
\beq 
V_{\theta }(x)= V (xe^{\theta })
\eeq   
\begin{lem} There exists $\theta _0>0$ and $R>0$ such that, for each 
$\theta \in B^+ (\theta _0) $, there exists 
$A_{ \theta } >0$ such that, if $ | x |  \geq R$  
\beq 
<x> ^k \leq A_{\theta }   \Im \, V_{\theta } (x) 
\hskip 1cm 
<x> ^k \leq A_{\theta }   \Im e^{\theta -  \overline { \theta } } 
 V_{\theta } (x) 
\eeq  
\end{lem} 

{\it Proof. } By the hypotheses (H1) and (H2), we can write, 
if $\theta = \sigma + i \tau \in B^+ (a/2)$ 
\beq 
V_{ \sigma + i \tau }(x) = V_{\sigma }(x) + i \tau e ^{ \sigma } \sum 
_{j=1}^3   x_j \frac { \partial V} {\partial x_j} (xe^{\sigma })
+ {\cal  O} ( \tau ^2  <x>^k ) 
\eeq 
 If  $ | x |  $ is large enough and $0 < \Im \theta < \theta _0$, 
(where $\theta _0$ depends  on the constants of 
hypotheses (H2) and (H3)),  there exists $A_{\theta } >0$ such that 
(27) is valid. \fp    

For each $\varepsilon >0$, we set 
\beq 
\Delta _{\varepsilon }= 
\left(\begin{array}{rrrr}
1&0\cr
0&\varepsilon \cr
\end{array}\right)
\eeq 

The points i) and ii) of Theorem 2 are consequences of the  
points ii) and iii) 
of the following Lemma. 

\begin{lem}
i) Let $K$ be a compact set of $\C $ and  $\theta  $ such that 
$0 < \Im \theta  < \theta _0$. Then there  exists 
$B_{\theta }>0$ (independent of $c$) such that, if $c$ 
is large enough, for each 
$u = \left(\begin{array}{rrrr}
u_1\cr
u_2
\end{array}\right)$ in ${\cal S}(\R^3, \C ^4)$ 
$(u_j \in {\cal S}(\R^3, \C ^2))$, for each $z\in K$ and $c\geq 1$, 
 we have 
\[ 
\Vert   <x>^{k/2}u_1 \Vert   + \Vert   <x>^{-k}\sigma .D u_1 
\Vert   
+  \Vert   u_2 \Vert   \leq \ldots 
\] 
\beq   
\ldots \leq  B_{\theta } \left ( 
\Vert   \Delta _c^{-1} (H(\theta , c)-z) \Delta _c^{-1}u \Vert   + 
 \Vert   u_1 \Vert   \right 
) 
\eeq  
 
 

ii) If $K$ contains no eigenvalue 
of $H_{\infty }$,  
there  exists $A_{\theta }>0$ (independent of $c$) such that, 
if $c$ is large enough  
\beq 
\Vert   u \Vert   _{L^2(\R ^3 , \C ^4)} \leq A_{ \theta }  
\Vert   \Delta _c^{-1} (H(\theta , c)-z)\Delta _c^{-1}u \Vert   
_{L^2(\R ^3 , \C ^4)} 
\eeq 
for all $u\in {\cal S} (\R ^3, \C ^4)$, and 
$z\in K$, and 
therefore,
\beq 
\Vert   u \Vert   _{L^2(\R ^3 , \C ^4)} \leq A_{ \theta }  
\Vert   (H(\theta , c)-z)u \Vert   _{L^2(\R ^3 , \C ^4)} 
\eeq 


iii) If $D$ is a disc centered at an eigenvalue of $H_{\infty }$, and 
containing no other eigenvalue, then, if 
$0 < \Im  \theta < \theta _0$, 
\beq 
\lim _{ c \rightarrow + \infty } 
 \sup _{z \in \partial D} \Vert   (H(c , \theta)-z)^{-1} 
 - R_{z \theta  \infty} \Vert   = 0
 \eeq   
\end{lem}

{\it Proof of point i).} 
The equality    
$\Delta _c^{-1} (H(\theta , c)-z) \Delta _c^{-1}u
= \left(\begin{array}{rrrr}
f\cr
g
\end{array}\right)$ is equivalent to  
\beq 
f= (V_{\theta } -z) u_1 + e^{ -\theta }\sigma . D u_2 
\eeq 
\beq 
g= e^{ - \theta } \sigma . D u_1 + \left ( \frac {V_{\theta } -z} 
{c^2} -2 \right ) u_2  
\eeq 
It follows from the two last equalities that 
\beq 
\langle u_1  , e^{ \theta - \overline { \theta }} (V_{\theta }-z)  
u_1  
\rangle 
 \, - \, \langle \left ( 
 \frac {V_{\theta }-z}{c^2}-2
 \right )  
  u_2 , u_2 \rangle 
 \,
 \, = \, \langle u_1  , e^{\theta - \overline { \theta } } f \rangle 
 \, - \, \langle g , u_2 \rangle 
\eeq 
and therefore, taking the imaginary parts in the last equality 
and applying Lemma 4,  
\[ 
\Vert  < x> ^{ k/2} u_1 \Vert   ^2 + c^{ -2} \Vert   <x>^{ k/2} u_2 
\Vert   ^2 
\leq \ldots 
\] 
\beq 
\ldots \leq B_{ \theta } \left [ \Vert  f \Vert ^2 +   
\Vert   u_1 \Vert ^ 2   
+ \Vert   g \Vert   \Vert   u_2 \Vert   + c^{ -2} \Vert   u_2 \Vert   
^2   \right ] 
\eeq 
Taking now the real parts in (36), we obtain, with another $B_{ 
\theta 
}$
\[ 
\Vert   u_2 \Vert   ^2 \leq B_{ \theta } 
\left [ \Vert  f \Vert ^2 +   
\Vert   u_1 \Vert ^ 2   
+ \Vert   g \Vert   \Vert   u_2 \Vert   + c^{ -2} \Vert   u_2 \Vert   
^2   \right ] 
\] 
The inequality (30) (with another $B_{ \theta }$) follows easily, if 
$c$ 
is large enough, from the two last ones.  
 


{\it Proof of point ii).} 
Suppose that the inequality () were false. Then there would exist 
 a sequence $(u_n)$ in ${\cal  S}(\R ^3, \C ^4)$, a sequence 
$(z_n)$ in $K$, and a sequence $c_n \rightarrow +\infty $ 
such that 
\beq 
\Vert   u_n \Vert   = 1 
\hskip 1cm 
\Vert  \Delta _{c_n}^{-1} (H(c_n , \theta )-z_n) 
\Delta _{c_n}^{-1}u_n \Vert   \rightarrow  0
\eeq 
Taking a subsequence, we can assume that $z_n \rightarrow  z\in K$. 
Let us set $u_n = \left(\begin{array}{rrrr}
\varphi _n\cr
\psi _n
\end{array}\right)$ and 
$\Delta _{c_n}^{-1}H(c_n, \theta)\Delta _{c_n}^{-1} u_n = 
\left(\begin{array}{rrrr}
f _n\cr
g _n
\end{array}\right)$. If we set $V_{\theta }(x)= V(xe^{\theta })$, 
we have the relations (34) and (35) with $f$, $u_1$, $u_2$ replaced 
by 
$f_n$, $\varphi _n$, $\psi _n$. By (30) the sequences  
$<x>^{k/2}\varphi _n$  
and  $<x>^{-k} \sigma . D \varphi _n$  are bounded in 
$L^2(\R ^3, \C ^2)$.
 By these properties, we may assume (after taking subsequences) that 
there 
exist $\varphi $ and $\psi $ in $L^2( {\bf R} ^3, \C ^2)$ such that 
$\varphi _n \rightarrow  \varphi $ (strongly) and 
$ \psi _n \rightarrow  \psi $ (weakly) in $L^2( {\bf R} ^3, \C ^2)$. 
We have 
\beq 
 (V_{\theta } -z)\varphi   + e^{-\theta } \sigma .D \psi =0
\eeq 
and 
\beq 
 e^{-\theta } \sigma .D \varphi   -2  \psi  =0
\eeq 
and therefore $(H_{\theta }-z)\varphi =0$.  If $\varphi = 0$, 
it follows that $\Vert  \varphi _n\Vert  \rightarrow 0$, and, since 
$\Vert  f_n\Vert   + \Vert   g_n\Vert   \rightarrow 0$, the 
point i)  shows that $\Vert   \psi _n\Vert   \rightarrow 0$, and 
this 
gives a contradiction since $\Vert   \varphi _n\Vert   ^2 + \Vert  
\psi _n\Vert   ^2 = 1$. Therefore, there exists $\varphi \not=  0$ in 
$L^2 (\R ^3, \C ^2 )$ such 
that $(H_{ \theta } -z) \varphi = 0$, and there is a contradiction 
since $z\in K$ and $K$ contains no eigenvalue of $H_{ \theta }$. 
The inequality (31) is proved, and (32) follows easily. 



{\it Proof of point iii)} 
Suppose that there exist 
$\theta $ such that $0< \Im  \theta < \theta _0$,  a 
sequence $(F_n)$ in $L^2(\R ^3, \C ^4)$, a sequence 
$(z_n)$ in $\partial D$,  a sequence $c_n \rightarrow +\infty $, 
and $\delta >0$  such that 
\beq 
\Vert   F_n \Vert   = 1 
\hskip 1cm 
\Vert  (H(c_n , \theta )-z_n)^{-1}F_n  - R_{z_n \theta \infty } 
F_n \Vert   \geq \delta 
\eeq 
Let us set 
\beq 
F_n = \left(\begin{array}{rrrr}
f _n\cr
g _n
\end{array}\right)
\hskip 1cm 
U_n= (H(c_n  , \theta )-z_n )^{-1} F_n = \left(\begin{array}{rrrr}
\varphi _n\cr
\psi _n
\end{array}\right)
\eeq 
By the point ii) (applied to the compact $\partial D$), the sequence 
$\Vert   U_n\Vert  $ is bounded. By the point i) (applied to the 
function $\Delta _{c_n} U_n$), we have 
\beq 
\Vert  <x>^{k/2} \varphi _n \Vert   + 
\Vert   <x>^{-k} \sigma . D \varphi _n \Vert   + 
c_n \Vert   \psi _n \Vert   \leq B_{ \theta } [ 
\Vert   F_n\Vert   + \Vert   \varphi _n\Vert   ]= {\cal  O}(1)
\eeq 
Therefore $\Vert   \psi _n \Vert   \rightarrow 0$, which implies, 
together with (41), that, for $n$ large enough 
\beq 
\Vert   \varphi _n - (H( \theta)-z_n)^{-1} f_n \Vert   \geq 
\frac { \delta } {2} 
\eeq 
By (43), we may assume, (after taking a subsequence), that there 
exist 
$\varphi $ and $\psi $ in $L^2( {\bf R} ^3, \C ^2)$ such that 
$\varphi _n \rightarrow  \varphi $ (strongly) and 
$c_n \psi _n \rightarrow  \psi $ (weakly) in $L^2( {\bf R} ^3, \C 
^2)$.  
We may assume 
also that $z_n \rightarrow  z \in  \partial D$ and that $f_n $ 
 weakly converges to $f \in L^2(\R ^3, \C ^4)$. 
 It follows that 
 \[ (V_{\theta }(x)-z)\varphi + e^{-\theta } \sigma . D \psi = f 
 \] 
 \[ e^{-\theta } \sigma . D \varphi - 2 \psi = 0
 \] 
 and therefore $ (H_{\theta } -z )\varphi = f$. 
 Since the 
 operator $(H_{\theta }-z)^{-1}$ is compact, we may assume also that
 \beq 
(H _{\theta } -z_n)^{-1} f_n 
 \rightarrow  
\widetilde \varphi \in L^2(\R ^3, \C ^2) 
\eeq 
(strong convergence). We have $ (H_{\theta } -z) \widetilde \varphi 
= (H_{\theta }-z) \varphi = f$, and there is a contradiction with 
(44) 
since $\Vert   \varphi - \widetilde \varphi \Vert   \geq \delta / 2$ 
and $z$ is not in the spectrum of $H_{\theta }$. 

{\it Proof of of theorem 2.} 
The point i) is a consequence of Lemma 5 (point ii). 
For the point ii), let $E_0$ be a simple eigenvalue of $H_{ \infty 
}$. Let $D$ 
be a disc, centered at $E_0$, with radius $\rho >0$, containing no 
other eigenvalue of $H_{ \infty }$ inside it, and $\Gamma $ be the 
boundary of $D$. By the point i), we know 
that, for $c$ large enough, $H(\theta , c)-z$ is invertible for all 
$z \in \Gamma $. We define then an operator 
$\Pi _{ \theta c} $ by 
\beq 
\Pi _{ \theta c } = \frac {1} {2i\pi } 
\int _{ \Gamma } (H(\theta, c)-z)^{-1} dz 
\eeq 
Similarly we define $\Pi _{ \theta \infty }$ by 
\beq 
\Pi _{ \theta \infty } = \frac {1} {2i\pi } 
\int _{ \Gamma } R_{z \theta \infty }  dz 
\eeq 
where $R_{ z \theta \infty }$ is defined in (25). 
It follows from Lemma 5 (point iii) that 
\beq 
\lim _{c \rightarrow + \infty } 
\Vert   \Pi _{ \theta c } - \Pi _{ \theta \infty} \Vert  = 0
\eeq
The point ii) follows easily.  \fp  
 
 

 \section{Proof of theorem 3.}\label{s4}

If $D= B(E_0, \rho )$ is a disc like in the theorems 2 and 3, and if 
$E_0$ is a simple eigenvalue of $H_{ \infty }$, we know, by 
 theorem 2, that, for $c$ large enough, $H(\theta, c)$ has 
only one eigenvalue $\lambda (c)$ of multiplicity $2$ in 
$B(E_0, \rho )$. Since $E_0$ is also a simple eigenvalue of 
the dilated Schr\"odinger operator $H_{\theta }$ defined in (24) 
(section 3), let $ \varphi _{\theta } $ be a normalized eigenvector 
($H_{\theta } \varphi _{ \theta } = E_0 \varphi _{ \theta }$, 
$\Vert   \varphi _{ \theta } \Vert   =1$). By the 
global ellipticity of $H_{ \theta }$, we know that 
$\varphi _{ \theta }$  is in ${\cal  S}(\R ^3)$. Let  
\beq 
\psi _{ \theta }  = \left(\begin{array}{rrrr}
\varphi _{\theta }\cr
0 \cr 
0 \cr 
0 
\end{array}\right)
\eeq 
If $\Pi _{\theta c } $ is defined in (46), (where $\Gamma $ is the 
boundary of $D$), 
$\Pi _{ \theta c } \psi _{\theta }$ is 
in the 
eigenspace of $H(\theta, c)$ corresponding to the eigenvalue 
$\lambda (c)$ and, 
by (48), if $c$ is large enough,  
$ \Pi _{ \theta c } \psi _{\theta } \not=  0$. Therefore 
\beq
\lambda (c) = \frac 
 {(H(\theta, c) \Pi _{\theta  c} \psi _{\theta }, \Pi _{\theta c} 
\psi 
 _{\theta })} 
 { \Vert   \Pi _{\theta c} \psi _{\theta }\Vert   ^2} 
\hskip 1cm 
E_0 = \frac {(H_{\theta } \Pi _{ \theta \infty } \psi _{ \theta } , 
\Pi _{ \theta \infty } \psi _{ \theta } )}
{\Vert   \Pi _{ \theta \infty } \psi _{ \theta }\Vert   ^2 }
\eeq  
(since $ \Pi _{ \theta \infty } \psi _{ \theta } = \psi _{ 
\theta } $).    

\begin{lem}
Let $\psi $ be a function in ${\cal S}( \R ^3, \C ^4 )$ and $\Gamma $ 
be the boundary of $D= B(E_0, \rho)$. Let $F(\varepsilon , z)$ 
be the function defined, 
for $\varepsilon $ small enough and $z\in \Gamma $ by 
\beq 
F(\varepsilon , z) = (H(\theta , 1/\varepsilon ) -z)^{-1}\psi 
\hskip 1cm if \ \ \ \varepsilon \not=  0
\eeq 

\beq 
F(\varepsilon , z) = R_{z \theta \infty } \psi 
\hskip 1cm if \ \ \ \varepsilon =  0
\eeq 

where $R_{z \theta \infty }$ is defined in (25). Then 
$\varepsilon \rightarrow F(\varepsilon , z)$ is $ C^{ \infty }  $ 
from 
some neighborhood of $0$ to ${\cal  H} = L^2( {\R } ^3, {\C }^4)$, 
and 
depends 
continuously of $z$ in $\Gamma $. 
\end{lem}

{\it Proof.} 
If $\Delta _{ \varepsilon } $ is the operator defined in (29), 
we can write, by (34) and (35) 
\beq \Delta _{c}^{-1} (H(\theta, c)-z) \Delta _c^{-1} = 
A + c^{-2} B  \eeq 
where 
$$
A = \left (
\begin{array}{cc}
V_{ \theta } -z&e^{ -\theta } \sigma . D\cr
e^{ -\theta } \sigma . D&-2 
\end{array} \right )
\hskip 1cm 
B= \left (
\begin{array}{cc}
0&0\cr
0&V_{ \theta } -z 
\end{array} \right )
$$
By Lemma 5, there is $t_0>0$ such that  $A+tB: B^1 \rightarrow 
{\cal  H}$ is invertible if $0<t\leq t_0$, 
and there exists $K>0$ such that 
 \beq \Vert   (A + t B)^{-1}f \Vert   \leq K \Vert  f\Vert   
\hskip 1cm 
0< t \leq t_0
\ \ \ \ \ \ 
\forall f \in {\cal  H} \eeq 
Moreover, if we set $(A+tB)^{-1}f = \left(\begin{array}{rrrr}
u(t)\cr
v(t)
\end{array}\right)$, we have, by Lemma 5  
$$\Vert   <x>^{k/2} u(t)\Vert   + 
\Vert   <x>^{-k}\sigma . D u(t) \Vert   + \Vert  v(t)\Vert   \leq 
\ldots $$
$$ \ldots \leq K ( \Vert  f\Vert   + \Vert  u(t)\Vert   )
\hskip 1cm 
0< t \leq t_0
\ \ \ \ \ \ 
\forall f \in {\cal  H}$$
 In the other hand,  if 
$H_{ \theta }$ is the operator defined in (24), and $z\in \Gamma $, 
the operators $D^{\alpha } (H_{\theta } -z)^{-1} D^{ \beta }$ are 
bounded in $L^2(\R ^3)$ if $ | \alpha + \beta  |  \leq 2$ 
(we construct easily a parametrix of this 
operator in a suitable class). Therefore, the following operator $S$ 
is bounded in ${\cal  H}$ 
$$
S= \left (
\begin{array}{cc}
(H_{\theta }-z) ^{-1} & 
\frac {e^{- \theta } }{2} (H_{\theta }-z) ^{-1}\sigma . D \cr
\frac {e^{- \theta } }{2} \sigma . D(H_{\theta }-z) ^{-1} 
&\frac {e^{- 2 \theta } }{4} \sigma . D(H_{\theta }-z) ^{-1}\sigma . 
D 
- \frac {I} {2} 
\end{array} \right )
$$
and it satisfies $A S = I$. Moreover $u  \in {\cal  H}$ and 
$(A+ tB) u=0$ imply $u= 0$ $(0\leq t \leq t_0)$. It follows easily 
from 
these properties that, if $f\in {\cal  H}$,  the function $G(t)f$ 
defined by 
\beq G(t)f = (A + tB)^{-1}f
\ \ \ \ \ \  
\mbox {   if   } \ \ \ 0<t\leq t_0
\hskip 1cm 
G(0)f= Sf  \eeq  
is continuous in $[0, t_0]$ to ${\cal  H}$. 
Let $E$ be the space of $f \in {\cal  H}$ such 
that, for each $m$, $<x>^m u$ is in ${\cal  H}$. 
Using the commutation relation 
$$
x_j (A+tB)^{-1} = (A+tB)^{-1} x_j -i e^{-\theta } (A+tB)^{-1} 
\alpha _j (A+tB)^{-1} $$
where $\alpha _j= \left ( \begin{array}{cc}
0 & \sigma _j \cr
\sigma _j &0
\end{array} \right )$, it follows that, for each integer $m$, there 
is 
$K_m$ such that  
$$\Vert  <x>^m (A + t B)^{-1} f \Vert   \leq K_m 
\Vert   <x>^m f \Vert   
\hskip 1cm 
\forall f\in E 
\ \ \ \ \ 0\leq t \leq t_0$$
and that, for each $f\in E$, the function $<x>^m G(t)f$ is continuous 
in $[0, t_0]$ to ${\cal  H}$. 
It follows  that, for each $f\in E$, the function $G(t)f$  
is $ C^{ \infty }  $ on $[0, t_0]$ to ${\cal  H}$, and that 
\beq G^{(p)}(t)f= (-1)^p (A+tB)^{-1} \left ( 
B (A+tB)^{-1} \right )^p 
\ \ \ \ \  
\mbox { if  } \ \ \  0< t \leq t_0 \eeq 
and $G^{(p)}(0)f = (-1)^p S (BS )^pf$. This property can be proved, 
by 
induction on $p$, using the previous remarks. The Lemma follows 
easily since $F(\varepsilon , z) = \Delta _{\varepsilon } 
G(\varepsilon ^2) \Delta _{\varepsilon } \psi$. 






{\it Proof of Theorem 3.} Since $\psi _{ \theta }$ defined in (49) is 
in ${\cal  S}(\R^3, \C ^4)$, (this can be proved by using a 
parametrix of $H_{\theta }$), it follows from 
(50) and Lemma 6 
that the function $g$ defined 
in some neighborhood of $0$ by 
\beq 
g(\varepsilon ) = \lambda (1/ \varepsilon ) \hskip 1cm if 
\ \ \ \varepsilon \not=  0
\eeq 
\beq 
g(0) = E_0 
\eeq 
is $ C^{ \infty }  $. We remark that 
\beq 
J H(\theta , c) J = D_{\theta , -c}
\hskip 1cm 
J= \left(
\begin{array}{cc}
I&0\cr
0&-I
\end{array}\right)
\eeq  
Since $\psi _{ \theta }$ defined in (49) satisfies $J \psi _{ \theta 
} 
= \psi _{ \theta }$, it follows that $g$ is an even function of 
$\varepsilon $, and there exists a $ C^{ \infty }  $ function 
$f$ in a neighborhood of $0$ such that 
$g(\varepsilon ) = f(\varepsilon ^2)$, which proves theorem 
3.                                                                                                                                                                                                                                                                                     










\section{Imaginary part of the resonances.}\label{s5}
 In this section, we need another definition of the resonances, 
 using the exterior scaling. We are very grateful to X.P. Wang for 
 this suggestion. For each $\varepsilon >0$ and $c>1$, we have to 
 introduce two auxiliary Hamiltonians : one of them (denoted by 
 $D(\theta , c)$) is obtained from $D(c)$ by an exterior complex 
scaling (cf. Hunziker [6]), and 
 the other one, denoted by $H_0(c)$, is obtained from 
 $D(c)$ by a modification of the potential (cf. Wang [17] and Parisse 
[9]). 
 
 For the construction of the distorted operator $D(\theta , c)$, we 
 use, for each $\varepsilon \in ]0, 1[$,  a  function 
 $\varphi \in C^{ \infty }(\R)$  such that $\varphi (t)= 0$ if 
 $t\leq 2 - \frac {\varepsilon} {2}$ and $\varphi (t)=1$ if $t\geq 
2$. 
  For each $\theta \in \C$ and $x\in \R ^3$, we set 
 \beq 
 \varphi _{\theta } (x)= x + \theta X_c (x)
 \hskip 1cm 
 X_c(x) = x  \varphi \left ( \frac {V(x)} {c^2} \right ) 
 \eeq 
 If $ | \theta  | $ is small enough, we can define  
  a system $p_{\theta } = 
  (p_{\theta , 1},   p_{\theta ,2},   
  p_{\theta ,3} )$ of differential operators by 
  \beq 
  p_{ \theta } = ^t (\varphi ' _{\theta }(x)) ^{-1} D_x - 
  \frac {i} {2} \nabla (\ln J_{\theta }(x))
  \hskip 1cm 
   J_{ \theta }(x)= {\rm det} \, \varphi ' _{\theta } (x)
  \eeq 
  and a distorted Dirac operator $D(\theta , c)$ by 
  \beq 
   D(\theta , c) = 
\left(
\begin{array}{cc}
V( \varphi _{\theta }(x))&c\sigma \cdot p_{\theta }\cr
c\sigma \cdot p_{\theta } & V( \varphi _{\theta }(x))-2c^2
\end{array}\right)
\eeq 
 
\begin{prop} 
With the previous notations, if $ | \theta  | $ 
is small enough, if $D$ is a disc as in theorem 2 (point ii), and if 
$c$ is large enough, the spectrum of $D(\theta , c)$ in 
$D$ is the 
same sequence of eigenvalues $E_j(c)$ as for the operator $H(\theta , 
c)$ defined in (7), with the same multiplicities. 
\end{prop}  

  
For the proof of this proposition, we shall use the following Lemma. 


 \begin{lem} There exist $A>0$ and $\theta _0>0$ with the following 
 properties. If $z \in \C $, $\Im \ z <0$, $c\geq 1$, if 
 $\theta \in \Omega $,  where 
 \beq  
 \Omega = \{ \theta\in \C , \ \ \ 
  | \theta  |  < \theta _0, 
  \ \ \ \ \ 
 0 < \Im \ \theta < \frac 
 {  | \Im \ z  |  } 
 {A (c^2 +  | Re \ z |)  } 
 \}
 \eeq  
 then 
 $z- H(\theta , c) : B^1 \rightarrow {\cal H}= L^2 (\R ^3, \C ^4)$ 
 is invertible and 
 \beq 
 \Vert   (z- H(\theta , c))^{-1} \Vert   _{{\cal L}({\cal H})}
  \leq  \frac {A} { |  \Im \ z | }
 \eeq 
 Moreover, for each $f\in {\cal  H}$, the function 
 $\theta \rightarrow (z- H(\theta, c))^{-1}f$ 
 $(\theta \in \Omega)$, extended by $(z-D(c))^{-1}f$ for real $\theta 
$, 
 is holomorphic in $\Omega $ 
 and weakly continuous in $\overline { \Omega }$. 
 \end{lem}
 
 
{\it Proof of the Lemma.} If we set 
$u= \left(\begin{array}{rrrr}
u_1\cr
u_2 
\end{array}\right)$, the equality $(H(\theta, c)-z)u= f$ implies 
\[ 
\Im \left  [ e^{\theta} \langle f, u \rangle  \right ] = 
\Im \left [ e^{ \theta } \int V_{ \theta } (x) 
 | u(x) | ^2 dx \right ] - 
 \Im \left ( e^{ \theta } z \right ) \Vert   u\Vert ^2 
 - 2 c^2  \Im \left ( e^{ \theta }  \right ) \Vert  u_2 \Vert   ^2 
\]
By the hypotheses on the potential $V$, there exist $R$, $A$ and 
$\varepsilon _0$, independent on all the parameters, such that 
\[ 
\Im \theta  <x>^k \leq A \Im \left [ e^{ \theta } V_{\theta }(x) 
\right ] 
\hskip 1cm 
if \, \, \, \, \, 
 | \theta  |  \leq  1, 
 \, \, \, 
 0< \Im \theta < \varepsilon_0, 
 \, \, \, \, 
  | x |  \geq R
\] 
and 
\[ 
 | \Im \left ( e^{ \theta } V_{\theta }(x) \right )  |  
 \leq A   \Im \theta 
 \hskip 1cm 
if \, \, \, \, \, 
 | \theta  |  \leq 1, 
 \, \, \, 
 0< \Im \theta < \varepsilon_0, 
 \, \, \, \, 
  | x |  \leq  R
\]  
It follows that,  with other constants $A$ and 
$\varepsilon _0$, if $\Im z <0$, $ | \theta  |  < 1$, 
$0< \Im \theta < \varepsilon _0$, and if 
$(D(\theta, c)-z)u=f$, we have   
\[ 
 | \Im z  |  \Vert   u\Vert   ^2 \leq 
 A \left [ \Vert  f\Vert   \Vert   u\Vert   + 
  |  \Im \theta  |  (c^2 +  |  Re \ z |  ) 
  \Vert   u\Vert   ^2 \right ] 
\] 
If moreover $0 \leq \Im \theta \leq  | \Im z  |  / 
(2A (c^2 +  | Re \ z |  ))$, then 
\beq 
\Vert  u \Vert _{ {\cal  H}}  
\leq \frac {2A} {  |  \Im \ z |  }
\Vert   (z- H(\theta , c)) u \Vert   _{{\cal  H}}
\eeq 
By the results of  section 2, it follows that, for each 
$\theta \in \Omega $ (with another $A$), 
$z- H(\theta , c) : B^1 \rightarrow {\cal  H}$ is invertible and that 
the inverse depends holomorphically on $\theta $ in $\Omega $. 
The result about weak continuity follows from (64), 
using the implication (13). \fp 


{\it End of the proof of the Proposition.}
Once the lemma 7 is established, the proof of Proposition 1 follows 
 the classical proof of the Aguilar-Balslev-Combes theorem [1]  (see
 Hislop-Sigal [5] or Laguel [8] for more details).  
For real $\theta $, small enough, we define an operator 
$U_{\theta } : {\cal H} \rightarrow {\cal H}$ by 
\beq 
(U_{\theta } f) (x) = e^{ 3 \theta /2} f(xe^{\theta })
\eeq 
and  an  operator 
$\widetilde U_{\theta }  : {\cal H } \rightarrow {\cal H}$ by 
 \beq 
 (\widetilde U_{\theta }f)(x)= J_{\theta }(x)^{1/2}  f(\varphi 
_{\theta } (x))
 \hskip 1cm 
  \eeq
Then   $ U_{\theta }$  and $\widetilde U_{\theta }$  are unitary, and 
we have 
\beq 
H(\theta , c) =   U_{\theta } D(c)  U_{\theta } ^{-1}
\hskip 1cm 
D(\theta , c) =  \widetilde U_{\theta } D(c) \widetilde U_{\theta } 
^{-1}
 \eeq 
 There exists a  subspace ${\cal  A}$ in 
 ${\cal  H}= L^2 (\R ^3, \C ^4)$ and $\theta _0>0$ such that, 
 for each $f \in {\cal  A}$, the functions 
 $\theta \rightarrow  U_{ \theta }f$ and 
  $\theta \rightarrow  \widetilde U_{ \theta }f$ extend to  
  holomorphic functions from $B(0, \theta _0)$ to ${\cal  H}$, 
  and such that, for each $\theta \in B(0, \theta _0)$, 
  $U_{\theta } {\cal  A}$ and $\widetilde U_{\theta } {\cal  A}$
  are dense in ${\cal  H}$. 
If $f,g\in {\cal A}$,  
$ | \theta  | < \theta _0$ and $\Im \theta >0$, we set 
\beq 
F_{fg}(z, \theta) = < U_{\overline {\theta }}f, \, 
(z-H(\theta , c))^{-1} U_{\theta }g>
\eeq 
\beq 
\widetilde F_{fg}(z, \theta) = < \widetilde U_{\overline {\theta }}f, 
\, 
(z-D(\theta , c))^{-1} \widetilde U_{\theta }g>
 \eeq 
 By the results of section 2 and their analogous for $D(\theta, c)$, 
 we know that, if $c\geq 1$, these functions of $z$ are meromorphic 
in 
 $D$. Let $A$ and $\theta _0$ be the constants of Lemma 7. There is 
an 
 analogous of Lemma 7 with $H(\theta, c)$ replaced by $D(\theta, c)$, 
 and we may assume that the constants $A$ and $\theta _0$ are the 
same. 
 If $E_0$ is the center of $D$ and $\rho $ its radius, let 
 \[ 
 \omega = \{ \theta \in \C, \ \ \ \ \ 
  | \theta |  < \theta _0, \ \ \ \ 
  0< \Im\ \theta < \frac { \rho } {2A(c^2 +  | E_0 | + \rho )} \} 
 \]
By Lemma 7, if $z\in D$ and $\Im z < - \frac {\rho } {2}$, the 
functions $\theta \rightarrow  F_{fg}(z, \theta) $ 
and $\theta \rightarrow \widetilde F_{fg}(z, \theta) $ are 
holomorphic in $\omega $ and continuous in $\overline {\omega }$. 
By (68), they are equal  in  $\overline {\omega } \cap \R$, and 
therefore they are equal in $\omega $. Now, if $\theta \in \omega $, 
the functions $z \rightarrow F_{fg}(z, \theta) $ and 
$z \rightarrow \widetilde F_{fg}(z, \theta) $ are meromorphic in 
$D$ and equal in $\{ z\in D, \ \ \ \Im z < - \frac {\rho } {2} \} $, 
and 
therefore they are equal on $D$.  
A point $z_0\in D$ is an eigenvalue of $H(\theta , c)$ 
(resp. of $D(\theta , c)$) iff there are $f$ and $g\in {\cal A}$ 
such that $z_0$ is a pole of $z \rightarrow F_{fg}(z, \theta)$ 
(resp. of $z \rightarrow \widetilde F_{fg}(z, \theta)$). Therefore, 
these eigenvalues are the same. \fp 

Therefore, under the hypotheses of theorem 2, if $D$ is a disc 
centered at $E_0$, of radius $\rho $, and containing no other 
eigenvalue of $H_{ \infty}$, if $E_j(c)$ $(1\leq j \leq 2\mu )$ are 
the resonances in $D$, there exists an orthonormal system of 
functions $\psi _j$ in 
$L^2(\R ^3, \C ^4)$ $(1\leq j \leq 2\mu)$, such that, if $c$ is large 
enough,  
\beq 
D(\theta , c) \psi _j = E_j(c) \psi _j
\eeq 

Now we shall define a modified real-valued potential, like in 
Wang [17] and Parisse [9] in the semiclassical study of multiple 
wells 
or resonances for the Dirac operator. 
For that, we can choose a function 
$\psi  \in  C^{ \infty }  (\R )$, nondecreasing, such that $\psi (t)= 
t$ 
if $t \leq 2 - \frac { \varepsilon } { 2}$, $\psi (t)\leq t$ for all 
$t$, and $\psi (t)= 2 - \frac { \varepsilon } { 4 }$ if $t \geq 2$. 
Using 
this function, we define a modified potential $V_0$ (depending on 
$\varepsilon $ and $c$) by 
\beq 
V_0(x) = c^2 \psi  \left ( \frac {V(x)} { c^2} \right ) 
\eeq 
 Let $d(x,  V_0, c)$ be the distance from $x\in \R ^3$ 
to the origin for the Agmon metric defined as in 
section 1, but with the potential $V_0$ instead of $V$. We set 
\beq 
\Sigma (c, \varepsilon ) = 
\inf _{V(x)\geq (2- \frac {\varepsilon } {2})c^2 } d(x, V_0, c)
\eeq 

\begin{lem}If $\varepsilon < 1/2$, there exists $K_{\varepsilon } >0$ 
such that 
$$V(x) \geq \frac {3} {2} c^2 \Rightarrow 
c \leq K_{\varepsilon } d(x, V_0, c)\leqno i)$$  
$$<x> \leq K_{\varepsilon } (1 + d(x, V_0, c))
\hskip 1cm 
\forall x \in \R^3 \leqno ii)$$ 
$$S(c, \varepsilon )\leq \Sigma (c, \varepsilon)\leqno iii)$$ 
\end{lem} 

{\it Proof.} Let $x\in \R ^3$, and $t\rightarrow x(t)$ be a 
$C^1$ curve such that $x(0)=0$ and $x(1)=x$. Suppose that 
$V(x) \geq (3/2)c^2$. Let $t_0$ and $t_1$ such that 
\[ 0<t_0<t_1<1
\hskip 1cm 
V(x(t_0))= \frac {1}{2}c^2 
\hskip 1cm  
V(x(t_1))= c^2
\] 
and 
\[   
\frac {1}{2} c^2 \leq V(x(t)) \leq c^2 
\hskip 1cm
\forall t\in [t_0, t_1] 
\] 
For each $t\in [t_0, t_1]$, we have $V_0(x(t))\geq \frac {1}{2}c^2$ 
and $2c^2 - V_0(x(t))\geq \frac {\varepsilon } {4} c^2$, and 
therefore 
\[ 
\frac {1}{c} \int _0^1 \left [ V_0(x(t)_+ (2c^2 - V_0(x(t)) \right 
]^{1/2} 
 | x'(t) |  dt 
 \geq 
 \frac {c \sqrt {\varepsilon } } {4}  | x(t_1)-x(t_0) |  
 \] 
 By the hypotheses on the potential $V$, there exists $K>0$ 
 and $K'>0$ such that, if $c$ is large enough,  
 \[ 
 \frac {1}{2} c^2 \leq  | V(x(t_0))-V(x(t_1)) |  \leq 
 K  | x(t_0)-x(t_1) |  \left [ 
 <x(t_0)> + < x(t_1)> \right ]^{k-1} 
 \] 
 \[ 
 \ldots \leq K' | x(t_0)-x(t_1) |  V(x(t_1))^{(k-1)/k} 
 \leq K' | x(t_0)-x(t_1) |  c^{2 - 2/k} 
 \] 
 The point i) follows from the last inequalities. For the point ii), 
 we can find $R>0$ such that $V_0(x) \geq 1$ if $ | x | \geq R$.  
 If $ | x | \geq R$ and  if $x(t)$ is a curve as above,  
 there exists $t_0\in [0, 1]$ such that 
 $ | x(t_0) |  \leq R$ and $ | x(t) | \geq R $ if $t\in [t_0, 1]$. 
 It follows that 
 \[ 
 \frac {1}{c} \int _0^1 \left [ V_0(x(t)_+ (2c^2 - V_0(x(t)) 
 \right ]^{1/2}  
 | x'(t) |  dt 
 \geq 
\frac {\varepsilon } {2}  | x-x(t_0 ) |  
\] 
and therefore $ | x |  \leq R + \frac {2} {\varepsilon } d(x, V_0, 
c)$. The proof of the point iii) is straightforward. \fp 



We denote by $H_0(c)$ the  modified Hamiltonian corresponding to the 
modified potential $V_0$ 
\beq 
H_0(c)=
\left(
\begin{array}{cc}
V_0( x)&c\sigma \cdot D_x\cr
c\sigma\cdot D_x& V_0( x)-2c^2
\end{array}\right),
\eeq
We see easily  that $H_0(c)$ is essentially self-adjoint and, 
using the arduments of section 3, we see that, if $D$ is a 
neighborhood of $E_0$ like in the theorem 2 (point ii), $D\cap \R $ 
contains, for $c$ large 
enough, $2\mu $ eigenvalues $\lambda  _j(c)$ $(1 \leq j \leq  2\mu )$ 
of $H_0(c)$ (if they are repeated according to their multiplicities). 
Let $\varphi _j= \varphi _j(c)$ $(1\leq j\leq 2\mu )$ be 
 an orthonormal system of corresponding 
eigenfunctions. 
\beq 
H_0(c) \varphi _j = \lambda _j(c) \varphi _j 
\hskip 1cm 
\Vert   \varphi _j \Vert   = 1 
\eeq 
and we have, if $\rho $ is the radius of $D$ and if $c$ is large 
enough 
\beq 
 |  \lambda _j(c)-E_0 |  \leq \frac {\rho } {2}
\eeq  

 

The following result about the exponential decay at 
infinity of the functions $\varphi  _j(c)$ is well-known (see 
Wang [17]). 

\begin{prop}
 With the previous notations, for each $\varepsilon >0$, there exists 
 $C_{\varepsilon }>0$, independent of $c$ such that the functions 
 $\varphi _j$ $(1\leq j \leq 2\mu )$ satisfy 
 \beq 
 \Vert  e^{(1-\varepsilon ) d(., V_0,c)} \varphi _j  \Vert ^2 
+   \frac {1} {c^2} 
\Vert  e^{(1-\varepsilon ) d(., V_0,c)} \nabla \varphi _j  \Vert ^2 
\leq C_{\varepsilon } 
\eeq
 \end{prop}
 
{\it Proof.} The proof is the same as in 
Wang [17] but, since it is written in [17] in the semiclassical 
context, we give a sketch of the proof here.
By a direct calculus, we see, like in Wang [17] (Proposition 2.1) 
that, for 
each real-valued function 
$\Phi $, bounded, uniformly lipschitzian on $\R^3$, we have 
\beq 
c^2 \int _{\R ^3}  | \nabla (e^{\Phi }\varphi _j) | ^2 \, dx + 
\int _{\R^3} \delta (x, c) 
 | e^{\Phi } \varphi _j  |  ^2 \, dx = 0
 \eeq 
where 
\beq 
\delta (x, c) = \left [ V_0(x) - \lambda _j (c)  \right ]  
\left [ 2 c^2 - V_0(x)  + \lambda _j(c) \right ]  - c^2  |  \nabla 
\Phi (x) | ^2
\eeq 
There exists 
$R_{ \varepsilon } >0$ such that, if $0\leq \varepsilon \leq 1$ 
\[ 
 | x |  \geq R_{ \varepsilon } \Rightarrow 
 V_0(x) \geq \frac {8( | E_0 | + (\rho /2))+4} { 2\varepsilon ^2- 
\varepsilon ^3 } 
\] 
If $\Phi $ satisfies $\Phi (0) = 0$ and 
\beq 
c^2  |  \nabla \Phi  | ^2 \leq V_0(x)_+ \, (2c^2 - V_0(x))\, (1 - 
\varepsilon )^2 
\eeq 
using (76), we see that 
\beq 
   \delta (x, c )  \geq c^2   
\hskip 1cm if  \, \, \, | x | \geq R_{\varepsilon } 
\eeq 
We can find $K_{ \varepsilon }>0$, independent on $c$, such that 
\beq 
c^{-2}  | \delta (x, c ) |  +  | \Phi(x) |  \leq K_{\varepsilon }
 \hskip 1cm if \, \, \,  | x |  \leq R_{\varepsilon } 
 \eeq 
It follows that 
\beq 
 \int _{ \R ^3}  | \nabla (e^{\Phi (x)}\varphi _j (x) )| ^2 \, dx 
+  \int _{ | x |  \geq R_{ \varepsilon }}  |  e^{ \Phi (x)} \varphi 
_j(x) |  ^2 \, dx \leq  \ldots 
\eeq 
\beq 
\ldots \leq K_{\varepsilon } 
\int _{ | x |  \leq R_{\varepsilon }}   
 |  e^{ \Phi (x)} \varphi _j(x) |  ^2 \, dx  
 \leq K _{\varepsilon } e^{ K_{\varepsilon }}  
 \eeq 
 Since, for $c$ large enough,  
 $ | \nabla \Phi (x) | ^2 \leq  6  c^2$, it follows 
 from (84) and (82) that 
 \beq 
 \int _{  | x | \geq R_{\varepsilon } } 
  | e^{ \Phi (x)} \nabla \varphi _j(x) | ^2 \, dx 
  \leq 
 (2 + 12 c^2 ) K_{ \varepsilon } e^{K_{\varepsilon }}
    \eeq 
   Since $\varphi _j$ satisfies (75), we remark also that 
   \beq 
   \int _{  | x | \leq R_{\varepsilon } } 
  | e^{ \Phi (x)} \nabla \varphi _j(x) | ^2 \, dx \leq 
  3 \frac {e^{2K_{\varepsilon }}}{c^2} 
  \Big [ 
  \Vert H_0(c) \varphi _j  \Vert ^2   + 
     \Vert V_0 \varphi _j   \Vert ^2   +  \Vert c^2 \varphi _j  \Vert 
^2   
  \Big ] 
  \eeq 
  \beq \leq K'_{\varepsilon } c^2 
  \eeq 
  where $K'_{\varepsilon }$ is independent on $c$. We used 
  $ | \lambda _j(c) |  \leq  | E_0 |  + (\rho /2)$ and $V_0(x) \leq 
  (2- (\varepsilon / 4))c^2$.  Therefore, 
  with $K''_{\varepsilon }>0$ independent on $c$, and on the function 
  $\Phi $ satisfying (80) 
 \beq 
 \frac {1} {c^2} \Vert e^{ \Phi } \nabla \varphi _j \Vert ^2 
 + \Vert e^{ \Phi } \varphi _j \Vert ^2 \leq K''_{\varepsilon } 
 \eeq 
The proposition follows by the   argument of [17]. \fp 
 

Now we shall study the decay at infinity of the orthonormal system of 
functions $\psi _j$ satisfying (71), following the technique of Sigal 
[13]. 
For that, we 
set 
\beq 
\widetilde d (x, V_0, c) = {\rm inf} ( d(x, V_0, c), \, \Sigma (c, 
\varepsilon ))
\eeq 



\begin{prop}
 With the previous notations, for each $\varepsilon >0$, there exists 
 $K_{\varepsilon }>0$, independent of $c$ such that the functions 
 $\psi _j$ $(1\leq j \leq 2\mu )$ satisfy 
 \beq 
 \Vert  e^{(1-\varepsilon ) \widetilde d(., V_0,c)} \psi _j  \Vert 
\leq K_{\varepsilon } c^{ (1-2/k )_+} 
\eeq
 \end{prop}
 
In the proof, and also later, we shall use a cut-off 
function defined as follows. 
 We can choose a function $h\in  C^{ \infty }  (\R )$ such that 
$0\leq h(t) \leq 1$ for all $t$,  $h(t) = 1 $ if $t\leq 2- 
\varepsilon $ and 
$h(t) = 0$ if $t \geq 2 - \frac {\varepsilon } {2}$. We set 
\beq 
\chi (x) = h \left ( \frac {V(x)} {c^2} \right  ) 
\hskip 1cm 
\forall x \in  \R ^3
\eeq 
We remark that, with $A_{\varepsilon }$ independent on $c$  
\beq 
 |  \nabla \chi (x) |  \leq A_{\varepsilon }  c^{-2/k }
 \eeq 
 We remark also that 
 \beq 
 \chi D(\theta, c) = \chi H_0(c)
 \eeq 
 and therefore 
 \beq 
D(\theta , c) \chi - \chi H_0(c) = [H_0(c), \chi]= c ( D  \chi ). 
\alpha   
\eeq 
where 
\[
( D\chi) . \alpha   = \left(
\begin{array}{cc}
0&\sigma \cdot  ( D \chi )\cr
\sigma \cdot ( D \chi )& 0
\end{array}\right),
\]  

 
 
{\it Proof of Proposition 3.} Let $\gamma $ be the boundary of $D$ 
(a circle with center $E_0$, and with  radius $\rho $). 
If $c$ is large 
enough, all the resonances $E_j(c)$ $(1\leq j \leq 2\mu)$ are 
contained in $B(E_0, \rho /2)$. The same arguments as for 
Lemma 5 (point ii) show that, for $c$ large enough 
\beq 
\Vert   (z- D(\theta, c))^{-1} \Vert   \leq K 
\eeq 
for all $z\in \gamma$, where $K$ is independent on $c$. Let $P$ be 
the projection defined, 
for $c$ large enough,  by 
\beq 
Pf = \frac {1} {2i \pi } \int _{\gamma } (z - D(\theta , c))^{-1}f \, 
dz 
\eeq 

First, we shall prove that the functions $P\varphi _j$ 
satisfy the estimations of the proposition. 
It follows from (94) that, for each $z \in \gamma $, and for all 
$f\in 
L^2(\R ^3 , \C 
^4)$ 
 \beq 
 (z-  D(\theta , c))^{-1} (\chi f ) = \Big [  \chi 
 + c (z-  D(\theta , c))^{-1} ( D\chi). \alpha  
 \Big ] \,  (z-  H_0(c))^{-1} f
 \eeq 
 Applying this equality with $f = \varphi _j$ and integrating over 
 $\gamma $, we obtain, by (96) 
 \beq 
 P( \chi \varphi _j)= \chi \varphi _j + g_j 
 \hskip 1cm 
 g_j = \frac {c} {2i \pi } \int _{\gamma } 
 \frac 
 { (z- D(\theta , c))^ {-1} ( D\chi ). \alpha   \varphi _j } 
 { z- \lambda _j(c)} \, dz
 \eeq 
We can write 
\beq 
\Vert  e^{(1-\varepsilon ) \widetilde d(., V_0,c)} P \varphi  _j  
\Vert 
\leq 
 \, e^{(1-\varepsilon ) \Sigma (c, \varepsilon )} 
\Vert P((1 -\chi )  \varphi  _j ) \Vert 
+ \Vert  e^{(1-\varepsilon ) \widetilde d(., V_0,c)}\chi \varphi  
_j ) \Vert + \ldots 
\eeq 
\beq 
\ldots + \Vert e^{(1-\varepsilon ) \widetilde d(., V_0,c)} g_j \Vert 
\eeq 
By (95),  the $L^2$ norm of the projector $P$ 
is bounded by some constant $K$ independent of $c$. 
By the definition of $\Sigma (c, \varepsilon )$ and by the 
proposition 
2, 
\beq 
e^{(1-\varepsilon ) \Sigma (c, \varepsilon )} 
\Vert P((1 -\chi )  \varphi  _j ) \Vert 
\leq K_{\varepsilon } 
\eeq 
for  some constant $K_{\varepsilon }$, 
 independent on $c$. If $c$ is large enough, 
 using (95) and (76), we see  that
 \beq 
\Vert  e^{(1-\varepsilon ) \widetilde d(., V_0,c)} g_j \Vert 
 \leq K_{0 } c e^{ (1-\varepsilon ) \Sigma (c, \varepsilon)} 
 \Vert (\nabla \chi ) \varphi _j \Vert 
 \eeq 
 with  some other constant $K_{0 } $. Therefore, using also (92) and 
the 
 definition of $\Sigma (c, \varepsilon)$, we obtain,  
  \beq   
\Vert  e^{(1-\varepsilon ) \widetilde d(., V_0,c)} g_j \Vert 
 \leq  K'_{\varepsilon } c^{ 1- (2/k )} 
 \Vert  e^{(1-\varepsilon )  d(., V_0,c)} \varphi _j
 \Vert 
 \leq K''_{\varepsilon } c^{1-(2/k )}
 \eeq 
 where $K'_{\varepsilon } $ and $K''_{\varepsilon }$ are independent 
 on $c$.  
 We used Proposition 2, which shows also that 
 \beq 
 \Vert  e^{(1-\varepsilon ) \widetilde  d(., V_0,c)} (\chi \varphi _j)
 \Vert  \leq  
  \Vert  e^{(1-\varepsilon )  d(., V_0,c)}  \varphi _j \Vert  
  \leq C_{\varepsilon } 
  \eeq 
  Summing up, we proved that, for some other $K_{\varepsilon }$ 
  independent on $c$  
  \beq 
  \Vert  e^{(1-\varepsilon ) \widetilde d(., V_0,c)} P 
  \varphi _j \Vert  \leq K_{\varepsilon } c^{(1-(2/k))_+}
  \eeq 
  Now we shall orthogonalize the system $(P\varphi _j)$ 
  $(1\leq j\leq 2\mu )$. We remark that 
  \beq 
  P \varphi _j - \varphi _j = 
  \frac {1} {2i \pi } \int _{\gamma } 
  \frac 
  { (z - D(\theta , c))^{-1} \big ( D(\theta , c) - H_0(c) \big ) 
  \varphi _j }
  {z -\lambda _j(c)} 
  \, dz 
  \eeq 
  It follows that 
  \beq 
  \Vert P \varphi _j - \varphi _j \Vert \leq 
  K_0  
  \Vert \big ( D(\theta , c) - H_0(c) \big ) \varphi _j \Vert 
  \eeq 
  where $K_0$ is independent of $c$.  
  We have, if $V_{\theta }$ is defined in (26) and 
  $V_0$ in (72)   
  \[ 
  \Vert \big ( D(\theta , c) - H_0(c) \big ) \varphi _j \Vert  
  \leq K \int _{V(x) \geq (2 - \varepsilon /2)c^2 } 
   |  \nabla \varphi _j (x) | ^2 \, dx  + \ldots 
   \]
   \beq 
    \ldots + K \int _{V(x) \geq (2 - \varepsilon /2)c^2 } 
    [ 1 + |  V_{\theta } (x) 
  -V_0 (x)  |  ^2  |]   \varphi _j (x) | ^2  \, dx 
  \eeq  
for some constant $K$, and we have also 
 $ | V_{\theta }(x)- V_0(x) |  \leq K <x>^{k }$. 
 By Lemma 8 and  
 proposition 2, it follows that, for some  $K_{\varepsilon }$
  \[ 
  \Vert P \varphi _j - \varphi _j \Vert \leq  
  K_{\varepsilon } e^{-\Sigma (c, \varepsilon)}
  \]  
  By Lemma 8, 
  $\Vert P\varphi _j - \varphi _j\Vert \rightarrow 0$ 
  when $c \rightarrow + \infty $. Hence the Gram 
  matrix $S = (P\varphi _j , P\varphi _k)_{1\leq j, k \leq 2\mu }$ 
  tends to identity when $c \rightarrow +\infty $. Therefore, if 
  $c$ is large enough, $T = S ^{-1/2}$ is defined, and bounded 
  independently of $c$. If we set $T = (a_{jk})$, the system 
  of functions $\psi _j = \sum a_{jk} P\varphi _k$ is an 
  orthonormal basis of ${\rm Im} P$, which satisfies the 
  estimations (90). \fp 
  
  {\it End of the proof of theorem 4.} We consider again the function 
  $\chi $ defined in  (91) and an orthonormal 
  system of eigenfunctions $\psi _j$ 
  satisfying (71).  
  By Proposition 3, we can write 
  \beq 
  \int _{ {\rm supp }\, (1-\chi ) }  |  \psi _j(x) | ^2 \, dx 
  \leq K_{\varepsilon }^2 c^2 e^{-2(1-\varepsilon ) \Sigma (c, 
  \varepsilon)}
  \eeq 
It follows by Lemma 8 (point i)) that,  if $c$ is large enough
  \beq 
  \int (1-\chi (x))   |  \psi _j(x) | ^2 \, dx \leq 
  \frac {1} {2}
  \eeq 
  If we write the imaginary part of the scalar product of both sides 
  of 
  (71) with $\chi \psi _j$, we obtain, using (93)  
  \[ 
  (\Im E_j(c)) \, \int _{\R ^3} \chi (x)  |  \psi _j(x) | ^2 \, dx 
  = \Im \, \langle  D(\theta, c) \psi _j , \chi \psi _j \rangle 
   = \ldots 
   \] 
   \beq 
   \ldots = \Im \, \langle H_0(c) \psi _j , \chi \psi _j \rangle 
   =-  \frac {1} {2} \langle  [H_0(c), \chi ] \psi _j , \psi _j 
   \rangle 
  \eeq 
  Using (110) and (92), we have, for some constants $K$, $K'$ 
  and $K''_{\varepsilon }$   
  \[ 
    |  \Im E_j(c) |  \leq 
     |  \big ( [H_0(c) , \chi ]\psi _j , \psi _j \big )  |    
      \leq K c \int  |  \nabla \chi (x)  | | \psi_j(x) | ^2 \, dx 
      \leq \ldots 
      \] 
      \[ 
   \ldots \leq K' c^{1-(2/k )} \int _{{\rm supp }(1-\chi )}  
   | \psi _j(x) | ^2 \, dx 
  \leq K''_{\varepsilon } c^3 
   e^{-2(1-\varepsilon ) \Sigma (c, \varepsilon)}
  \] 
   The estimation (12) of Theorem 4 follows, with another 
   $\varepsilon $, using Lemma 8.  
   
  \vskip 1cm


\centerline{\bf References}

[1]J. AGUILAR, J.M. COMBES, A class of analytic perturbations 
for one-body Schr\"odinger Hamiltonians, 
{\it Comm. Math. Physics,} 
22 (1971), 280-294. 
\par 
\noindent 
[2]E. BALSLEV, B. HELFFER, Limiting absorption principle 
and resonances 
for the Dirac operator. {\it Advances in Appl. Math,} 
13 (1992), 186-215. \par 
\noindent
[3]D. GRIGORE, G. NENCIU, R. PURICE, On the nonrelativistic limit of 
the Dirac Hamiltonian, {\it Ann. Inst. H. Poincar\'e, Phys. Th. } 
51 (1989) 231-263.  \par 
\noindent  
[4]B. HELFFER, J. SJOSTRAND, R\'esonances en 
limite semi-classique. {\it 
M\'emoire de la S.M.F. } 24/25, 1986.
\par 
\noindent 
[5]P.D. HISLOP, I.M. SIGAL, {\it Introduction to spectral theory, 
with 
applications to Schr\"odinger operators}, Springer, 1996. 
\par 
\noindent 
[6]W. HUNZIKER, Distortion analyticity and molecular 
resonance curves,  {\it Ann. Inst. H. Poincar\'e, Phys. Th. }, 
{\bf 45}, 4 (1986), 339-358.  
\par 
\noindent 
[7]T. KATO, {\it Perturbation theory of linear operators, }
Springer, 1980.
\par
\noindent 
[8]M. LAGUEL, {\it R\'esonances en limite semiclassique et 
propri\'et\'es de l'Hamil\-tonien effectif.} Th\`ese, Reims, 1999. 
\par
\noindent 
[9] B. PARISSE. R\'esonances pour l'op\'erateur de Dirac. 
{\it Helvetica Physica Acta,} 
64 (1991), 557-591.\par
\noindent 
[10] B. PARISSE. R\'esonances pour l'op\'erateur de Dirac II. 
{\it Helvetica Physica Acta, } 
65 (1992), 1077-1118.\par
\noindent 
[11]P. SEBA, The complex scaling method for Dirac resonances, 
{\it Lett. Math. Phys.} 16 (1988), 51-59. 
\par 
\noindent
[12]M. A. SHUBIN, {\it Pseudodifferential operators and 
spectral theory,} Springer, 1987.
\par 
\noindent
[13]I.M. SIGAL, Sharp exponential bounds on resonances states and 
width of resonances states, {\it Advances in Applied Math.}, {\bf 9} 
(1988), 127-166. 
\par 
\noindent
[14]B. THALLER, {\it The Dirac equation.} Springer, 1992\par 
\noindent 
[15]E. TITCHMARSH, A problem in relativistic quantum 
mechanics, {\it Proc. London Math. Society}, 
41 (1961), 170-192.
\par 
\noindent 
[16]K. VESELIC, The nonrelativistic limit of 
the Dirac equation 
and the spectral concentration, {\it Glasnik Math.} 4 (1969), 
231-240. 
\par 
\noindent
[17]X.P. WANG, Puits multiples pour l'op\'erateur de Dirac, 
{\it Ann. Inst. H. Poincar\'e,} 42 (1985), 269-319.
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