Content-Type: multipart/mixed; boundary="-------------0009121221817" This is a multi-part message in MIME format. ---------------0009121221817 Content-Type: text/plain; name="00-358.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-358.keywords" Resonances, Dirac ---------------0009121221817 Content-Type: application/x-tex; name="resonances.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="resonances.tex" \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 $\{z\in\C^3,\ \vert {\rm Arg } z_j\vert^{-s})$ $(s>0)$, 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 ^k \leq A_{\theta } \Im \, V_{\theta } (x) \hskip 1cm ^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 ^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 \[ \Vert ^{k/2}u_1 \Vert + \Vert ^{-k}\sigma .D u_1 \Vert + \Vert u_2 \Vert \leq \ldots \] \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, \[ \Vert < x> ^{ k/2} u_1 \Vert ^2 + c^{ -2} \Vert ^{ k/2} u_2 \Vert ^2 \leq \ldots \] \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 $^{k/2}\varphi _n$ and $^{-k} \sigma . D \varphi _n$ 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 ^{k/2} \varphi _n \Vert + \Vert ^{-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 $00$ 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 $$\Vert ^{k/2} u(t)\Vert + \Vert ^{-k}\sigma . D u(t) \Vert + \Vert v(t)\Vert \leq \ldots $$ $$ \ldots \leq K ( \Vert f\Vert + \Vert u(t)\Vert ) \hskip 1cm 0< t \leq t_0 \ \ \ \ \ \ \forall f \in {\cal H}$$ 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^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 $$\Vert ^m (A + t B)^{-1} f \Vert \leq K_m \Vert ^m f \Vert \hskip 1cm \forall f\in E \ \ \ \ \ 0\leq t \leq t_0$$ and that, for each $f\in E$, the function $^m G(t)f$ 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 \[ \Im \theta ^k \leq A \Im \left [ e^{ \theta } V_{\theta }(x) \right ] \hskip 1cm if \, \, \, \, \, | \theta | \leq 1, \, \, \, 0< \Im \theta < \varepsilon_0, \, \, \, \, | x | \geq R \] 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 $$V(x) \geq \frac {3} {2} c^2 \Rightarrow c \leq K_{\varepsilon } d(x, V_0, c)\leqno i)$$ $$ \leq K_{\varepsilon } (1 + d(x, V_0, c)) \hskip 1cm \forall x \in \R^3 \leqno ii)$$ $$S(c, \varepsilon )\leq \Sigma (c, \varepsilon)\leqno iii)$$ \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 \[ 00$ and $K'>0$ such that, if $c$ is large enough, \[ \frac {1}{2} c^2 \leq | V(x(t_0))-V(x(t_1)) | \leq K | x(t_0)-x(t_1) | \left [ + < x(t_1)> \right ]^{k-1} \] \[ \ldots \leq K' | x(t_0)-x(t_1) | V(x(t_1))^{(k-1)/k} \leq K' | x(t_0)-x(t_1) | c^{2 - 2/k} \] 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 ^{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. 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