ISOZAKI, Hiroshi

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Title On the S-matrix for three body Schrodinger OperatorsAuthor(s)ISOZAKI, HiroshiCitation数理解析研究所講究録 (1992), 779: 76-84Issue Date1992-04URL http://hdl.handle.net/2433/82478RightType Departmental Bulletin PaperTextversionpublisherKyoto University76On the S-matrix for three bodySchr\"odinger OperatorsHiroshi ISOZAKIDepartment of Mathematics, Osaka University, Toyonaka, 560,JAPAN1.1 Collision processes in the three-body problemWe consider collision processes of quantum mechanical three particles la-belled by 1, 2, 3. Suppose in the initial state the two of them form a boundstate, denoted by $(1,2)$ , and the third particle collides with this pair. Thenthere occurs one of the following five phenomena:$(1, 2)+(3)\Rightarrow\{\begin{array}{l}(a)(l,2)+(3)(b)(1,2)^{*}+(3)_{*}(c)(1,2)’+(3)(d)(1,3)+(2)(e)(1)+(2)+(3)\end{array}$(a) is an elastic process. In (b), the energy of the pair changes. In (c), theenergy of the pair does not change, but this pair takes a different state(which happens when the eigenvalue is degenerate). (d) is a rearrangementprocess. Finally in (e), all of the three particles move freely after the collision.The first four cases are treated in essentially the same way as in the 2-bodyproblem. So, we consider the properties of the S-matrix associated with thecase (e).1.2 S-matrixIn $R^{3}$ we consider three particles with mass $m_{i}$ and position $x^{i}$ . We choosea pair $(i,j)$ and denote it by $\alpha$ . Let$\frac{1}{m_{\alpha}}=\frac{1}{m_{i}}+\frac{1}{m_{j}}$ , $\frac{1}{n_{\alpha}}=\frac{1}{m_{k}}+\frac{1}{m_{i}+m_{j}}$be the reduced masses and779 1992 76-8477$x^{\alpha}=\sqrt{2m_{\alpha}}(x^{i}-x^{j})$ , $x_{\alpha}= \sqrt{2n_{\alpha}}(x^{k}-\frac{m_{i}x+m_{j}x^{j}}{m_{1}\cdot+m_{j}})$be the relative coordinates. Let$X= \{(x^{1}, x^{2}, x^{3});\sum_{1=1}^{3}m_{i}x^{i}=0\}$ .Then in $L^{2}(X)$ the Schr\"odinger operator is given by$H=H_{0}+ \sum_{\alpha}V_{\alpha}(x^{\alpha})$ , $H_{0}=-\triangle_{x^{\alpha}}-\triangle_{x_{\alpha}}$ . (1.1)If the pair potentials are short-range, the wave operators are known to exist:$W_{0}^{\pm}= s-\lim_{\ellarrow\pm\infty}e^{itH}e^{-itH_{0}}$ , (1.2)$W_{\alpha}^{\pm}= s-\lim_{tarrow\pm\infty}e^{itH}e^{-itH_{\alpha}}J_{\alpha}$, (1.3)where$H_{\alpha}=H_{0}+V_{\alpha}$ , $(J_{\alpha}f)(x^{\alpha},x_{\alpha})=u_{\alpha}(x^{\alpha})f(x_{\alpha})$ , (1.4)$u_{\alpha}$ being a normalized eigenfunction of $h^{\alpha}=-\triangle_{x^{\alpha}}+V_{\alpha}(x^{\alpha})$ with eigenvalue$E^{\alpha}<0$ . The scattering operator $S_{0\alpha}$ is defined by$S_{0\alpha}=(W_{0}^{+})^{*}W_{\alpha^{-}}$ . (1.5)To introduce the S-matrix, we use unitary operators $\mathcal{F}0$ : $L^{2}(R^{6})arrow$$L^{2}((0, \infty);L^{2}(S^{5}))$ and $\mathcal{F}_{\alpha}$ : $L^{2}(R^{3})arrow L^{2}((E^{\alpha}, \infty);L^{2}(S^{2}))$ defined by$( \mathcal{F}_{0}f)(\lambda,\theta)=C_{0}(\lambda)\int_{R^{t}}e^{-;\sqrt{\lambda}\theta\cdot x}f(x)dx$ ,(1.6)$C_{0}(\lambda)=(2\pi)^{-3}2^{-1/2}\lambda$ ,$( \mathcal{F}_{\alpha}f)(\lambda,\omega)=C_{\alpha}(\lambda)\int_{R^{3}}e^{-i\sqrt{\lambda-E^{\alpha}}x}4l\cdot f(x)dx$,(1.7)$C_{\alpha}(\lambda)=(2\pi)^{-3/2}2^{-1/2}(\lambda-E^{\alpha})^{1/4}$ .Let$\hat{S}_{0\alpha}=\mathcal{F}_{0}S_{0\alpha}\mathcal{F}_{\alpha}^{*}$ . (1.8)Then as is well-known, $\hat{S}_{0\alpha}$ is decomposable, namely, for any $\lambda>0$ , thereexists a bounded operator $\hat{S}_{0\alpha}(\lambda)\in B(L^{2}(S^{2});L^{2}(S^{5}))$ such that$(\hat{S}_{0\alpha}f)(\lambda,\theta)=(\hat{S}_{0\alpha}(\lambda)f(\lambda, \cdot))(\theta)$for a.e. $\lambda>0,$ $\theta\in S^{5}$ and all $f\in L^{2}((E^{\alpha}, \infty);L^{2}(S^{2}))$ . This $\hat{S}_{0\alpha}(\lambda)$ iscalled the S-matrix. Note that this definition contains a sort of ambiguity.Two families of operators $\{\hat{S}_{0\alpha}(\lambda)_{i}\}_{\lambda=0}^{\infty}(i=1,2)$ define the same scattering78operator $S_{0\alpha}$ , if $\hat{S}_{0\alpha}(\lambda)_{1}=\hat{S}_{0\alpha}(\lambda)_{2}$ for a.e. $\lambda>0$ . The study of this family ofoperators $\{\hat{S}_{0\alpha}(\lambda)\}_{\lambda 0}^{\infty_{=}}$ is not an easy problem. The general result known sofar is that of Amrein-Pearson-Sinha [1] and Enss-Simon [5] asserting that$\hat{S}_{0\alpha}(\lambda)$ is a Hilbert-Schmidt operator for $a.e.\lambda>0$ , if the pair potentialsdecay faster than $|x^{\alpha}|^{-2}$ . Our aim is to show that there is a representative$\{\hat{S}_{0\alpha}(\lambda)\}_{\lambda=0}^{\infty}$ continuous in $\lambda>0$ and to investigate its detailed properties.1.3 Main results (1)We assume that $V_{\alpha}$ is a real $c\infty$-function such that for a constant $\rho>0$$|\partial_{y}^{m}V_{\alpha}(y)|\leq C_{m}(1+|y|)^{-\rho-m}$ , $m=0,1,2,$ $\ldots$ , (1.9)where $\partial_{y}^{m}$ denotes an arbitrary derivative of m-th order with respect to $y$ ,and $C_{m}$ is a constant. This assumption is stronger than actually needed.One can also allow certain local singularities for $V_{\alpha}$ . Let$X_{\beta}=\{x\in X;x^{\beta}=0\}$ ,and define$M=S^{5}\backslash \cup\rho X\rho$ , $N=S^{5}\cap(\cup\rho X\rho)$ .Theorem 1.1 (1) Suppose $\rho>4+1/2$ . Then $\hat{S}_{0\alpha}(\lambda)$ has a continuo$\iota\iota s$ kerneloutside $N$ :$\hat{S}_{0\alpha}(\lambda;\theta,\omega)\in C((0, \infty)\cross M\cross S^{2})$ .(2) $Sup$pose $\rho>5+1/2$ . Let $\beta$ be any pair and decompose $\theta\in S^{5}$ as$\theta=(\theta^{\beta},\theta_{\beta})$ in accordance with the choice of the Jacobi-coordinates. Thenas $|\theta^{\beta}|arrow 0$ ,$\hat{S}_{0\alpha}(\lambda;\theta,\omega)\simeq|\theta^{\beta}|^{-1}A_{\beta,-1}(\lambda;\frac{\theta^{\beta}}{|\theta^{\beta}|},\theta_{\beta},\omega)+A_{\beta,0}(\lambda;\frac{\theta^{\beta}}{|\theta^{\beta}|},\theta\rho,\omega)$,where$A_{\beta,-1}( \lambda;\frac{\theta^{\beta}}{|\theta^{\beta}|},\theta_{\beta)}\omega)$$= \sum_{j}^{fini\ell\epsilon}C_{\beta 1}^{(j)}(\lambda;\theta_{\beta},\omega)\cross\int_{R^{S}}\frac{\theta^{\beta}}{|\theta^{\beta}|}\cdot x^{\beta}V_{\beta}(x^{\beta})u_{\beta^{j)}}^{(}(x^{\beta})dx^{\beta}$$+C_{\beta 2}( \lambda;\theta\rho,\omega)\cross\int_{R^{S}}V_{\beta}(x^{\beta})\varphi\rho(x^{\beta})dx^{\beta}$ ,$u_{\beta}^{(j)}$ being the eigenfunction with zero eigenvalue for $h^{\beta}$ , and $\varphi\rho$ the zero-resonance. $A\rho,0$ is continuous with respect to all of its arguments. $A_{\beta,-1}=$$0$ , if $\theta$ is neither an eigenvalue nor the resonance for $h^{\beta}$ . In this case,$\hat{S}_{0\alpha}(\lambda;\theta,\omega)$ is continuous at $\theta^{\beta}=0$ .79In the course of the proof, we shall see that $\hat{S}_{0\alpha}(\lambda)$ is a $B(L^{2}(S^{2});L^{2}(S^{5}))$-valued continuous function of $\lambda>0$ if $\rho>5+1/2$ . The zero-resonance $\varphi\rho$is the solution of the equation $h^{\beta}\varphi\rho=0$ which behaves like $\varphi\rho\simeq C/|x^{\beta}|$ as$|x^{\beta}|arrow\infty,$ $C\neq 0$ . From our proof, one can see that if the pair potentialsdecay sufficiently rapidly, $\hat{S}_{0\alpha}(\lambda;\theta,\omega)$ is smooth on $M\cross S^{2}$ , but the zeroeigenvalue and the zero-resonance are known to exist even if the potentialis compactly supported. (See e.g. [3]). For the S-matrix from 2 to 3 clusterscattering, it is therefore the zero-eigenfunctions and the zero-resonances ofsubsystems that determines its singularities.As for the coefficients $C_{\beta 1}^{(j)}(\lambda;\theta\rho,\omega)$ and $C_{\beta 2}(\lambda;\theta\rho,\omega)$, we haveTheorem 1.2 Up to a multiplicative constant depending only on $\lambda$ and $E^{\alpha}$ ,$C_{\beta 1}^{(j)}(\lambda;\theta\rho,\omega)$ and $C_{\beta 2}(\lambda;\theta\rho,\omega)$ coincide with the scattering amplitudes for2 cluster scattering.More precisely, $C_{\beta 1}^{(j)}(\lambda;\theta\rho,\omega)$ and $C_{\beta 2}(\lambda;\theta\rho,\omega)$ are the scattering ampli-tudes for 2-cluster scattering in which, after the collision, the pair $\beta$ becomesthe bound state with zero energy or the zero-resonance, respectively. Oneshould note that the notion of 2-cluster scattering associated with the zero-resonance is somewhat ambiguous since it has not yet been introduced inmathematical literatures. So, we shall begin by explaining the meaning ofTheorem 1.2.1.4 Generalized eigenfunctionsWe first recall the 2-body problem. The generalized eigenfunction $\varphi(x,\xi)$ of$-\triangle+V$ in $R^{n}$ is written as$\varphi(x,\xi)=e^{ix\zeta}-v$ ,$v=v(x,\xi)=(-\triangle+V-|\xi|^{2}-i0)^{-1}(V(x)e^{ix\xi})$ .The first term, $e^{ix\xi}$ , represents the incident wave, and the second term, $v$ , thescattered wave. The scattering amplitude $A(\lambda;\theta,\omega)=\hat{S}(\lambda;\theta,\omega)-\delta(\theta-\omega)$is derived from $v$ in the following way:$v(x, \sqrt{\lambda}\omega)\sim C(\lambda)r^{-(n-1)/2}e^{;/\overline{\lambda}r}A(\lambda;\theta,\omega)$ ,$\theta=x/|x|$ , $r=|x|arrow\infty$ .In the case of the three body problem, our generalized eigenfunction isgiven by$\varphi(x, \lambda,\omega)=e^{i\sqrt{\lambda-E^{\alpha}}\omega\cdot x_{\alpha}}u_{\alpha}(x^{\alpha})-v$ , (1.10)$v=R(\lambda+iO)f$, $R(z)=(H-z)^{-1}$ , (1.11)$f=f(x, \lambda,\omega)=\sum_{\gamma\neq\alpha}V_{\gamma}(x^{\gamma})u_{\alpha}(x^{\alpha})e^{i\sqrt{\lambda-E^{\alpha}}\omega\cdot x_{\alpha}}$. (1.12)80We relate the coefficients $C_{\beta 1}^{(j)},$ $C_{\beta 2}$ appearing in Theorem 1.1 to the asymp-totic properties of $v$ .For small positive constants $\epsilon$ and $\epsilon_{2}$ , we take smooth functions $\chi\rho(x)$and $\psi_{\beta}(t)$ defined by$\chi\rho(x)=\{\begin{array}{l}lif\frac{|x^{\beta}|}{|x|}<2\epsilon,|x|>l0if\frac{|x^{\beta}|}{|x|}>3\epsilon\end{array}$ (1.13)$\psi_{\beta}(t)=\{_{0}1$ $ifif$ $|\begin{array}{ll}t- \lambda t- \lambda\end{array}|$. (1.14)We put$w(x \rho)=\int_{R^{3}}u_{\beta}^{(j)}(x^{\beta})\psi_{\beta}(D_{x\rho})\chi_{\beta}(x)v(x)dx^{\beta}$ , (1.15)where $\psi_{\beta}(D_{x\rho})$ is a pseudo-differential operator (Ps.D.Op. in short) withsymbol $\psi_{\beta}(|\xi\rho|^{2})$ , and $u_{\beta}^{(j)}(x^{\beta})$ is an eigenfunction of $h^{\beta}$ with zero eigen-value. Note that the integral is convergent since it is actually performed onthe set $\{|x^{\beta}|\leq 2\epsilon(1-4\epsilon^{2})^{-1}|x\rho|\}$. Then we haveLemma 1.3 The following asymptotic expansion holds :$w(r\theta_{\beta})\sim C(\lambda)r^{-1}e^{\oint\sqrt{\lambda}r}C_{\beta 1}^{(j)}(\lambda;\theta\rho,\omega)$ ,as $r=|x\rho|arrow\infty$ in $L^{2}(S^{2})$ . Moreover, up to a constant depending only on$\lambda,$ $C_{\beta 1}^{(j)}(\lambda;\theta_{\beta},\omega)$ coincides with the scattering amplitude introduced by thetime-dependent method.One should note that it is not an obvious problem to construct theintegral kernel of the 2-cluster scattering amplitude when the final state hasthe zero energy. The above lemma means that the kernel does exist and isequal to $C_{\beta 1}^{(j)}$ .Now, one can think of the collision process, in which, after the colli-sion, the pair $\beta$ takes the zero-resonance state. It is not easy to define theassociated scattering amplitude by the time-dependent method, since thezero-resonance, $\varphi\rho$ , does not belong to $L^{2}(R^{3})$ . However, the stationarymethod explained above works equally well for this case. We define $\tilde{w}$ by$\tilde{w}(x\rho)=\int_{R^{3}}\varphi\rho(x^{\beta})\psi_{\beta}(D_{x\rho})\chi\rho(x)v(x)dx^{\beta}$ . (1.16)By the same reasoning as above, this makes sense. We then haveLemma 1.481$\tilde{w}(r\theta_{\beta})\sim C(\lambda)r^{-1}e^{i\sqrt{\lambda}r}C_{\beta 2}(\lambda;\theta\rho,\omega)$ ,as $r=|x\rho|arrow\infty$ in $L^{2}(S^{2})$ .Comparing the above two lemmas, it seems to be natural to call$C_{\beta 2}(\lambda;\theta\rho,\omega)$ the scattering amplitude associated with the zero-resonance,up to a constant factor. Actually, the above two lemmas are the content ofTheorem 1.2.1.5 Main results (2)We continue the above point of view of studying the spatial asymptoticproperties of $v$ .Theorem 1.5 If $\rho>4+1/2$ , for any $\lambda>0$,$s-\lim_{rarrow\infty}r^{5/2}e^{-i\sqrt{\lambda}r}v(r\cdot)=C(\lambda)\hat{S}_{0\alpha}(\lambda;\cdot,\omega)$ ,$C(\lambda)=e^{\pi i/4}2\pi\lambda^{-1/4}(\lambda-E^{\alpha})^{-1/4}$ ,in $L_{loc}^{2}(M)$ .So, the scattering matrix $\hat{S}_{0\alpha}(\lambda)$ is obtained from the asymptotic be-havior of $v$ in the same way as in the two body problem. However, it is noteasy to replace $M$ by $S^{5}$ in the above theorem, since the behavior of $v$ in aneighborhood of $N$ is,rather complicated. What we can expect is the limitin an averaged sense.Let $\chi_{\beta}(x)$ be as in (1.13). We take $\rho+(t)\in C^{\infty}(R^{1})$ such that $\rho+(t)=1$if $t>1-\epsilon,$ $\rho+(t)=0$ if $t<1-2\epsilon$ . We also take $\rho(t)\in C_{0}^{\infty}((0, \infty))$ suchthat $\rho(t)=1$ near $t=1$ and $\int_{0}^{\infty}\rho(t)dt=1$ . In order to facilitate the proof,the pair potentials are assumed to be rapidly decreasing, but the followingtwo theorems can of course be proved for more slowly decreasing potentials.Theorem 1.6 Suppose that $V_{\alpha^{Z}}s$ are rapidly decreasing functions. Then$s-\lim_{Rarrow\infty}\frac{1}{R}\int_{R^{6}}e^{-i\sqrt{\lambda}\theta\cdot x}\theta\rho\cdot\hat{x}\rho\rho+(\frac{\theta\rho}{|\theta\rho|}\cdot\frac{x\rho}{|x\rho|})\chi\rho(x)\rho(\frac{|x\rho|}{R})v(x)dx$$=C(\lambda)\hat{S}_{0\alpha}(\lambda;\theta,\omega)$ ,$C(\lambda)=-(4\pi)^{-1}\lambda^{-1/2}C_{0}(\lambda)^{-1}C_{\alpha}(\lambda)^{-1}$ ,in $L^{2}(\tilde{N}_{\beta})$ , where $\tilde{N}\rho$ is a small neighborhood of $N\cap X\rho$ and $\hat{x}\rho=x_{\beta}/|x_{\beta}|$ .In the neighborhood of the $X\rho$-plane, there are two sorts of scattering,the 3-cluster scattering and the 2-cluster scattering. We can distinguishbetween them by changing the way of taking the limit at infinity of $v$ .82Let $u\rho\iota(x^{\beta})$ be a normalized eigenfunction of $h^{\beta}$ with eigenvalue $E_{l}^{\beta}\leq 0$ .Let $A_{\beta l}(\lambda;\theta\rho,\omega)$ be the 2-cluster scattering amplitude associated with theprocess in which, after the colloision, the pair $\beta$ takes the bound state $u\rho\iota$ .Theorem1.7 Suppose that $V_{\alpha}s$ are rapidly decreasing functions. For a small$\epsilon>0$, let $\psi_{\beta}(t)\in C^{\infty}(R^{1})$ be such that $\psi_{\beta}(t)=1$ if $t>\lambda-\epsilon,$ $\psi\rho(t)=0$if $t<\lambda-2\epsilon$ . Let $\psi_{\beta}(D_{x\rho})$ be the Ps.D. $Op$ . with symbol $\psi_{\beta}(|\xi_{\beta}|^{2})$ . Fix$a>0$ arbitrarily. Then as $r=|x_{\beta}|arrow\infty$ , we have the following asymptoticexpansion:$\psi_{\beta}(D_{x\rho})v\simeq\sum_{l}C_{\beta l}(\lambda)u_{\beta l}(x^{\beta})r^{-1}e^{i\sqrt{\lambda-E_{l}^{\beta}}r}A_{\beta l}(\lambda;\theta\rho,\omega)$ , $\theta_{\beta}=x\rho/r$ ,$C_{\beta l}(\lambda)=2\pi i(\lambda-E^{\alpha})^{-1/4}(\lambda-E_{l}^{\beta})^{-1/4}$,uniformly for $|x^{\beta}|<a,$ $\theta_{\beta}\in S^{2}$ .1.6 Some remarksAmrein, Pearson and Sinha [1] showed that, for the N-body problem, thetotal cross-section with 2-cluster initial state is finite for almost $aU$ energyand derived its asymptotic properties in an averaged sense under the as-sumption that the potentials decay faster than $|x^{\alpha}|^{-2}$ . See also Enss-Simon[5]. Amrein and Sinha [2] also showed that, for the three body problem, thetotal cross-section is finite for all $\lambda>0$ under the assumption that each2-body subsystem has neither the zero $ei$genvalue nor the zero-resonance.Ito and Tamura [11] studied the semi-classical asymptotics for the totalcross-section in distributional sense. All of these works treats the case ofthe initial state of 2-clusters, while Newton [18] and Yafaev [20] studied thestructure of the S-matrix coressponding the collision process from 3-clustersto 3-clusters. Asymptotic properties of generalized eigenfunctions of three-body Schr\"odinger operators have been so far studied mainly by physicists.In the work of Newton [17], Theorems 1.5 and 1.7 were derived by intuitivearguments. If we consider the collision process of initial state of 3-clusters,we are led to consider the generalized eigenfunction formally defined by$\psi(x, \lambda,\theta)=e^{i\sqrt{\lambda}\theta\cdot x}-R(\lambda+i0)\sum_{\alpha}V_{\alpha}e^{;\sqrt{\lambda}\theta\cdot x}$ .The rigorous study of this generalized eigenfunction seems to be muchharder. Nuttal [19] and Newton [17] gave precise explanations. One shouldalso note the work of Mercuriev [16] of the three-body scattering theory forthe Coulomb potential based on the stationary theory.To prove the above theorems, following Isozaki-Kitada [10], we first lo-calize the S-matrix in the phase space. An important role is played by the83estimate of the resolvent of $H$ multiplied by Ps.D.Op’s. For that purposewe utilize the estimate of Skibsted [15] on propagation properties in thephase space of $e^{-itH}$ , whose method is based on the work of Sigal-Soffer[14]. These estimates have been further refined by G\’erard [6]. The singular-ities of $\hat{S}_{0\alpha}(\lambda)$ arise from the low-energy asymptotics of 2-body subsystemsstudied by Jensen and Kato [12]. Another important tool is the spectral rep-resentation theory developed for 2-body Schr\"odinger operators, the key ideaof which is to relate the generalized eigenfunction to the spatial symptoticsof the resolvent of Schr\"odinger operators.References1 W.O.Amrein, D.B.Pearson and K.B.Sinha, Bounds on the total scatteringcross-section for N-body systems, Nuovo Cimento 52 A (1979), 115-131.2 W.O.Amrein and K.B.Sinha, On the three body scattering cross sections,J. Phy.A : Math. Gen. 15 (1982), 1567-1586.3 C.L.Dolph, J.B.Macleod and D.Thoe, The analytic continuation of the re-solvent kernel and scattering operator associated with the Schrodinger op-erator, J. Math. Anal. Appl., 16 (1966), 311-332.4 V.Enss, Quantum scattering theory of two and three body systems withpotentials of short and long range, Lecture Notes in Math. 1159, Springer,Berlin-Heidelberg-New York, (1985).5 V.Enss and B.Simon, Finite total cross sections in non-relativistic quantummechanics, Commun. Math. Phys. 76 (1980), 177-209.6 C.G\’erard, Sharp propagation estimates for N-particle systems, preprint(1991).7 G.M.Graf, Asymptotic completeness for N-body short-range quantum sys-tems : A new proof, Commun. Math. Phys. 132 (1990), 73-101.8 H.Isozaki, Structures of S-matrices for three body Schr\"odinger operators,(to appear in Commun. Math. Phys.).9 H.Isozaki, Asymptotic properties of generalized eigenfunctions for threebody Schrodinger operators, prepnnt (1991).10 H.Isozaki and H.Kitada, Scattering matrices for two-body Schr\"odinger op-erators, Scientific papers of the college of arts and sciences, Tokyo Univ. 35(1985), 81-107.11 H.T.Ito and H.Tamura, Semi-classical asymptotics for total scattering crosssections of 3-body systems, prepnnt (1990).12 A.Jensen and T.Kato, Spectral properties of Schr\"odinger operators and timedecay of the wave functions, Duke Math. J. 46 (1979), 583-611.13 I.M. Sigal and A. Soffer, The N-particle scattering problem : Asymptoticcompleteness for short range quantum systems, Ann. of Math. 125 (1987),35-108.14 I.M.Sigal and A.Soffer, Local decay and propagation estimates for time-dependent and time-independent Hamiltonians, preprint (1988).15 E.Skibsted, Propagation estimates of N-body Schr\"odinger operators,Commun. Math. Phys. 142 (1991), 67-98.8416 S.P.Mercuriev, On the three-body Coulomb scattering problem, Ann. Phys.130 (1980), 395-426.17 R.G.Newton, The asymptotic form of the three-particle wave functions andthe cross sections, Ann. Phys. 74 (1972), 324-351.18 R.G.Newton, The three particle S-matrix, J. Math. Phys. 15 (1974), 338-343.19 J.Nuttal, Asymptotic form of the three-particle scattering wave functionsfor free incident particles, J. Math. Phys. 12 (1971), 1896-1899.20 D.Yafaev, On the multichannel scattering in two spaces, Theo. Math. Phys.37 (1978), 867-874.

On the S-matrix for three body Schrodinger Operators

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On the S-matrix for three body Schrodinger Operators

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