Abstract. We prove asymptotic completeness for 3-particle systems with pair potentials satisfying the virial condition and decreasing at infinity as (x) -~ for. 1. #>~.
Invent. math. l14,333-397(1993)
Inventiones mathematicae 9 Springer-Verlag 1993
Asymptotic completeness for 3-particle long-range systems C. G~rard Centre de Math6matiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France Oblatum 9-IV-1992 & 14-VI-1993
Abstract. We prove asymptotic completeness for 3-particle systems with pair potentials satisfying the virial condition and decreasing at infinity as ( x ) -~ for # > ~ .1
1 Introduction We study in this paper the problem of asymptotic completeness for some classes of 3-body systems with pair potentials decreasing at infinity like ( x ) - " for p > 7. These systems are described by a Hamiltonian of the form: H =
A~, + ~ i=1 ~
~
Vi j ( x i -
x j),
i ~a, Vb r a}, and equal to 1 in:
{xeXIIx"l
< e0/2, Ixbq > 2~1, Vb qi a}.
Then as in [Ki.Ya], we set:
( x log t'~ la.t(x) : E q , \ ~ - j Vb.t bq:a Note that using the hypotheses (H) below, we see that I,.t satisfies the estimates:
I O ~ I . , , ( x ) l < C k , , ( ( x ) + ( t ) ) -Ikl-"+~
re>0,
keN".
(1.1)
Since the conditions that we will eventually impose on/z will be of the type/~ > go, we can as well assume, by decreasing/~ a little, that ( 1.1 ) holds with ~ = 0. Then we will denote by S,(t, ~,) a solution of the Hamilton-Jacobi equation: 1 ~ OtSa -B- 2r
2
-[- Ia,t(O, O~aSa),
( f o r I~al > e, Itl > T~. A construction of such a solution can be found for example in [H6]. Note that one can as well use other modifiers, for example the stationary ones used in [I.K]. The choice of modifiers is actually irrelevant to the problem of asymptotic completeness for N-body Hamiltonians and should be more considered as a part of the 2-body theory, which is now well understood. We can then define the modified wave operators: Definition 1.1 Under the hypotheses (H), the modified wave operators defined by: f2~ = s. limt~ + | e itH e- iSa(t, D~)Epp(Ha ) exist.
Proof Using Proposition 2.3, we get that under hypotheses (H), 0 is not an eigenvalue of the 2-body subsystems H". Hence all the eigenvectors of H" decay exponentially, from which the existence of the wave operators follows by a well known argument (see for example [ D e l , Lemma 5.1] for a proof). [] Let us now formulate our hypotheses. We assume that the potentials V, satisfy the following conditions: (HI):
[O~Va(X)] < C ~ ( x ) -I'l-u,
VaeN" and ~1 1 +/~/2. (H3): for a r ami., V~(x) satisfies the virial condition: 2V,(x) + x . V,,V.(x) < - C ( x ) -u,
for C > 0.
We would like to emphasize that the virial condition (H3) has nothing to do with the repulsive condition x . O ~ V ( x ) 1. Although the use of the Weyl-H6rmander calculus is not essential for our proof, it provides a natural framework. Let gt, x.~(fx, 64) be a metric on T*X. F o r a given weight function m(t,x, 4)eC~176t x T ' X ) we say that a function f(t, x, 4) belongs to the class S(m, g) if: k
Vk~IN,
sup If(k)(t,x, 61,... ,6k) lI-Igt,:~,~(~i) -1/2 1. Here f(k) is the kth differential of f i n the variables (x, 4). To have a good symbolic calculus in the class S(m, g), m has to be slowly varying and 9 has to be slowly varying and a-temperate. We refer to [H6] for a precise description of these conditions which will always be satisfied in our examples. Associated with a metric 9, there is a natural Planck constant which depends on the point (x, 4) and characterizes the properties of the symbolic calculus in the metric g. This Planck constant is defined by:
h2(x, 4) = sup ax,~(t) ,~r*x gT,.r where g~,~ is the dual metric to gx,~ for the symplectic form cr on T*X. Roughly speaking a smaller Planck constant means a better symbolic calculus but a worse localization in phase space. F o r a symbol a~S(m, g), we will denote by A = OpWa(t, x, Dx) the Weyl quantization of a defined by:
[ x+y ) OpWa(t,x, Dx)u(x) = ( 2 n ) - " ~ S e ' a~t,---~---,4_ u(y)dCdy.
(2.1)
We recall that O p w a is formally selfadjoint if a is real-valued. We will now describe how to obtain Theorem 1 from the result about 2-body time-dependent Hamiltonians obtained in Theorem 10.2 In the end of the section,
Asymptotic completeness for 3-particle long-range systems
337
as a guide for the rest of the paper, we will explain the classical intuition behind the quantum estimates that we are going to prove. The first standard step (see [S.$2], [Gr], [ D e l ] ) consists in replacing the evolution e -i'n by a collection of evolutions Ua(t) associated with effective timedependent Hamiltonians. This result is described in the next Proposition. Let us first introduce some notations. We will denote by Ua(t) the unitary evolution generated by the time-dependent Hamiltonian Ha(t ) = Ha + Ia,t(x), where we recall that Ia.t(X) is the time-dependent potential introduced in (1.1). Then one has the following result : Proposition 2.1 Let u~L2( X ) such that Ea(H) u = u,for some interval A away from z w apv(H ). Then there exist vectors uaeL2(X)for ae~r a 4= a . . . . such that: e-itnu =
~,
U,(t)ua + 0(1) when t ~
+ ~.
a :~ a m a x
For the proof of the Proposition, we refer for example to [Gr, Lemma 5.1]. A similar result appeared also in the earlier work of Sigal-Softer [S.S1], [S.$2]. The next result we use is also well known and concerns the existence of asymptotic energy. This result appears in the works of Enss (see [ E l i , [E2]) and of SigalSofter (see [S.$4]). A proof can be found for example in [S-$3, Theorem 4.1].
Proposition 2.2 For any x~Cff (IR) , the following norm limit exists: g2z def= lira U*(t)z(Ha)Ua(t). Moreover there exists a selfadjoint operator Ha+ such that: Qz = z ( H~+)9 The operator Ha+ is called the asymptotic energy of the cluster a ~ d . Other asymptotic observables have been used successfully to study the long range problem, in particular the asymptotic velocity (see [ D e l l ). However for the problem we handle, it is important to consider this asymptotic energy operator. Let us now explain how we can use these two well known results and the Theorem 10.2 to prove asymptotic completeness for potentials satisfying hypotheses (H). We start with an easy consequence of the virial condition.
Proposition 2.3 Let H be a 3-particle Hamiltonian satisfying the hypotheses (H). Then for any a 4= a . . . . one has E~o~(H a) = O. Proof The fact that 0 is not an eigenvalue of H a, if H a is a 2-body Hamiltonian with a potential satisfying the virial condition is a direct consequence of the virial Theorem (see [ M o ] or [C.F.KS] for a proof). Indeed by computing the commutator [Ha, iAa], where A" is the generator of dilations on X a defined by Aa = OpW(xa" Dxo), we get: [H a, iZ a] = 2H a _ 2V~(xa) - xaVxaVa(xa). Using the virial condition, we get: E~o~(H") [H a, iAa]Eto~(H ~) > cEio~(H ~) ( x a) -UEio~(na). Since the operator of multiplication by ( x a) -~ is injective, this implies that 0 is not an eigenvalue of H ~, by the virial argument. [] We can now give the proof of Theorem 1.
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C. G6rard
Proof of Theorem 1 By Proposition 2.1, it suffices to consider an evolution of the form Ua(t) u, for a vector u~Lz(X) and a :~ amln- The case when a = amln can be handled using (2.3). Using the spectral Theorem, we will now decompose u with respect to the asymptotic energy H%. Indeed for any e > 0, we find c > 0 such that : u = Etc' + ~ot(na+ ) u + E l_ oo,-c~(H~+ ) u + E{o}(H~-)u + re,
(2.2)
where IJr,[[ =< 5. We claim that the following limits exist : lim
Ua*min(t)Ua(t)Etc , +oot(n~-)u,
t~+oo
lim
tJ*(t)Ua(t)Ej-|
-c~(na+)u.
r~+oo
Here tT,(t) is the evolution generated by the time-dependent Hamiltonian
I4a,t = H, + la,t(x~). The existence of the first limit is a consequence of the fact that:
Ua(t)Etc, +~t(H~+) u = F(~-~ > Co)U~(t)Etc,+~t(H')u+o(1), for some Co > 0 (see [Del, Theorem 5.3 c)]), and of the propagation estimates of Graf (see [Gr, Theorem 4.4]). The existence of the second limit can be found in [E2, Proposition 4.7] and is a consequence of the exponential decay of negative eigenvectors of a 2-body Schr6dinger Hamiltonian. Finally we claim that using Theorem 10.2 in Sect. 10, one has E~ot(H"+) = 0, for potentials satisfying hypotheses (H). Indeed we remark that the estimates proven in the following Sections for an evolution U (t) associated with a 2-body Hamiltonian with time-dependent potential apply also to the evolution Ua(t ) , if one considers propagation observables depending only on the internal variables (x ~, ~) . In particular we obtain that E~o~(H~+)= 0. From this result and (2.2), we get that for any e > 0, one has u = Ua. . . . "-[-Ua,c At- r~, where Ilr~lJ< e, and the following limits exist: lim U~m,,(t ) Ua(t)u ........ t~+oo
lim U*(t)
Ua(t)u~,~.
t~+co
By standard 2-body theory (see for example [Del]), we see then that the following strong limits exist: l i m e is~'~(t'/)*) U . . . . (t), t~+ao
lira e is~
o..)+i,n~ t~.(t)
(2.3)
So we obtain that u...,.,, belongs to Im O+r.,.' and that u.,~ belongs to Im O +. This proves Theorem 1 since (~)a. . . . . I m Q + is closed. [] In the rest of this Section, we will give some results from Classical Mechanics, which explain the quantum estimates of the following Sections. These results can be used to prove a natural analog of asymptotic completeness for 3-body classical systems with pair potentials satisfying hypotheses close to (H), as long as # is greater than x/~ - 1 (see [Ge]).
Asymptotic completeness for 3-particle long-range systems
339
We consider a classical 2-body Hamiltonian with a time-dependent potential: ht(x, 4) = ~1 42 + V ( x ) + I,(x) = h(x, 4) + I,(x), where Vsatisfies IV(x) I + (x)lOxV(x) I < c(x)-U, and the time-dependent potential satisfies lit(x) l + ( ( x ) § ( t ) ) Ic~xlt(x) 1< C ( ( x ) § ( t ) ) -~. We will assume that the potential V is radial. We denote by (Xo, 40) ~ T * X an initial point and by ~bt(Xo, 40) the Hamiltonian flow for ht starting at (Xo, 40) at time 0. As in Quantum Mechanics, there is a notion of Heisenberg derivative. Namely for a function f ( t , x, 4) ~CI(IRt x T ' X ) , we denote by D t f t h e total time derivative o f f equal to O,f+ {ht, f } , where {., .} denotes the Poisson bracket on T*X. The first observation is that the energy h has a limit along the classical trajectories. Indeed we see that the Heisenberg derivative of h is of size O ( t - l - u ) , from which it follows that: lira h(dpt(Xo, 40)) =:/~(Xo, 30) exists.
t~+Qo
We are interested in the "exceptional ~ates", i.e. the points (Xo, 30) such that /~(Xo, 30) = 0. Let us denote by t2o the set h-l({0}). The first observation is that if (Xo, 40) ~f2o, one has: [xt(xo, r
< Ct%
where cto = (1 + #/2) -1.
(2.4)
This can be proved by exactly the same arguments used in Sect. 4. It is also a consequence of a more general result of Derezinski (see [De3] ). Another easy remark is that: V(xo, 4o)~f2o,
one has: h(q~t(Xo, 40)) = O(t-~).
(2.5)
Next we use the virial condition, and compute the Heisenberg derivative of a(x, 4) = (x, 4). We get:
Dta(x, ~) = 2h(x, 3) + V(x) - x . Vxlt(x),
(2.6)
where IT(x) = - 2V(x) - x V~V(x). By the virial condition, we get:
Dta(x, ~) > 2h(x, 3) + c ( x ) - " - O(t-").
(2.7)
If we apply (2.7) along a trajectory (at(Xo, ~o) for (Xo, ~o)et2o, using (2.4) and (2.5), we get that: a(~t(Xo, 40)) > c tl -~o, (2.8) Using then the fact that (x, 3) is the Heisenberg derivative of x2~2, we obtain from (2.8) that: Ixt(xo, r > ct~~ (2.9) Next we use (2.9) to improve (2.5). Indeed if we compute the Heisenberg derivative of h, we get: Dth(x, 4) = 4" V~It(x). (2.10) If we use that h(~bt(xo, 4o) ) = O(t-u), and [xt(xo, 4o)l > c t ~~ we see that 4t(Xo, 40) = O(t-'~ which gives using (2.10) that:
u
~o)~t2o,
h(q~t(Xo, 30)) -- O(t-u-~~
.
(2.11)
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C. G6rard
Finally we compute the Heisenberg derivative of the angular momentum J = x A ~, and find using again (2.4) and integrating in t that:
V(Xo, ~o)~'2o,
J(~t(Xo, Go)) = O(t~~
(2.12)
In the next step we use polar coordinates. We put r = Ixh and denote by p the dual variable to r. Using (2.9), (2.12) and the fact that ~2 = p2 + j2/r2 ' we get easily that: V(Xo ' ~O)e~-~O' -2 1 Pt2 + V(r,) = O ( t - # - % . / 2 ) . (2.13) Using the fact that V(r) is negative for r > Ro (a consequence of the virial condition), we can introduce the following function:
K ( r ) = i ( - 2V(s))-t/2ds" Ro
Using (2.13), it is very easy to check that if (Xo, ~o)~f2o, one has: d ~ t K ( r t ( x o , ~o)) = 1 + O(t-u+~~ F r o m this we deduce that:
rt(xo, 40) -- F(t) = O ( t l - u ) ,
(2.14)
where F(t) is the inverse function to K ( r ) . A quantum analog to this estimate is obtained in Theorem 9.1. Using again (2.13), we can also prove that pt(Xo, 4o) - F'(t) = O(t -u) . So the exceptional trajectories live in a region of the form: Jr, -- F(t)[ < ct x-", IP~- F'(t)[ < ct -u. Using the fact that r and p are conjugate variables, we see that if p > 89 the uncertainty principle forbids that a quantum wave packet lives in such a region. Although due to the non commutativity of observables in the quantum case, we cannot really prove an estimate as sharp as (2.14), this is the heuristic explanation for the absence of exceptional states for the quantum problem, which is proven in Theorem 10.2.
3 A collection of technical results We collect in this Section some technical results which will be used in Sect. 5, 6. We will also explain two methods introduced originally by Sigal-Soffer in [S.S1], [S.$3] which we will use to obtain various plropagation estimates in the following Sections. We consider a Hamiltonian H --- ~D 2 + V(x), where V satisfies: [skV(x)l < C k ( X ) -u-lkl,
Vk~N".
(3.1)
and a time-dependent potential It satisfying: I,~/,(x)l < C k ( ( x ) + ( O ) - t - " ( ( x )
+ (t)~) -Ikl+l, u
(3.2)
for some 0 < ct < 1. We will denote by H ( t ) the time-dependent Hamiltonian H ( t ) = H + I,. We are interested in various commutation properties between functions of different self adjoint operators and also between classes of pseudodifferential operators. To s t u d y f ( A ) for an unbounded selfadjoint operator A, we
Asymptotic completeness for 3-particle long-range systems
341
will use formulas relatingf(A) with the resolvent (z - A)-1. They are described in Proposition 3.2. We first need a Lemma about almost analytic extensions of functions. We will denote by go the metric d22/(2) 2, and in general by F~(2) a function in the symbol class S((2)~,90). We will denote by F~(2 > 1) a symbol in S ( ( 2 ) ~, 00) supported in {2 > 1}. Lemma 3.1 Let f eC~(lR). Then there exist a funetion f r analytic extension o f f such that:
called an almost
IOef(z)l < CklImzl k, V k ~ N . Let f ~ S ( ( 2 ) - ~, #o ), for v > O. Then there exist a function f ~ C~ ( ff~) called an almost analytic extension o f f such that:
IOzfl 1)(H + i)Rdz)Fo([-~ < ~) = O(t-Nllmzl-~tk'N)), Vk > O, n~N, a# < 5 < ~ + ~l~/2.
(ii)
(x)l/2Fo(l~Jg >= 1)R,(z)D~(x) -1/2 =O(t-~u/2,Imzl-k), re# < 6 < ~ + ~1~/2,for some keN.
(iii)
D~Fo(I~ > l)R,(z) =O(t-'U/2llmz,-l),
(iv)
R,(z)SDk Fo(I~ > 1) = O(t-N'u/2)llmzl-~tN),
Va/~l)Rt(z)Fo(~J
o
=
x
t
t-~
0
> ~
1)o X
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C. G~rard
where for simplicity of notations we denote by/~o a function as Fo supported in ix[ => t ~. Using (iii) to estimate Rt(z)Fo(IXl/t ~ >=1)Dx, we obtain
1 Rt(z)D~(x) -1/2 = (x)a/ZFo
(:)
> 1 R,(z)D~Fo
x (x)-1/2 + ( x ) l / 2 F o ( ~ > xD~Fo = 11 +
I2.
(~
l'X ,/2 >->_~)(x)-
1)Rt(z)
Asymptotic completeness for 3-particle long-range systems
345
By (3.6), we have
III, II = O(t-~'[Imzl-C(N)), VNeN,
(3.11)
so it suffices to estimate 12. We have
X) I/2 Fo
>=
t z)D x l)Dx(x)-l/Z Fo(~-J > ~) + Rt(z)t~(D~(x)-'/2ffo(~>= 1 ) + (x) -3/2 Xffo
>=
, z)Dx( x )- l/2 Fo
>=
~gt(Z)ffo(~~)(X) -1+Rt(g)DxVo(L~a~-~) + t~R,(z)D~ffo
> 1 x)-I/2R,(z)D~Fo
(x) -1/2
+t~R,(z)(x)-3/2Fo( ~ >l)R,(z,DxFo(~J > ~ ) ( x ) -1/2 = O(t-~llmzl -x) + O(t-~"/ZlIrnz[-1 ) + O(t~-'-~UlImz[-2 ) + O (t ~-~"/2- 2~1im z[- 2)
= 0 (t-~"/2[Imzl-2), using (iii) and the fact that fi < c~+ c~#/2, # < 1. This and (3.11 ) completes the proof of (ii).
Proof of(iv): for N = 1, the result reduces to (iii). Assume by induction that we have proven (iv) for [kl = N. We write for k~N ~, Ikl = N:
Rrz 'N§,
~,
F o ( ~ > = 1 ) = R,(z)ND~R,(z)D~,R,(z)Fo(~ >=1)
+ R,(z)N[D~,R,(z)]D~Fo(~> 1) = R,(z)ND~R,(z)D~zR,(z)Fo(~ >=1) + R,(z)n§
1).
(3.12) Note that
IDa, HI =
E
ctD~-%V(x).
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C. G6rard
So we have
Rt(z)N+ ~t~[Dk, H]R,(z)D~Fo(~ > 1) = Rt(z)N+lt~Fo(]~ > ~)[D~,H]Rt(z)Dx, Fo(~J > l ) + O(t-N~176 (3.13) for all NoEIN, using (i) and the fact that ]]D~-ZRt(z)Nl] < C[Imz] -N, for Ill < g. Similarly, we have
z~g,
Rt(z)ND~R(t z )D x,Rt(z)F(~-jo > 1) = R, (z)D~F(~J>-~)R(z)DN o =1 t x,R,(z) xFo(~
>=
1)+O(t-N~176 (3.14)
for all No~N. Using (3.13) and (3.14) in (3.12), we get
Rt(z)U+lDkxDx'F~
> I)=Rt(z)ND~F~
>l)Rt(z)Dx'F~ =2J \t
+ ~ R,(z)N+IDk-'Fo t 1, one has
ct# + Ilia - Ilia#~2 - 6 > O, 5 < a + a#/2, # < i. This completes the proof of (iv). Proof of (v): we assume here only that e# < 6. As in the proof of (ii), we compute: since
i~t Rt(z)(x)Fo(~-~j - k~-Ik-jl~/2, 9) and that ctp/2 < ~, it is easy to check that the estimate of the Lemma holds for k + 1. [] Proposition 3.6 Let us denote by Ht the Schr6dinger V(x).Fo(IXl/t ~ > 1). Then one has: (i, VzeC~(IR,.(X(t~H, - Z ( t ' H , ) ) F o ( ~ >
operator 51 D ~ +
2 ) = O(t - ~ ,
for c~# < 6 < ~ + a#/2. (ii) Z ( t ~ H ) F o ( ~
2)
Fo(~
I)x'Dx)z~(H),
and we estimate separately the two terms [Z(tOH),AJ. Let us first estimate [Z(t~H), A~]. We will estimate the norm of the c o m m u t a t o r by the n o r m of each of its terms. Using Proposition 3.6(iii) , we get that Z(tOH) A1 and A~Z(tOH) have a n o r m O(ts-st~/2) , if s# < 5. So we get :
[z(t~n), A~] = O(t~-~"/2),
(3.19)
if s# < (~. Let us now estimate [Z(t~H), A2]. Using formula (3.3), we write: [Z(tOn), A2]
=
i ~ I 6~e~(2') Rt(z) t~zl(n) KzI(H) Rt(z) dz a di,
(3.20)
where K is equal to: K = OpW(~ZFo + (x. 4) 2lxl-at-'Fo' - Fox" VxV(x)). Using L e m m a 3.4(iii), we get that: [)~(t~n), A2] = O(tO-s"),
(3.21)
if 6 > s#. Next we put together (3.19) and (3.21), and take the optimal s which is equal to s = Cr and satisfies clearly s# < 6. F o r this value of s, we obtain that: [)C(t~n), A] = O(t m-~~
(3.22)
F r o m (3.22), we get that the first term in (3.18) has a norm O(t m-'ou)-~) . Let us now estimate the second term in (3.18). Using again formula (3.3), we write it as: i
~
(t-rA + i) ~ ~ OeF_~(z) (z -- t-rA) -~t -~ [Z(t~n), A](z - t-rA)-~dz ^ de. (3.23) By (3.22), we know that [Z(t~H), A] has a norm O(t m-'~ Next we use the fact that [[(t-VA + i) (z - t-rZ)-~[[ has a norm O((z)llmz[ -1) uniformly in t, and we see that the integrand in (3.3) has a norm O ((z)] Im z I-z t m -,o~)- r). Using then the estimates of L e m m a 3.1 on d e f t - , we get that: II(t-rh + i) I-z, F_~(t-rA)]ll = O(t~"-~~ which completes the proof of (i).
(3.24)
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C. G6rard
Proof of (ii): the proof of (ii) is similar to but simpler than the one of (i), using instead the fact that [H, A] has a norm O(1).
Proof of (iii): as usual we start by considering resolvents. We write:
Fo(l~ < 1)](z-t-rA)-lFoC-~
=0+(z-t-'A)-lt-'[A,
>2).
Using the result of Lemma 3.8 (ii) below, we know that III-a, Fo] tt = O(1). Applying then formula (3.3), we see that (iii) holds.
Proof of(iv): we write Fo(2) = (2 + i) F_ 1(2), which gives [Fo(t-'A),Fo(~ < 1 ) ] = t-'IA, Fo(l~ < 1 ) ; F - I ( t - ' A , + (t-rA + i ,
= I1 + I2. By Lemma 3.8 (ii), we know that [A, fo(Ix[/t ~ < 1)] = O(1), so that I1 = O(t -~). To estimate 12, we write using formula (3.3)
[
(t-~A + i) f _l(t-~A),Fo
" is b o u n d e d by Ilmzl -~") for z in a compact set of ~. The details are left to the reader.
[]
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C. Grrard
In the rest of the Section, we recall two abstract methods introduced by Sigal-Soffer (see [S.S1], [S-$3]), which we will use to prove propagation estimates. We will denote by U(t) the unitary propagator associated with an abstract timedependent Hamiltonian H ( t ) = H + I,, where H is an unbounded self-adjoint operator and for example It is a C 1 bounded self-adjoint operator. The first method allows to prove "weak" propagation estimates:
Proposition 3.9 Let t~--~eb(t) a C 1 map from ~ + into the Banach space B ( W ) o f bounded selfadjoint operators on an Hilbert space ~r such that supt=>o II~(t)II < oo. For a time-dependent selfadjoint operator H(t) , we denote by Dqb(t) the Heisenberg derivative of ~(t) defined by: d~ DcP(t) = - ~ + [H(t), i+]. Assume that there exist operators B(t) and Bi(t) for i = 1. . . . .
n such that :
Dcb(t) ~ B*(t)B(t) -- ~B*(t)Bi(t), i
+~[]Bi(t) U(t) ull2dt _- O.
4 Large velocity estimates In this Section, we will prove large velocity estimates for 2-body Hamiltonians with time-dependent potentials. We will also prove a rather easy propagation estimate about the energy operator H for such Hamiltonians. We will consider a timedependent Hamiltonian H ( t ) = H + It(x), where H = ~Dx 1 2 + V(x) is a two-body Schrrdinger Hamiltonian, and V and It satisfy the conditions in Sect. 3, and denote by U (t) the unitary propagator associated with H (t). We first need some auxiliary estimates, summarized in the next proposition.
Asymptotic completeness for 3-particle long-range systems
355
Proposition 4.1 Let us denote by g the metric: dx 2 (x) 2+t
d~ 2 2~ -~ 42 -
Then the following estimates hold: (i) a d ~ ( I , ) F o ( ~ (ii)
=O(t -1-u-(k-1)a) fromHk(lR")intoL2(lRn).
> 1)
a d ~ ( l , ) F o ( ~ > 1)(x)~~ = O ( t -1-~-~k-1)~+~~
Hk(IR ")
into L2 (IR"). (iii) ad~ Fp
>1
eOp S ~
-k,g , VpsN +.
k((x)+t~)
Proof. The proofs of (i) and (ii) are easy consequences of symbolic calculus and of /i
i
X
the fact that on supp F o ( ~ l >-_ 1), V satisfies the estimates: /
I ~ g l 0, Co > 0,
there exists M = n(#, 6, ~, Co) such that : Z(taH) f O\eo+ // [xl ~ => 1) U(t)u
=< C~.~or-~~
Proof. Our proof will be a modification of the abstract method due to Sigal-Soffer recalled in Sect. 3, which is very useful to obtain strong propagation estimates. Let us pick a function F,(2)r with F . ( 2 ) = F2/2(2), and F . ' ( 2 ) = F~,-~)/2(2), such that F, is supported in {2 > 1}. We also pick a cutoff function z~C~~ , such that Z > 0, 2Z'(2) < 0. We will consider the following (unbounded) propagation observable: M:= F.([~)Z(t~H)F.([~),
(4.1)
for ~ = ao + e. Our proof will consist in showing by induction on n that (for suitable vectors u ), the quantities: ( M ) t : = ( M U (t)u, U(t)u),
356
C. G&ard
are bounded. Let us denote by ut the vector U(t)u. Following the idea of Sigal-Soffer, we will first compute the Heisenberg derivative DtM of M. We have:
D t M = DtFnXFn + FnDtXFn + FnzDtF. = 11 + 12 + 13.
(4.2)
Step 1: in Step 1, we consider I2. We have:
Dtz(t~H) = 6t ~- 1HZ,(tOH) + [It, iz (tOH)].
(4.3)
Note that the first term in (4.3) is a negative operator. We will consider the second term, and put it in a form suitable for our future induction argument. Using the commutator expansion of Lemma 3.3 in Sect. 3, we may write: N-1
1
[I, iz(tOH)] = ~ ~.t
k6
k
Z ( I(tOH)adkn(It)+ Rm
(4.4)
k=l
where one has: i
RN = ~ ~~e~(z) Rt(z)-NtN~ad~(It)Rt(z) dz/x d~.
(4.5)
We write F.RNF. as:
F.RNF. = i ~ ~(z)F.Rt(z)_NtN~ad~(it)R,(z)F. dz/x d~
(4.6) where ]]B~oH = O (t - ~ ), using Lemma 3.4 (i). Using again Lemma 3.4(i), Proposition 4.1(i) and the fact that we choose 6 = ~ (F._ 1/2Xk (t~H)tkaB~zt (t~H) F._ 1/2 u,, u,)
k=l
+ O(t 2n(1-~)-l-~+(N-1)(~-~)) II UllM, 2 u,
(4.13)
and the operators Bk have a n o r m O ( t - l - " - ( k - 1 ) ~ ) . We will now compute It, the term I3 being similar. We have: D,F.= and:
= Ill + I12.
(4.14)
Step 2 : in Step 2, we consider 1~2. This term has the form:
where Bo = Bo(x) is a b o u n d e d function. As in Step 1, we will now insert (modulo small errors), terms of the form Zl (toH) for some cutoff function Zt eC~~ with ~ - 1 on supp 2- Indeed if C is equal to one of the differential operators ( X ) - 1/2 F0 x/lxl 9D~, ( x ) 1/2 F0 or ( x ) - 1/2 Fo, we may write using the c o m m u t a t o r
358
C. G~rard
expansion: N-1
1
11 (t~H)C = CXl (t6H) + ~=1~. tk6adk (C) z~k)(t~H) + RN,
(4.16)
where RN has a norm at most O(t ~/z+N~-~)) using Proposition 1.1 (iii). Since xtk)z = 0, we may write:
112 = t-~Fn-1/2)(AC1z1Fn-1/2 + t-2~Fn-1/2z1C2ZIFn-1/2 + Fn--1/2RNFn-u2, (4.17) where using Proposition 3.6, one has: X
C1 = ( x ) - t/2F o ~ . DxZ (t6H)(x) 1/2Fo = 0 (t -~u/2 ),
C2 = (x)-X/2FoZ(t~H) (x)1/2F0 = O(1),
(4.18)
and using again (4.12), we have:
[(Fn-1/2gNF.-1/2u,, u,)l < C ( t ) (2"-1)
~,+~-~ON
u
2 M,M,
(4.19)
for M = M(n). Summarizing, we have:
(lazut, ut) = t-~-~u/2(BoZ1Fn-uaut, ztF.-1/2ut) + O(tt2n-X~=+t~-=~N)IlUiI~,M.
(4.20) where Bo has a norm 0(1). Step 3: in Step 3, we will consider the term I t 1. By the properties of F,, we have: I11 = -- t - 1 Fn/2ZFn/2. ~2 2 Our goal is to symmetrize this expression to turn it into a negative operator modulo small errors. We write: F2/2 )(,F2/2 = gn/2 Fn/2zF,/2 F,,/2 + ft,~2 [Z, ft,~2]F2/2 + ft.~2 [L F./z ]fn/2 fin~2" (4.21) Note that the last two terms have the same structure. Next we use the commutator expansion Lemma to expand [;t(t~H), Fp]. By Proposition 4.1, we know that: ad~ (Fp)~OpWS (\ ( xt )~p" ( ~ ) k ( ( x ) + U) - k , g ) ,
(4.22)
where we recall that the metric g is equal to: dx 2 ( x ) 2 + t 2,
d~ 2 +-~2 "
By symbolic calculus, we notice that we can write ad~(Fp) as: adk (Fv) = t -~k Fp-kBk ( t , X, Dx), where the operator BR has a norm 0 ( 1 ) from Hk(IR ~) into O p w S ( ( ~ ) k, g). So we may write: N-t ~ ~ 2n/2 = Fn/2 [Z, Fn/2]f
E
k=l
L2(~xn) and
is in
1
~..Fn-kt(~-~)kBkz(k)( t~H)F" + Rm 9
(4.23)
Asymptotic completeness for 3-particle long-range systems
359
where the remainder RN is equal to: i
~ O~(z) ff,/2(R,(z))Nad~(ff,/z) tN~R,(z) F.dz^d2.
Using again Proposition 4.1 and (4.12), we get that:
KRNu,, u,)l < Ct 3"r
N IlulIM, 2 M,
(4.24)
for N > n/2, M = M ( N ) . Let us now examine one term in (4.23). By c o m m u t i n g again Fk/2 through X(kl(taH) , we may write: N-1
Fn-kBkX(k)(t~H) Fn = Z Fn-k/2-iBk, iXi(t~H)Fn-k/2 -i -1- RN, k,
(4.25)
i=0
where RN, k c a n be estimated as in (4.24) and the operators 0 ( 1 ) . Summing up (4.21 . . . . . 4.25), we have:
Bk, i have norms of size
N
Ill = -- t-lF.xFn + t -1+~-~ ~ F.-k/2Zk(t~H)BkZk(t~H)F.-k/2 k=l
+ O (t z,~ 1 - ~ + ( ~ - ~ , ~l~,,~2 )H~e~IM,
(4.26)
M -
Here Bk is a bounded operator of n o r m O (1), and )~keC~ (IR). The fact that we may insert these cutoff functions to the left and right of Bk follows from the same commutation argument explained in the Steps 1 and 2. Step 4: in Step 4 we prove the Theorem by induction on n. Let us first summarize the consequences of the Steps 1 to 3. We have: N-1
(D, Mu,,u,) < t-l+~-~ ~ (Fn_k/2ZkBkZkFn_k/2Ut,tit) k=l N-1
+ t -i-u+~ ~, (F,-1/zZkBkZkFn-i/2ut, Ut) k=l
+ t - ~- ~,/2 ( F , _ ~/2Z1BoX~ F . _ 1/2 u~, u~)
+ O(t2"t~-~+~t~-~')llull~, M.
(4.27)
where all the operators B~ are b o u n d e d with norms O ( 1 ) . Start of induction: we pick n < ~. Then if ueH c~"~)'cr all the terms in the right hand side of (4.27) are in L ~ (dr) since 6 < ~t < a, and applying Sigal-Soffer's argument of Sect. 3, we get:
Z(t~H)F,(I~ >=1)u,
< Cllullc,,~,,c(,,).
(4.28)
Assume now that we have proven that:
x(tnH)F.(I-~ >=l )ut ao,
VN~IN, ~M = M(a, N)
Z(t~H) F.(I-~ >
1)ut
such that:
~CNIlulIM, M,
Vn c%, VNeN, 3M = M(c~, N) such that:
Indeed this follows again from the commutator expansion Lemma, by writing: N-1 1
F,X = xF, + k~=l~ t~a-~)k)(k)(tnH) BkF,-k + Ru.
(4.32)
Here IIRNut II is of size O(t "(1 -')+u(~-')) IlullM,~, and BR is a differential operator with a symbol in S ( ( ~ ) k, 9) supported in {Ixl/t ~ > 89 We may again commute Bk with z(k)(t~H) and we finally obtain: N-I
N-1
F.Z = )~F, + E
E kt-lt(~-"kBk, jzJk)(t~
j=l
F,_, + RN,
(4.33)
k=l
w h e r e / ~ can be estimated as Rs. Then (4.31) follows from (4.30) and (4.33). To complete the proof, we remark that the estimate (4.31) can be written as:
(F, zu,, Zu,) < Cllull~,~. We use then that: >=1)>= t("-')"Fo(l~l, >1),
F,(~
(434)
if c( > ~. So (4,31) implies that: t"(''-',
1)Zu.zu,> t-lX(tOH) + R(t), where llR(t) [[~L~(dt). from which the Proposition follows by Proposition 3.9. []
5 The generator of dilations In this Section we prove propagation estimates for the generator of dilations 89 Dx + Dx. x), which will be used in the next Section. We assume in this Section that the potential V(x) satisfies the virial condition:
2v(x) +
x. vxv(x)
0. Recall from Sect. 3 that we defined the cutoff generator of dilations A to be: A: = •, (H) OpW(x 9Dx))~l (H), for a cutoff function 7~1e C ~ (N) equal to 1 near the origin. The result of this Section is the following Theorem: Theorem 5.1 Let zeC~ (]R ) be a cutoff function equal to 1 near the origin. Then for any 6,# with Crop=Co(2) ~-~
in { 2 < 0 } .
(5.2)
We also pick a function x e C ~ (N) such that 2Z'(zl) < 0, and ~(2) ~ 1 near 0, and a cutofffunction Fo EC~~ (~-) also equal to 1 near the origin. To simplify notations, we will still denote by A the operator ~1 ~(~( H ) O p W ( x ' D ~ ) x l ( H ) - ct a-~" for some function Ha as ~ and some small constant c > 0. We consider the following propagation observable:
M =Z(t~H)Fo([~
Cto, we obtain:
Z( t~ H)DtFoF I-vFoz( t~ H) =O(t-~)Z~(t~H)B~176 = O(t-1)Fo
>
>i]
1
Xl(t~H)BoFo
~1
Z(t'n)'{'-O(t -~ )
>=
• n . /'lxJ~- --< 1I/ X(taH) + O ( t - ~ ) ,
(5.9)
where Bo is a multiplication operator of norm O (1) and B, is an operator of norm O(t ~-~"/2-~) by (5.6). Step 3: in Step 3 we will compute the term D,F,_v. We have:
DtFI_ v = -- 7 t - ~ - l A F ' l _ v ( t - ~ A ) + D, Fl_~(s-~A)l~=t.
(5.10)
We claim that: t-y'
[H (t), iF1 _~ ((t>-~A)] = ~-(F'~ _v(t-~A) [H (t), iAJ
+ [n(t),iA]F'l_,(t-rA) + O(t-3r).
(5.11)
This will be a consequence of the commutator expansion. Indeed to use Lemma 3.3, we write as usual F 1 - , ( 2 ) = (2 + i ) F _ , ( 2 ) , and:
2[H(t),iF,_~(t-~A)] = t - ~ [ n ( t ) , iA]F_~(t-~A) + t - ~ F _ ~ ( t - ~ A ) [ n ( t ) , i A ] + ( t - r A + i)[H(t), iF_~(t-~A)] + [H(t), i F _ , ( t - r A ) ] ( t - r A + i). Next we use Lemma 3.3 to get:
(t-~A + i)[H(t), iF_~(t-rA)] + [H(t), iF_,(t-rA)](t-~A + i) = t-7(t-rA + i)F'_~(t-~A)[H(t),iA] + 1 t-2~,(t-rA + i) F,. (t-rA)i ad](H(t)) + (t-rA + i)R3 + t-~[H(t),iA]F-~(t-rA)(t -rA + i) -- ~1 t-2~iad2(H(t)) F"_,(t-~A) + R3(t-rA + i) = t-~(t-~A + i) F_,(t -r A ) [ n ( t ) , iA] + t-r[H(t), iAJF'~(t-~A)(t-~A + i) + 89
[ad2(n(t))]] + (t-~A + i)R~ + R3(t-~A + i),
364
C. G6rard
where we denoted by F_~_ 1(2) the function (2 + i) F ' J 2 ) . Using Lemma 3.8, we see that: [F_,_,(t-VA), [ad2(H(t))]] = O(t-'). (5.12) Let us now estimate the term (t-VA + i)R3, the term R3(t-VA + i) being similar. We have: i
-
(t-~A + i)R3 = ~ - ~ a ~ F - , ( z ) ( t - ~ A + i)(z -- t-~A)-3t-3~ad ] x ( H (t))(z - t-VA)- 1 dz ^ dS.
(5.13)
The integrand in (5.13) has a norm O ( ( z ) [ I m z[-*) by Lemma 3.8(iv), so one has: ][(t-~A + i)R311 = O(t -3~)
(5.14)
Putting together (5.12) and (5.14) gives (5.11). F r o m (5.11) and (5.10), we get: D, F I - v = 1 t - ~ ( F ,
v(t-M)D,A
+ D , A F ' a _ J t - M ) ) + O(t-3r).
(5.15)
Let us now compute DtA. We have, using L e m m a 3.80): DtA = z I ( H ) ( 2 H + lT)xl(n) -- c(1 -- r
-~" + r~,
where l?(x) = - 2V(x) - x . V x V ( x ) , and r, = [I,, iA]. Using Proposition 3.6(ii), (ii'), we may write: t - ~Z(t~H) FoF'l - ~ ( t - r A ) Xl (H)(2H + ~'(x)) Z1 (H) FoZ(t~H) = t - r Z ( t Z H ) F o F ' ~ _ J t - r A ) ( 2 H z I ( t Z H ) + VJx))FoZ(t~H) + O ( t - ~ ) .
(5.16)
where l?,(x) = IT(x) F(Ixl/t ~ < 1) + ct-~"F(Ixl/t ~ >= 1) satisfies V,(x) > ct -~", using the virial condition, and the cutoff XI is equal to 1 near supp X- So we get: XFo D, F1-v(s-~A)ls=, FoX = t - r Z F o F ' I - J Z ~ ( t ~ H ) 2 H + Vt - c t - ' " + r,)FoX + t-TXFo(ZI(t~H)2H + V, -- ct -~" + rt) x F'l -vFoZ + O(t-3~). Next we use again L e m m a 3.3 to symmetrize this term. We will put: B = 2HxI(tZH) + ~',(x) = B1 + B2. We get that: ZFoDtF1-v(s-~A)ls=t FoX = tYZFo(F'I-v) 1/2 (B - ct -'u + rt)(F'l _~)I/2FoZ + t - rXFo [ [(F'I - ,)~/2, B + rt], (F'l - ~)l/2]FoX + O(t-3~). So we need to estimate terms of the form: [[F_~o, B,], F _ J
= I1,
[[ F _ ~o, B~], F_~o] = 12, [ [ F - v o , r,], F _ , o ] = Is,
Asymptotic completeness for 3-particle long-range systems
365
for some vo > 0. Let us first consider I2. Using formula (3.3), we get: i
-
[ F-vo(t-'~A),B2] = ~ S O~F-~o(Z)(Z - t - r A ) - a t-~x 9 VxV,(x)(z - t - ~ A ) d z A d~. (5.17) We deduce from (5.17) that [F_~o(t-eA), using again (3.3):
B2] has
a n o r m O(t -~) . Next we write
[[F-~o(t-~A), B2], F-~o(t-~A)] i ~ =~jc~F_~o(z)(z
- t-rA)
it ~'[x. VxVt(x),F_~o(t ~A)](z - t - ~ A ) - l d z A d 2 .
We remark that x. V~Vt(x) is a potential satisfying the same estimates as lT~(x), so the term
[ x . V~Vt(x), F-~o(t-VA)] has a n o r m of size O(t -~) . F r o m this we deduce that I2 has a n o r m O(t-2~). Let us now estimate I1. If we write Ba as t - ~ ( r a H ) for some ~ C ~ ( I R ) , we get: i [F_~o(t-~A), ~t(taH)] = ~ ~ O~F_~o(Z)(Z - t - ~ A ) - l t - ~
•
- t-~A)dzAd2.
(5.18)
Using the estimate (3.22), we get that [F-~o(t-~A),BI] (and hence [ [ F .... (t-~A), B1], F _ J ) has a n o r m of size O(t-~-a~~ Finally let us estimate 13. We have: rt = D(I(H), I t ] O p W ( x ' D x ) z I ( H ) + h.c. + z I ( H ) I t x , ( H ) , where ~ ( x ) = x . Vfl,(x). A computation similar to the one done in the proof of Lernma 3.8(iii) shows that [F_,o(t-YA),r,] (and hence [[F_,o(t-rA),r, ], F_,o(t-~'A)] ) is of n o r m O(t -~-~) . Summing up, we have obtained:
Z FoDtFI - 9(s- rA )Is=tFoZ = t-~ZFo(F'I_v)I/2(B - ct-'u + rt)(Fl_v')l/2FoZ + O(t -2~-~ou) + O(t -3y) > c~zFot-r-'~'F'~ _ , ( t - r A ) F o z + O(t -2r-o~o") + O(t-3~).
(5.19)
To complete Step 3, we transform the factor: R ( t ) : = - 7t-~'-~AF'~-~(t-~A) + c~t-~-~UF'~-~(t-~A) in the following way: We write it as G(t, t-~A) for the function G(t, 2 ) = - yt-~2F'l_~(2) + Clt-~'-"-F'~-~(2). We claim now that if 7 + a# < 1, one has:
G(t, 2) > ct-~'-~"F'~ _~(2)
(5.20)
for t > T. Indeed since F'~ _~ is positive and supported in ] - oo, C], the term -Vt-~2F'~_~(2) is clearly positive for 2 < 0, and G(t, 2) is als6 greater than ct-~-'"F'~ _~(2) for t large enough for 0 < 2 -< IE. F r o m (5.20), we get for t > T:
R(t) > ct-~-~UF'~-,.(t-~A). This completes Step 3.
(5.21)
366
C. G6rard
Step 4: in Step 4 we prove the Theorem. Let us first summarize the results of the previous steps : Putting together the estimates of (5.7, 5.8, 5.9, 5.15, 5.21), we obtain:
DiM > O(t-')Fo
> 1 BoXI(t~H)Fo
>
-- ct ~-~u/2+'~-l-~t-~ - Ct ~r176
nlro
< 1 zI(t~H)
- ct -2~-r
-- ct -3"r
(5.22)
for Bo, B1 as in (5.9). We take then a state u e H N'N, where N will be chosen later, and we consider the quantity (Mu,, u,). We have:
-~t (
u,, ut) = (D, Mut, u,) > -- Ct-t+~-~'/2-rllZl(t~H)fo _
>
u, llllull
ct~-~/2-~-l+e,-,llull2 _ ct~l-~o,)-r-a [lull2
- ct-2~-a~o~tlull m - ct-3ellull 2.
(5.23)
If we take u ~ H N'N with N = N(e, 6, # ) , we know by Theorem 4.2 that:
t - t +~-~/~-~ Z1 ( t~H)Fo ( ~
>=l ) u ,
e L ~ ( d t).
We claim now that for given 6, g, with % # < c5 < #, there exists eo and Yo < 1 - % # such that for e < eo, ~ = % + e, y = Yo, all the terms in (5.23) are in L l ( d t ) . This is easily seen using that %(1 + #/2) = 1 and that # < 1. By the Sigal-Soffer argument, we get:
IFt_f/2(t-'A)Fo(l~
< l)z,(t~H)ut
< CllullN, N.
(5.24)
for N = N(e, r P), and y = ~o. Let us now transform the estimate (5.24) into a decay estimate. We remark that by (5.2), one has: u
1,
[F,_~(2),>CoC'-~Fo(-~
< -1),
uniformly for 2elR, C >> 1. W e apply this inequality with 2 = t-r~ for v, to be chosen later. We obtain from (5.24):
t~,~-~) F o ( t - ~ o - ~ A
F o ( z ~ ( n ) O p W ( x . D ~ ) Z ~ ( n )
< ct~-~"),
and we obtain:
F o ( t - ~ + ~ A < c)Fo([-~ < 1 ) z ( t ' H ) u t ~ C t This completes the p r o o f of the Theorem.
[]
'l-~)t~l) ,IuIIN,~,.
(5.25)
Asymptotic completeness for 3-particle long-range systems
367
6 Minimal velocity estimates
In this Section we prove some minimal velocity estimates, using the results of Sect. 5. We have the following Theorem: Theorem 6.1 Let zeC~ ( R ) a cutoff function equal to 1 near the origin. Then for any
c}, I~ with eol~ < 6 < p, there exist eo and co such that : V0 < e < So, 3N = N(~, 6, #) with: fo(~lxl
< 1)" Z(tag) U(t)u
< C~.a.,t-Collulls.N.
Proof. We will again use the now familiar method of constructing p r o p a g a t i o n observables. W e denote by B the observable x2/2 - ct2-% for some ct = Cto + 5, and c ,~ 1. As in the previous Section we pick a function F l e S ( ( 2 ) , in {2 < C} for some C > 0, such that:
go) supported
F1 = _ F1/2, 2
Fi
=
FL
- F1(2) > Co(2)
in 2 < 0.
(6.1)
We also pick a function xeC~~ such that 2)((2) < 0, and )~(2) - l near 0, and a cutoff function F o ~ ; ~ equal to 1 near 0. We consider the observable M defined by:
M:=X(t~H)F (l~ = cl > c. T o estimate the terms containing R1, Rz we will use the propagation estimates on the generator of dilations obtained in Sect. 5. Let us first consider the easiest one, i.e. the one containing R : . Let us denote by ff~ the vector Fo(lXl/t')Zut. We claim that for fl > 1, we have:
[(Fo(t-l+~UA < ci)Fo(t-~B)~t, Fo(t-~B)~bt)] < C ~ o t - ~ l l u l l 2 M ,
(6.12)
for some c~ > 0 and M = M(fi,/~, z), c~ = % + e. Indeed this follows from T h e o r e m 5.1 and the fact that [Fo(t-~+~"A), Fo(t-aB)] = O(tl-P), which can be proven using formula (3.3). Let us now estimate the term containing R~. We have:
I ( A F o ( t - I + ~ A < cl)Fo(t-~B)~,, Fo(t-~B)~t)l < ]]Fo(t-t+~uA < ct)Fo(t-an)~otll IIAFo(t-aB)q/tl[ < U-~u/21[ Fo(t- 1 +~UA < Cl)Fo(t-aB)~b, II IIu II < C~ot~-~"/2-~ IIull~,M,
(6.13)
for M = M(6, #, e). We use here Proposition 3.6(iii) to estimate the norm of AFo(t-PB)Fo(Ix[/U)Z(tOH). Let us now summarize what we have obtained so far in Step 3: putting together the estimates (6.10 . . . . ,6.13), we obtain:
(ZFoDtFa (t-~B)FoZ), >=( Z F o ( - fit - ~- 1BF~ (t - ~B) + c t - ~+1 - ~"F~(t -~B))Fo Z )t -
O(t-~+~-~lz-~)I[
u [1~ , M ,
(6.14)
for some Cl > 0, and for some M as above. To complete Step 3, we notice that the function of B in the second line of (6.14) can be written as: -
fit - lZF,x (2) + c t - p + i - ~"F ~ ( 2 ) ,
370
C. G6rard
for 2 = t-r As in Sect. 5, we claim that this function is positive for t > T i f w e pick fl < 2 - ep. So we finally obtain:
, > O(t -#+a-~"/2-c') 11U]]~.M ,
(6.15)
for some cl > 0, and for some M as above. This completes Step 3. Step 4: in Step 4, we finish the p r o o f of the Theorem. Let us first put together the estimates of the previous steps. By combining the estimates (6.6), (6.8), and (6.15), we obtain:
( D t M ) t > - C t - X +n-"+r
2 - CtO-a-~/2ttull 2 - Ct-I +Z~-a-C~ -- Ct - # +'-~"/2 - ~ ' ti u li ~,N. (6.16) Here co > 0 can be chosen arbitrarily large, if we take N = N(6, ~, e, Co) large enough. We claim that for given 6,/~ with C~o# < 6 < #, there exists some eo > 0 and some 1 < flo < 2 - ~o~ such that for 0 < e < ~o, ~ = % + ~ and for fl = flo, all the terms in the right hand side of (6.16) are in Z l(dt). This follqws again from the fact that ~o(1 + p/2) = 1. If we put fl = flo, we can apply to ( M > , the argument of Sigal-Soffer, which yields:
< 1)x(PH)u,t < t ' ' for ~' < 7o. Indeed we use that: IFx(2)l >_- cFo(c-~2 > 1. We pick c = t ' ' for some e~ > 0 small enough such that ~ + flo < 2 - ~o#, and we notice that the cutoff Fo(t-P~ 2 - ct z - ' u ) ~ t '~) is of the form Fo(x 2 < clt z-~") for some ca > 0 since flo + ea < 2 - ~o/~. So by (6.18), we obtain:
Fo(x2 0
Asymptotic completeness for 3-particle long-range systems
371
The two first operations will use the minimal velocity estimates of Sect. 6. The last one will use another propagation estimate about the angular momentum operator for which we assume that the potential V(x) is radial. This estimate is described in Theorem 7.1 below. Let us denote by J = OpW(x A Dx) the angular momentum operator. Theorem 7.1 Assume that the potential V(x) is radial. Then for any 3, # with
% # < 6 < 1 t , for any f l > % - i t , there exist eo and constants c=c(e, fl), N = N(b, It, e) such that for any 0 < e < Co, one has:
) z(r
Lxl < 1 Fo(t- 2#J 2 >=1)Fo (\t~o+~
89
(7.4)
for a bounded operator Bo, which gives, using again Theorem 4.2:
[(zDtFoFI_v(t-zBj2)Fo~()t[ 0, N = N(6, It, e, Co). This completes Step 2.
(7.5)
372
C. G 6 r a r d
Step 3: in Step 3 we compute the term containing D , F , _ v . We have:
ZFo D, F1 - ~(t - 2#j 2)Fo X = - 2fit - 1 - 2axF~ j 2F '1- v(t - 20j 2 ) Fo X + XFo [It, iF1 _~(t - 2aj 2)] FoX 9
(7.6)
Let us compute the second term in (7.6). If we write F l - v ( 2 ) = 2F-v(2), we obtain:
Fo (b-~,
as:
< D t M + RtMU(t)u, U(t)v) . Here M = Z(t~H)Fo(Ixl/t ~1 0, if we choose cq = ~0 - ~ close enough to ~o. From Proposition 4.3, (which using (7.15) applies also to the evolution U(t)), we see that: t - 1/2 l[Z2(t6n) U ( t ) v I[a L 2 (dt). The same is true of t -1/~ [ l ~ 2 ( t 6 H ) M U ( t ) u II. Indeed one has: ~ 2 ( t 6 H ) M U ( t ) u = M ~ 2 ( t 6 H ) U (t)u + O ( t ~ - ~ - ~ , / 2 ) .
Using (7.23) and Cauchy-Schwartz inequality, we obtain: IIf(t2) -f(t~)lt 2 -< o(t2, q)llull I[vll + C
(tf )1/2 t-lCll~2(t~H)U(t)ull2dt \tl
= o ( t 2 , tx)II u I1 II v II ,
which proves the existence of the limit (7.16), and completes the proof of the first claim of the Theorem. The second claim is obvious by the definition of H+ and the fact that: ( z I ( H ) - z ~ ( I 4 ) ) U ( t ) ~ = (z~(H) - x ~ ( H ) ) F ( ~
> 1)U(t)~b+o(1)=o(1),
using Proposition 3.60). This completes the proof of the Theorem.
[]
In the sequel, we will still denote by H ( t ) and U(t) the Hamiltonian and propagator/t(t) and t~(t). We will use the fact that now the full potential V = E is radial to suppress the angular part of the Laplacian. Let us first recall some results about polar coordinates and introduce some notations. The change to polar coordinates will be implemented by the following unitary transformation: T: L2(]R n) ~ L2(IR +, L2(S "- 1)) , u: ~-~ Tu(r, oo) = c.r ~"- 1}/2u(r~o) .
Asymptotic completeness for 3-particle long-range systems
377
Formally one has the following identity (see for example [H6]): T~-=
D~2 +
n
1
(n-l) =
1
1L 2
7~+~
T,
where L 2 is the Laplace-Beltrami operator on S"- 1 and L 2 T = TJ 2. We define the time-dependent Hamiltonian H(t) to be: /4(t):= 89 2 + V(r) + l(t, ro3),
(7.24)
and its time-independent part / ~ : = l2 D r2 +
V(r)
= H z ( I R + , L 2 (S . - 1 )) n Hot (IR + , L z ( ~. - 1 )). It is well known that /q(t) is selfadjoint on D so that the unitary evolution U(t) generated by/4(t) exists. If u 9 D, one has:
with domain D
{
i#,~(t)u=#~)D(t)u,
i#, U* (t)u = U *(t)H(t)u .
We have the following Theorem: Theorem 7.4 Let H (t ) be a Hamiltonian satisfying the hypotheses of Sect. 3for # > 89 with a potential V which is radial. Then for any exceptional state ~b9 Eio}(H + ) (L 2(X)), the following limit exists: ~ : = lim ~)*(t)TU(t)dp. t~+oo
Moreover the vector ~p is in E{o}(H+)(L~(IR +, Lz(s"-t))), where the asymptotic energy Jq+ is defined as H+ by replacing U(t) with U(t) and H with H. Proof As in the proof of Theorem 7.3, we first replace modulo o(1) the vector U(t)(o by Z(t6H)U(t)ck. By density we can replace tk by a vector u e H M'M for M >> 1. Let us pick a = ao + s, a' = ~o - e and fl > % - # for ~ small enough. By Theorems 4.2, 6.1 and 7.1, we have:
Z(t~H)U(t)u= Fo(t-2oj2 1)Fo ( ~
< 1) Z(tbH)U(t)u+o(1),
for a function Fo 9 C~~(]R), equal to 1 near the origin. Since the proof is very similar to the one of Theorem 7.3, we will just indicate the main points.As before we pick a vector v 9 +, L2(S"-1)) and we differentiate f ( t ) = ( U * ( t ) T m U ( t ) u , v ) where:
M =Fo(t-2~J z < 1)Fo ( ~
> 1)Fo ( ~ _ -< 1)Z(t~H) 9
We first remark that due to the cutoff Fo (Ixl/U' > 1) present in M, TMU(t)u belongs to the domain D of/4(t), so we can differentiate f(t). We obtain:
if(t) = ( T D t M U ( t ) u , U (t)v) + (TR(t)MU(t)u, ~7(t)v), where R(t) = J2/[xlZ + c,/Ixl 2. The term (JE/[x[2)M has a norm O(t 2o-z') = O(t-2u+e~)where et can be taken arbitrarily small if we take fl close enough to % # and ~ small enough. So this term is in L~(dt) for et small enough, since
378
C. G6rard
/t > 89 The term I x I - 2 M has a norm O(t -2e' ) which is also in La(dt) if al is close to %, since ~t0 is always greater than 89 So IIR(t)MII is in Ll(dt). Let us consider now the terms coming from D,M. The terms DtFo(lXl/ff" > 1) and D, Fo (Ixl/t ~ ~t - / ~ . So the term coming from D,F(t-2oJ 2) is also integrable in norm. To complete the proof of the Theorem, it remains to consider the term coming from DtX(tOH). As in the proof of Theorem 7.3, we may write:
for two cutoff functions ~a,~aeC~~ We want to replace Tz~(taH) by y.l(tbH)T, so we estimate the difference (TzI(t~H) - za(t~H)T)M. We will use the cutoff F([ x [/t" > 1) contained in M to avoid domain problems when considering operators like J2/r2. Let us simply denote by Fa([x[,t) the function F(]xl/t" > 1)F(Ixl/t" < 1). By the same argument as in Proposition 3.6(i), we have:
ZI(tnH)M = za(t~H1)M + O ( t - ~ ) , where Ha is the selfadjoint operator defined by: I 2 j 2 Cn H1 := ~ D, + r - f f i ( [ x l , t ) + - ~ F l ( I x ] , t ) .
Here we still d e n o t e b y D 2 the operator T - a D Z T with domain T - 1 (D). Similarly we will denote by Ha the operator T-1HT. We now have:
(Zi (tn~) T - TZl (t6H1))M = T(Z1 (tOlta) - Zl (taHi))M = T ~ ~ O~2(z)(z - t~lqa) - ira -fi Fl({xl, t) + ~ Fi(Ixl, t) x(z-
t~H1)-lMdz A d~,.
Asymptotic completeness for 3-particle long-range systems
379
We may move the factor F(t-2~J z 0. So we may replace TXI(tOH) by X~(t~/4)T modulo in integrable error. We can now use the same argument as in the proof of Theorem 7.3. Namely the only estimate that we need is:
+fl o ~[)]l)&(toI4)U(t)vH2dt 1 }. Then for
any So# < 6 < So + Sop~2, one has: ~k-Jm~(t, X, Dx)
ad~,(F(t, x, D~)) = Op ~
where m j ~ S ( t -k(a~176
g~),
+ O(t-~),
and r(e) = e + e#.
Proof. We will prove the claim by induction on k. For simplicity, let us denote by 9 the metric 9~. Note that Ht belongs to the class Op wS((~ )2, 9). For k = 1, using symbolic calculus we get: [Ht,
F(t, x, Dx)] = OpW(OxFDx + ms),
where m t ~ S ( t - t,o- ~)-t,ou/2-,t~J), g), using that So + ~o #/2 = 1. The term m o = 0~F is clearly in S(t-t~~ 9), so the claim is proven for k = 1. Assume by induction that the claim is proven for k. Then we write: k
a d k ( F ) = ~ adm(OpW(mj, kDk-J)), j=l
where mj, k ~ S(t -kt~~176 g). Let us consider one of the terms in this sum. Using symbolic calculus, we can write modulo O(t -~tN)) the symbol of [//~,i OpW(~k-Jmj.k)] as: /v
~ k + l - J O x m j , k "b Z
L
e#,pt Ox# Vt~ k-j-13, mj, k,~, ,
#=l/h+#2=#
where: m j, k, ~ ~~ S (t - k (~o - *) - J (aou/2 -,(~) + ~2(~ou/2 +e))) .
Asymptotic completeness for 3-particle long-range systems
381
The term C~xmj.k is in S(t-(k+ 1~(~o-~)-j(,ou/2-r(~))), SO has the correct order. To prove that the other terms have the correct order, we have to check that: t -fl(~o - e ) - ( ~ o - e ) # tfl(aolt/2 +e) t - fll(ao,u/2 +e) ~
t - ( ~ o - e ) t-fl~(aoU/2 - r i O ) t - a o t t / 2 +r(e) .
It clearly suffices to check it for fl = 1, where it holds trivially. This completes the proof of the Lemma. [] Our first result is the following:
Proposition 8.2. Let f(t, x, 3) ~ S(1, g~) be a symbol supported in {ix] >= t "~ }. Then for any Ctol~ < 6 < 1, the following estimates hold: (i) [Z(tOH), F(t, x, Dx)] = O(ta-l+'(~)). (ii) Z(t~H)F(t, x, D~) = Z(taH)F(t, x, D~)Za(t~H) + O(t - ~ ) if z ~ - 1 on supp Z. (iii) Z(tOH)F(t, x, D~) = F(t, x, Dx)Z(t~H)V(t, x, D~) + O(t - ~ ) if F =- 1 on supp F. (iv) [z(H), V(t.x, D~)] = O(t-'~ (v) ( x ) ( 1 - F(t, x, D~))z(H)F(t, x, D~) = O ( t - ~ ) if F = 1 on s u p p F . Proof We first observe that due to the support of the symbol ofF, it suffices to prove the Proposition if we replace H by//t. Indeed this follows from symbolic calculus and Proposition 3.6(i). To prove the Proposition, we have to examine the commutator expansion. We have: [Z(t~HO, f(t, x, D~)] =
z(k)(t~Ht)t k~ad~),(F(t, x, D~)) + RN, k=l
(8.2)
"
where:
RN = ~
Oe~(z)(z -- t~H~)-Nt Ne a d ~ ( f ) ( z -- t~H~)- l dz A d~.
(8.3)
Using Lemma 8.1, we know that:
ad~(F(t,x,D~))=OpW(~
~N-Jmj(t,x,D~))+O(t-~), j=!
where mjeS(t-m~~176 Using again symbolic calculus with the metric g, we may write OpW(~N-Jmj) as: N-j l=1
where m~,zeS(t -(~§176
g~). Next we use Lemma 3.4(iv) to get:
II(z - t~Ht)-kD k II = O(t-~(~~
im zl-c(N) .
So the integrand in (8.3) is bounded by: t r~ 1} and for v < 1, we consider the multiplication operator Fv(t-aM). Let us denote by Fo(x, t) a cutoff function of the form Fo ( [xl/t ~' >= 1) Fo (Ix ]/t ~ < 1) for ~' = Cto - e, ct = ~o + e. Such a function is in S(1, dx2/t2('~ It is straightforward to check that if/7 < ~o + ~o#/2 = 1 and ife is small enough then one has the following estimates:
Idk(FoF~(t-aM))l < Ck(t )k(,o,/2 +~-Ol .
(8.6)
Let us also denote by ~ the metric dx2/t 2(v-~), for p = / 7 - ct0#/2, so that by (8.6),
ITxF~(t-PM)~ S(t ~~
t~).
We first need the following Lemma, similar to Lemma 8.1:
L e m m a 8.3 For CtolX < fl < 1, v < 1 and e small enough, one has: k
adk(FoF~(t-PM)) = ~ Opw(mk, j D k - J ) , j=O
where mk, j ~ S(t -k(o- ~)-j(~o- O1,/2,!~). Proof Let us prove the Lemma by induction on k. For k = 1, the symbol of [Ht, FoF~] is equal to: V~(FoF~)~ . We know that V~(FoF~)~S(t -(p-~), ~), so the result holds for k = 1. Assume by induction that the result holds for k. Then we write: k
adk,+ X(FoFv(t-aM)) = ~ [Ht, Op~(mj, kDk-~)] 9 j=l Let us consider one term in this sum. The full symbol of [Ht, OpW(mj, kDk-~)] is equal to: k--j
~k+ l-iV~m~,k + ~, Ck,~k-J-i(glxvi(x)mj, k .
(8.7)
i=1
Since m j. k ~ S(t-k(P-~), g-), the first term in (8.7) satisfies the correct estimates. Let us consider the others. Since FoF~(t-PM) is supported in {Ixl > t~~ we see that:
d i Vim.j, k ~ S(t-k(,-~)-i(~o-O- i(~o-,),/2-(~o-~),, ~)
Asymptotic completeness for 3-particle long-range systems
383
To prove the desired estimate, we have to check that t -"~~176 is bounded by t-~p-,)-,+ ~o-~),/2, for any i < k - j . It clearly suffices to prove it for i = 1, since ~o > C~op/2. For i = l, it holds since fl < 1. This completes the proof of the Lemma. [] Using Lemma 8.3, we will prove the following Proposition:
Proposition 8.4.
Let ~tol~ < fl < 1, v < 1, Crop < 6 < 1, 6 < fl and let ~o be small enough. Then there exists a constant Co = Co(fl, 6) such that for 0 < ~ < ~o, the following estimates hold: (i) Z(t~H)FoF~(t-~M) = FoF~(t-PM)z(t~H) + O(t-~~ (ii) z(?H)FoF~(t-OM) = Z(tOH)FoF~(t-aM)z~(taH) + O(t - ~ ) if z~ =- 1 on supp Z. Proof As in the proof of Proposition 8,2, it suffices to estimate the remainder term in the commutator expansion. Since the proof is exactly the same as above, we will just give the estimate of ad~(FoF~(t-aM)). By Lemma 8.3, one has: N
ad~(FoF~(t-~M)) = ~ OpW(mj, ND~-~). ~=~
Using symbolic calculus, we may rewrite OpW(m~,ND~N--j) as: N-j
OpW(mj, kDxS-j) = ~ r1 .~/ xn - . / - lrl~j, ~ N, 1 , 1=1
where mj, N, l a S(t -tN+~)~p-`)-j('~ 8) and is supported in {Ixl > t'~ Lemma 3.4, the operator (z - t~H)-ND~-J-Zmj, N,I will have a norm:
O(t-n"~~
+p-~))t- "'-~-~~
Using
m z l-~*N>.
Since ~ > ~o/~, the exponent p - ~ - (C~o- e)p/2 is positive for e small enough. O n the other hand if 6 < ~, then 6 - (~o - ~)1~/2 - P + ~ is strictly negative for e small enough, so that the norm of RN is finally O(t -~~ for some co > 0. By the same arguments as in the proof of Proposition 8.2, this completes the proof of (ii). The proof of (i) is similar, using formula (3.3) and the above estimates for N = 1. This completes the proof of the proposition. [] Our next Proposition is pure symbolic calculus.
Proposition 8.5
Let F, ~ S ( ( 2 ) ~, go), and F(t, x, ~) and f be two symbols in S(1, g~). Then if c%p < ~ < ~o + ~o/z/2, and ~, is small enough, one has: F(t, x, D~,)F~(t-au) = F(t, x, D~)F~(t-a M)ff(t, x, D~) + O(t- ~o) ,
tiff=- 1 on s u p p F . Proof We just use symbolic calculus to prove this result. By (8.6), we see that F~(t-aM) is in the symbol class S ( ( t ) 1 +~, ~), where ~ is the metric introduced before Lemma 8.3. So we can put both F(t, x, ~) and F~(t-aM) in a symbol class with the metric dx2/t 20 + d~2/t 2(a~ 1 -~), which has a Planck constant equal to t r+2e, where r = Cto/~/2 - fl + 1 - "o = Ctokt - fl is negative since C~o#< ft. The Proposition follows then from symbolic calculus. D Our next technical result will describe how a function of A, where A is the cutoff generator of dilations used in Sect. 5 localizes in phase space. Various estimates of
384
C. G&ard
this type have been proven in similar contexts. The proof we use is inspired by [GeSi]. As in L e m m a 8.3, we denote by Fo = Fo(x, t) a cutoff function of the form F([x[/U ~ >= 1)F([x[/t ~~ O, let ~ = 1 - % p - e, and let F ES(1, d~2/(),)2). Let F + ( t , x , ~ ) be a symbol in S(1,9a~) supported in {(x,~)b + x > - t ~~ sign(x)~
_
__
=
1)
J
1) F o /~ t ~+_x3 ~ 1~J f ~176 { Ixl
+Fo(-Ftl-(~~176
)
1)_ /
= ro( +_ t
=
1}, and denote by F the operator O p ~ ( ( x / ( x ) ) 9D~). Finally we pick a cutoff function Z e C~(IR) and a symbol F_~ 9 S ( ( 2 ) - ~ , go) for s > 0. Then our result is: Proposition 8.8 Let ffl-the operator defined by: 3~:= X(t~H)FF_~(t-~M)(( - 2 V,)-~I4F( - 2V~)-~/4 _ 1)F_~(t-PM)Fz(t~H). Then for % # < 6 < 1, and for e small enough, one has:
IIM II =
O(tc('~176
(8.18)
for c(ao, 6, #, e) = sup(ao~t - 6, 6 - ~o - ao#/2, - ao + a/~/2) + cefor some c > O.
388
C. G6rard
Proof The proof is divided in several steps. To simplify the notations, we will denote by ce, a term of the form Cste, where the constant will be allowed to increase a finite number of times. Step 1: in Step 1, we will simplify the operator/~r using the various localizations that are present. We first observe that due to the localization F, we can replace in M the cutoff Z(t~H) by Z(t~Hl,t) modulo an error O(t-~ Next we see that due to the support of F, we may write modulo an error O ( t - ~ ) , M as:
where:
iV1• = Z(t~Ha.,)FF • F _ , ( t - ~ M ) ( - 2Vz)- 1/a x (F -- ( -- 2 V~)1/2)(2Vt) - 1 / 4 F - s ( t - a M ) F • FZ(t~H1, t) , where F• is the operator of multiplication by F( -t- x / t ~~ >= 1/2). Let us consider for example the M + term. We first claim that it is enough to estimate: /V+ := ~(taHl.,)ffP+(r - ( -
2V,)l/2)ffff +~(teHl,,),
(8.19)
for some cutoffs ~, F, if+ with slightly larger supports than the corresponding )~, F, F+. Indeed using Propositions 8.4 and 8.5, we may write:
M+ = K * N + K a + O ( t - ~ ) ,
(8.20)
where:
K1 = (2 Vt)- a/4 F-s(t-tS M ) F • F z(t~H l,0 9 We will drop the tildas in (8.19) and denote F+ F simply by F to simplify notations. Using symbolic calculus, we can replace in M+ the operator F by Dx, modulo an error of size O ( t - ~ ) . So let us now denote by Ba the operator Dx, and by B2 the operator ( - 2Vt) ~/2. If we denote by K the localization operator K = F+Fz(t~Ha,t), we see that to estimate ]q+, we have to estimate:
K*(B1 - B z ) K . Step 2: in Step 2 we go from an estimate on K*(BZt - B ~ ) K to an estimate on K*(Ba - B z ) K . Heuristically, due to the presence of the cutoff z(teHa,,) in the localization operator K, we know that on the "support" of K, Btz - B z is small, and we want to estimate the size of Ba - B2. This can be done by writing:
K*(B~ - B2)K(Ba + B2) = K*(B 2 - BZ)K - K*[B~, B z ] K + K*(Ba -- B2)[K, B1] + K*(Ba - B2)[K, B2] = K * ( B 2 -- B2)K + Ra + R2 + R3 9 Let us first estimate Ra. If we use the fact that ( - 2Vt) w2 ~ Op S(t -~~ get: II[B1, B2] II = O(t-'~176 9 So we have:
liRa II = O(t-~~176
9
(8.211 g~), we
Asymptotic completeness for 3-particle long-range systems Let
us
now
estimate
R2.
We
will first
389
compute
[K, B1].
We
write
[K, B1] = [K, Dx] as: [Z(tOHI,,), D x ] F + X(tOHI,,)[F, Dx] = 11 + 12 . Using formula (3.3), we have: i 11 = [Z(t~HI,,), Dx] = ~ r
- tOHl,,) ltOr
- t~Hl,t) -1 dz A d2
= 0 ( ? - ~ o - ' , +") . By symbolic calculus, the term I2 is of n o r m O(t-~~ To estimate R2, it remains to estimate the n o r m of K*(B1 - B2). But Bz has a n o r m 0(t-('~ (see (8.17)), and the term K ' B 1 is equal to:
K*Bx = K * D x = K*zx(t~Hl,t)Dx + O(t -~~
= O(t -(~~
,
by Proposition 3.6 (iv') and Proposition 8.2 (ii). So we have: IIK*(B1 - Bz)II = O(t (~o-~,u/z).
(8.23)
Summing up, we have:
tl Rz II = O ( t - ' ~ 1 7 6
+ O(t~-~~176
9
(8.24)
Let us now estimate R3. We first compute [K, B2]. We have:
[K, B2] = [Z(t~Ht,,), ( - 2Vt)1/2] F + Z(t~Hl.t)[( - 2V0 x/2, F ] = 11 + I2 9 The term I~ can be written using formula (3.3) as: [Z(t~H1,0, ( - 2V~)1/Z]F = ~n~ Oe~(z)(z - t~Hl.t)-lt~OpW(t3x( - 2Vt)l/ZDx) •
t ~ H l , t ) - l F d z /x d i .
Using L e m m a 3.4(iii), we get that 11 has a n o r m O ( t ~ - ' ~ 1 7 6 Let us now compute I2. We may use symbolic calculus with the metric g~, and we get that: IIlz Ii = O( 11 -,o-~o,/2-,o +,) = O(t - ' ~ 2 4 7 . So using (8.23), we get: IIR3 II = 0(t-~~176
+") + O(t,~-~o-O/2)~ou +c~) .
(8.25)
Step 3: in Step 3, we finish the proof of the Proposition. To use (8.21), we first need to estimate the term K*(B~ - B2~)K. This is equal to K*(D~ + 2V,(x))K, so one has: IIK*(B~ - n ~ ) g II = O(t-~). (8.26) Using (8.21), (8.22), (8.24), (8.25), and (8.26), we get: ]IK*(Ba - B2)K(Bt + Bz)N = O(tc(~~ for c(~0, 6, #, ~) = sup( - 6, 6 - ~o - (3/2)Zo/~, - ao - ao#/2) + ce. Moreover, due to the support off(t, x, O, and using Proposition 8.5, we see that
KB1 = KDx is equal to K/~a + O(t-o0), where: /~1:= ct,O-l-~F(tl-('o-~)Dx < 89 + D x F ( t l - ( ' ~ 1 7 6
> 89
390
C. G6rard
Again this follows from the fact that the symbols o f / ~ and of D~ are the same on the support o f f The important property of B1 is that:
B1 >- ct~~
1.
(8.27)
So we have: IIK*(BI -- Bz)K(/~I + B2)II = O( ff~~ ~'u' ~)), for the same constant C(~o, 6, #, e). Next we use the fact that B2 > t -'~ and that (see (8.27))/31 > t ' ~ so that/~a + B2 is invertible and (/~ + B2) -1 has a norm O(t-~o~/2+"). So we finally obtain:
IIK*(B1 - B2)K II = O( t~'~ ~' "" ~)) ,
(8.28)
for C(ao, 6,/1, e) = sup(~op/2 - 6, 6 - ~o - ~o#, - ~o) + ce. If we now use (8.20), we finally obtain: }tM + II = O( ta'~ ~' u. ~)) (8.29) for C(~o, 6,/*, e) = sup(~o/~ - 6, 6 -- ~o - ~o#/2, - ~o + ap/2) + ce for some c > 0, which is the desired result. []
9 Sharp propagation estimates In this Section, we prove a sharp propagation estimate for the evolution of an exceptional state. This estimate will be enough to prove asymptotic completeness for potentials decaying like - " for # > 89in the next Section. We consider in this Section a time-dependent Hamiltonian H ( t ) = ~2 Dxz 4i- V(x) 4- It(x) on Lz(]R), where V and I, satisfy the estimates (3.1) and (3.2) of Sect. 3. We assume moreover that: I,(x) is supported in {[x[ > t "~ for some e > 0 ,
V(x) < - C -u, for C > O.
(9.1)
W e pick then a function f e S(1, g~) supported in the region {(x, 4)1 t ~~ t ~~ sign(x)-~ > W ~ 1}. More precisely we take f o r the form:
< Ixl
~, #o), such that: Fv < 0,
suppF~ c [c, + ~ [ ,
F'v(2) < 0 ,
(9.2)
for some c > 0. We consider the observable N defined by: N : = Z(taH)FF~(t-~M)Fz(taH), and we compute the Heisenberg derivative DtN. The exponent fi will be chosen later. We first observe for later use that: IIF F , 11= O( t~(1 -a+~)) 9
(9.3)
We pick a vector u e H M' M, for an M to be chosen later, and we will denote by ( B ) t the expectation value ( B u , ut) for ut = U(t)u. Step 1: in Step 1, we compute terms coming from Dtz(taH). We have: Dtx(taH) = 6t a- 1Hx'(taH) + [It, ix(taH)] .
(9.4)
Let us first estimate the term [It, X(taH)]. Using as usual (3.3), we get:
[It, i z ( t ~
i
] = 2n ~ Oe2(z)(z - t ~
t OpW(O~lt(x). D~)(z - taB)- 1 dz ^ d i .
Using that O~lt(x) is of size O(t-1-u) and supported in {Ixl >_- t=~ 3.4, we get: It lit, i z ( t ~ ] tl = o(t~-~-"-'~ .
and L e m m a (9.5)
Next we write as usual & n - l H x ' ( t a H ) as - t-~Z~(taH), and we prove that the contribution of this term is essentially a positive operator. Indeed using Proposition 8.40) and Proposition 8.2(i), we write: -
t-lz2(tan)FF~(t-~M)F~2(tan)
=
- t
-1 Z1 ( t a n ) 2 1 ( t a n ) F F ~ F x l ( t a n ) ~ l ( t b n )
+ 0 ( t - 1 -~o),
(9.6)
for 0 < Co = Co(B), as long as e0p < c5 < eo + ~0/~/2, and c5 - Ct-1-~o
(9.7)
Step 2: in Step 2, we estimate terms coming from DtF. Let us first look at the term dF/dt. We have:
dF
OpW(dtf(t, x, D~))
Op wm(t, x, D~),
dt where m(t, x, ~) is a symbol in S ( t - 1, 9,) supported in the region: f~ = {(x, ~)11xl > 2t ~~247or
Ixl _-< t ~~
or Ixl _>-~
t ~~
and sign(x).
~ >
t "0-1-~}
.
If we use Theorem 4.2, Theorem 6.1 and Proposition 8.7, we get that: IIOpW re(t, x, D~)x(tan)utll < t - l-c~
,
(9.8)
392
C. G6rard
for any ) < e < to, and for M = M(6, p, e). Let us now estimate the term [H(t), i F ] in DtF. Using symbolic calculus, we get that: [H(t), i F ] = OpW(Vxf(t, x, Dx).Dx) + [V(x) + I,(x), F] = 11 + 12. Let us first consider the term I 2. By symbolic calculus, IV(x) + I,(x), F ] is of the form OpW(ml(t, x, Dx)) modulo an error of n o r m O ( t - ~ ) , where ml ~ S(t -1+~, g~) and is supported in the region f2. So we can apply an estimate similar to (9.8) to control I 2. To estimate l t , we have to use the cutoff)~(t~H) contained in N. More precisely, we have: ]]X(t~H)OpW(c~xf(t, x, Dx). Dx)][ = O(t-~~176
+c~)
(9.9)
using that the support o f f is contained in {[xl > t ~~ and Proposition 3.6(iv). Let us now summarize what we have obtained in Step 2. Using (9.8) and (9.9), and the support of the symbol rnl, we get that if C~o# < 6 < 1, if v < Vo, there exists eo > 0 and Co = Co([1, 6), such that for 0 < ~ < ~o, one has: I(Z(tOH)DtFFv(t-#M)F)~(t~H) )t[ < C t - 1 -co it u H2
(9.10)
for M = M(6, #, e). Step 3: in Step 3, we compute the term D~F~. We have: D t F ~ ( t - ~ M ) = - fi t - p - 1MF'~(t-PM) + [H(t), i F ~ ( t - ~ M ) ] - t-PF'~(t-PM) . (9.11) Then term [H(t), i F , ( t - P M ) ] is equal to: _ t-P( _ F ' ) I / 2 ( t - P M ) ( _ 2V)-l/4OpW(sign(x)D,)( - 2V)-1/4( _ F , ) I / z ( t - O M ) . So by (9.2), (9.11), we have: Z(t~H)F(t, x, Dx)DtF~(s-PM)I~=tF (t, x, D~)Z(t6H) > t - P R , where: R = g(t6H)F(t, x, D~)SF(t, x, D~)Z(t6H),
(9.12)
and S is equal to: - ( - F',)I/2(t-PM)(( -- 2V)-1/4OpW(sign(x)D~)( - 2V)-1/4 _ 1)( -- F'~)l/2(t-tJM). Using then Proposition 8.8, we get that: liR ]l = IIZ(t~H)F( t, x, D~)SF(t, x, Dx)Z(t~H)II = O(t "~' "' ~J), for c(5, g, e) = supp(eog - 6, 5 - ~o - ao#/2, - ~0 + aog/2) + ce. F o r a given (5, the exponent fl has to satisfy, in order for the estimate (9.7) to hold, the condition 6 > [1. In order for the error term t - ~ R to have a n o r m in Ll(dt), [1 should moreover satisfy: - [1 - c(6, #, ~) < - 1 . To get the smallest possible value of [1 (i.e. the sharpest possible localization in our estimate), we look for the 6 which minimizes c(6, #). However, to have integrable errors in the terms (9.5), (9.6), we also need that 6 satisfies 6 ~o, and also fl > 1 + C(6o, It, e), i.e. we take fl > 1a + o~o#/2. If we put together (9.7), (9.10) (9.12) and use Sigal-Soffer's argument, we get that for fl = 89+ Crop~2 + e~, there exists v > 0, such that for e small enough one has:
IIF,(t-aM)F(t, x, Dx)z(tz~
< CllulIN, N.
(9.13)
for some N = N(fi, #, e). As in the previous Sections, we will now turn (9.13) into a decay estimate. We obtain that for fl = 89+ C~op/2 + e~, and e small enough, there exists a constant c(fi) > 0 such that:
liFo(t-aM > 1)F(t, x, Dx)Z(t~~
II < Ct-~ I)D~z'2(H)U(t)d?
394
C. Grrard
Then we estimate:
F(~ = =
> 1)D~z'2(H)U(t)q~ z
(
1 D2Z'2(H)U(t)d?, U(t
z2(H)F
__> 1
2nx'~(H)U(t)cb, U(t
,+) ,+)
+ O(t -(~~
+ O(t -~o"+~)
= O(t-=o.+,),
Using (9.t5) and the fact that I[ V(x)F(fxl/t~~ >=1)ll = O(t-~~ finally that: 11[I,, i n ] %'2(H)U(t)c~ 11= O(t - x -,-~ou/2 +~), a n d by integrating (9.16) from + ~
So we get (9.16)
to t, we obtain:
f(t) = O(t-"-~~
.
(9.17)
To complete the proof of the Proposition, we come back to our original cutoff function Z, and write l - X(2) = 2Z3(2) for some Zs ~ C ~ ( ~ ) , since 1 - Z()0 =- 0 near the origin. So we may estimate: II(1 - Z(t~H))U(t)49 II = ]1(1 - Z(tOH))zt (n)U(t)dp
tT(t)v> = ( D , M + M(I,(F(t), 0 9 ) - l,(Ixl, c o ) ) U ( t ) u ,tT
(10.3)
where D , M = arM + [Ht, iM]. We have used here the obvious fact that Ie(F(t), co) commutes with M. Let us first consider the second term in (10.3). Using that I I , ( F ( t ) , c o ) - I,(r, co)l < C t - t - ~ l r F(t)l, we get that the term M ( I , ( F ( t ) , c a ) L(r, co)) has a norm of size O(t-~-~§176 Since ~t > 89 we may take /3 = 89+ ao/i/2 + el for some el > 0, such that IIM(It(F(t), co) - It(r, co))II is in Ll(dt). It remains to estimate the terms coming from D,M. The term coming from D,F(t, x, D,) is in L~(dt) for e small enough and u 9 H M'M for M large enough, using (9.10). Let us now now consider the term coming from D , F ( t - a M ) . By the arguments used in Step 3 of the proof of Theorem 9.1, this term can be written as: O ( t - 1 ) F ( t - a l M l > l)F(t, x, Dx)z(ta~
+ O(t-fl-aou/2-(1/2)+e) .
Using Theorem 9.1 and the fact that we have chosen/3 = 89+ Cto#/2 + e~, this term is also in Ll(dt), for u 9 H u ' u and M large enough, e small enough. The last term we have to consider is the term containing DtZ(t~~ It can be written as: O(t - 1)Z1 (ta~H) + O(t - 1 +,~o-~- ~o~/2+~), for a function Z1 e C ~ ( ~ ) supported away from the origin, using (9.4) and (9.5). The second term has a norm in Ll(dt) since 3o
