RIETZ EXTRUCTOR

Vol. 71, No.

DUKE MATHEMATICAL JOURNAL (C)

July 1993

ASYMPTOTIC COMPLETENESS FOR N < 4 PARTICLE SYSTEMS WITH THE COULOMB-TYPE INTERACTIONS I. M. SIGAL

AND

A. SOFFER

CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Introduction Hamiltonians and kinematics

Asymptotic completeness Asymptotic energy cutoffs and sharp energy localization Proof of asymptotic completeness Time-dependent observables Propagation estimates revisited Asymptotic clustering revisited Many scales of configuration space Localization of momentum Evolution of threshold channels Estimates for free evolution Subballistic estimates Appendix Supplement References

243 247 249 251 254 260 265 273 275 278 282 286 290 295 296 297

1. Introduction. In this paper we study the scattering theory for many-body long-range systems. It was known since the foundation of quantum mechanics that the scattering theory for long-range systems is different from that for the short.range ones. However, only when one addresses the problem of many-body asymptotic completeness does one realize the extent of this difference. We discuss the latter within the framework of phase-space analysis, which is the only approach which so far has given access to the long-range many-body problem. There are two new problems one faces in passing from short-range systems to long-range ones. First, one has to prove sharper propagation estimates, namely, to show that in a certain sense there is no propagation outside a parabolic conical neighbourhood of the subset of the extended phase-space (i.e. including the time and energy axes) determined by the classical trajectories of quantum-mechanically Received 7 December 1991. Revision received 26 June 1992. Sigal is an I. K. Killam Research Fellow. Sigal supported by NSERC under Grant NA7901 and by NSF under Grant DMS-8808032. Softer is an A. Sloan Fellow in Mathematics. Softer supported by NSF under Grant DMS85-07040.

243

244

SIGAL AND SOFFER

stable clusters obtained in various breakups of the original system. This was done in [SigSof3]. (See Section 7 for discussion and straightforward extensions of this result and see [Gr] for a shorter and more direct proof of the key estimates.) Second, one must control the evolution for systems with effective long-range time-dependent potentials. These systems result from breaking up of the original N-particle system (see [SigSof3]) and are described by time-dependent Schr/Sdinger operators of the form H(t) tt + W(x, t), where H is a k-particle Hamiltonian with k < N- 1 and W is a smooth, real potential obeying Ic3x,0 W(x, t)l < C(1 + Ixl + Itl) -’-I1 with #, the decay rate of pair potentials at infinity (# 1 in our case). The problem here is that the evolution U(t) generated by H(t) does not conserve the energy (i.e. does not commute with H). As a result, starting at any initial-state orthogonal to the eigenfunctions of H, there are two mechanisms for a breakup of the system: one due to propagation, when the resulting fragments have nonzero relative velocities, and another due to spreading out of the wave packets (or uncertainty principle), when the fragments have zero relative velocities. In general, ___o, the system oscillates between these two modes of evolution. Controlling as these oscillations is a hard problem. The tools developed in the short-range scattering theory, in particular the powerful method of positive commutators, are applicable only to the analysis of purely propagative behaviour. In this paper we prove asymptotic completeness for N-particle Schr/Sdinger operators with two generations of thresholds (this, in particular, includes general four particle operators) and interacting via pair potentials vanishing at o as O(Ix1-1) (Coulomb-type potentials). Previously, asymptotic completeness for longrange quantum systems for 2 particles and various classes of potentials was proven by several authors (see [CFKS, HSr, RS1, Sig2] for references) and for 3 particles 1, in [En] (see with pair interactions vanishing at o as O(Ixl -) with # > also [SigSofS] for a different proof and see also [Mo, Sin Muth]). Additional comments and references can be found in the papers and books cited above. The main contribution of this paper, however, is a new technique developed in order to handle evolutions which are mixtures of diffusion and propagation. In particular, we introduce a refined version of phase-space (or microlocal) analysis by localizing in the shrinking with time neighbourhoods in the momentum and energy and in coordinate balls growing as fractional powers of time. The key point, though, is that we do such an analysis not on the entire Hilbert space, but on a small part of it which is singular from the point of view of the evolution. More precisely, this part is the smallest subspace on whose orthogonal complement, modulo a uniformly in time small piece, all possible breakups produce fragments with nonzero relative velocities. The evolution on the orthogonal complement can be controlled by tools developed in [SigSof 1-3]. The singular subspace in question is isolated by means of asymptotic projections with shrinking spectral intervals. We explain the ideas mentioned above on a more technical level. For an open set fl c IR, we denote by Eu(2) a C cutoff function supported in fl and equal to 1 on a slightly smaller subset of Let f(2) Eu(2) for (e, o) and let A be the dilation generator, i.e. (- i/2)(x.gradx + gradx" x). Denote @t U(t)@. A cosmetic

x/-

.

245

ASYMPTOTIC COMPLETENESS

adjustment of the methods of [SigSofl-3] allows us to control EA(H), and f(A/t), for any A away from the thresholds of H, for any e > 0 and for any L z, provided the decay rate of potentials # (see Section 8). Thus it suffices to concentrate on g(A/t)En(H),, where f is a sufficiently small neighbourhood of the thresholds, 9,(2) 1 f(2) and e > 0 sufficiently small. Ultimately, we introduce the asymptotic energy cutoffs (Section 4)

E(H)= lim U(t)*En(H)U(t), whose existence can be easily shown for any open f and for # > 0. An abstract monotonicity argument implies that the strong limits

Q+

lim

E(H),

exist, where the limit is taken over open sets f containing the threshold set of H and shrinking to it. (Ran Q+ (or Ran Q-) is the singular subspace discussed above.) On U(t)[Ran Q+ ]_L, H is localized, modulo a uniformly small in term, away from the thresholds (equations (4.7) and (5.4)-(5.5)) so that the results of [SigSofl-3] apply (Section 8). We show that on U(t) Ran Q+, H is sharply localized (Section 4), i.e. that

f(Itl dist(H, threshold set))ff,

O(Itl -u+a)

for any smooth bounded function, f, supported away from 0 and for any q Ran Q+/-. Here fl < # and dist(H, f) infvn IH vl. The proof of this estimate is rather general, does not require a detailed analysis of H, and holds for any # > 0. The hard job begins after that. Its outcome is that, on 9(A/t)t with ff Ran Q+/- and e > 0 sufficiently small, the coordinate is localized in {Ixl < Itl } with a > 1- (#/2), provided/ > (1 + 2-k-1) -1. Part of the proof of this result (Sections 9-11) is done for arbitrary k-particle systems and uses an analysis with many time scales (depending on the direction in the configuration space). However, the remainder of the proof (Section 13) is conducted under a restriction on the threshold set (or the number of particles; see equation (13.3)). If # > (kt/2), then the above result yields that we can pass from the evolution U(t) generated by H + W(x, t) to the evolution Uo(t) e-rote ’o wo,)a generated by H + W(O, t). Namely, we show that the wave operators

-

lim

Uo(t)*g(-)U(t)

exist on Ran Q-+, provided e > 0 is sufficiently small (Section 5). Indeed, the difference of the generators, W(x, t)- W(O, t), is integrable on the set { xl < ) with a < #. Again, the last argument is rather general and proven for arbitrary k-particle systems with # > x/3- given the result of Section 13. Thus it remains

246

SIGAL AND SOFFER

to control the evolution e -int generated by a k-particle time-independent Hamiltonian H with k < N 1. This completes the inductive step which leads to the proof of asymptotic completeness for multiparticle systems with the decay rate of the potentials/ 1. We comment on the composition of the paper. The main result is formulated in Section 3 (Theorem 3.1) and proven in Sections 4-13. Section 5 sums up the proof based on the results of Sections 7-13. This section can serve as a guide for the rest of the paper. The results of Sections 4 and 6-11 are formulated and proven for general many-particle systems with various assumptions on the decay rate of pair potentials. The results of Sections 7 and 8 concerning propagation estimates and asymptotic clustering are revisions of the corresponding results of [SigSofl-3] and straightforward adoption of them to the time-dependent Hamiltonians. We do not reproduce their proofs but discuss necessary (rather cosmetic) adjustments. Only Section 13 uses a restriction of the structure of the threshold set. In this paper we use the following convention for cutoff functions. In general, F( f), for a Borel set f c IR", stands for a smoothed out characteristic function of f. More precisely, if A is an interval from IR bounded from below such that AI > 0, then we define

if2 A and dist(2, dA) if2 q A.

>

If A is an interval bounded from above but not bounded from below, then we define

F(2 A)

1

F(2 (IR\A)).

Moreover, we introduce

The parameter 6 -1 will be called the sharpness of the cutoff function in question. Let Ifl denote the Lebesgue measure off c IR. We have that 6 Isupp F’(2 > 20)1 and similarly for F(2 > 20). Symbol (x) stands for a smooth function equal to Ixl for Ixl > and not less than 1/2 for Ixl < 1. Note that (x) is positive, homogeneous, degree-1 for Ixl > and is invertible. This notion should not be confused with the related one, (A) (1 + Ial2) x/2, for operators, also used in this paper. We denote x(x) -1 and z + for > 0. The main statements are formulated for only. +, while all the other statements as well as the proofs are given for Oi(t -) stands for an operator such that (-A + 1)-iOi(t -) and O(t-’)(-A + 1) -i are bounded by const (for > 1).

Acknowledgement. The bulk of this paper was done in 1985-86. It was written while the first author was visiting Princeton University and the IAS and ETH-

247


Ziirich in 1989 and the second author was visiting UC-Irvine and ETH-Ziirich in 1989. The authors are grateful to all these places for their hospitality. It is a pleasure to thank I. Laba for many helpful remarks.

2. Hamiltonians and kinematics. Consider an N-body system in the physical space R v. The configuration space in the center-of-mass frame is ([SS]) mixi

where x

(x l,

(2.1)

0

i=

,

xN) with x IR with the inner product N

(x, y)= 2

(2.2)

mxi’yi. i=1

Here m > 0 are masses of the particles in question. The Schr6dinger operator of such a system is A

H

+ V(x)

on Lz(x).

Here A is the Laplacian on X and

v(x)

y v,(x,- x),

where (ij) runs through all the pairs satisfying < j. We assume that the potentials V are real and obey: V(y) are At-compact. It is shown in [Com] (see also [CFKS], [RSII]) that under this condition the Kato theorem applies and H is selfadjoint on D(H)= D(A). Moreover, by a simple application of the H61der and Young inequalities [RSII] and by a standard approximation argument (see [CFKS], [’RSII]), one shows that if Vj are Kato potentials

[Ka 1], i.e.

V

Lr(R) + (L(R)),

where r >

ifv > 4 and r

2 ifv

< 3,

and the subscript e indicates that the L-component can be taken arbitrarily small, then V is Laplacian compact. Now we describe the decomposed system. Denote by a, b, partitions of the set {1,..., N} into nonempty disjoint subsets, called clusters. The relation b < a means that b is a refinement of a and b 4: a. Then ami, is the partition into N clusters (1),..., (N). Usually, we assume that partitions have at least two clusters, lal denotes the number of clusters in a. We also identify pairs l= (ij) with partitions having N- clusters: (ij)--{(ij)(1)...(i)...(j)...(N)}. We emphasize that the relation a (resp. (ij) is equivalent to saying that and j belong to different a) with clusters (resp. to same cluster) of a.

_

248

SIGAL AND SOFFER

We define the intercluster interaction for a partition a as I linking different clusters in a, i.e.

Ia

14a

sum of all potentials

Vt.

(2.3)

For each a we introduced the truncated Hamiltonian

H,,

H- I,,.

(2.4)

These operators are clearly selfadjoint. They describe the motion of the original system broken into noninteracting clusters of particles. For each cluster decomposition a, define the configuraton space of relative motion of the clusters in a to be

Xa {X XIx

xj if and j

belong to same cluster of a}

and the configuration space of the internal motion within those clusters to be mjx

0 for all

jCi

Ce

a}.

Clearly, Xa and X" are orthogonal (in our inner product) and they span X:

x xa @ Xa Given a generic vector x X, its projections on X and Xa will be denoted by x and Xa, respectively. If and j belong to some cluster of a, then xi(rcax)i- (rcax), where z is the orthogonal projection in X on X This elementary fact and the fact that -A (p, p) with p -iVx (see equation (2.6) and the sentence after it) yield the

.

x

decomposition

H =/-/a(R) +

(R)

T

on L2(X)

L2(Xa) ( L2(X).

(2.5)

Here H is the Hamiltonian of the noninteracting a-clusters with their centers of mass fixed at the origin, acting on L2(xa), and T -(Laplacian on Xa), the kinetic energy of the center-of-mass motion of those clusters. The eigenvalues of Ha, whenever they exist, will be denoted by e ", where (a, m) with m, the number of the eigenvalue in question countin9 the multiplicity. For a amin, we set e" 0. The set {e ", all } is called the threshold set of H, and e" are called the thresholds of H. For (a, m) we denote I1 lal and a() a. Our method is based on localization of operators in the phase-space F X x X’. Here and henceforth, the prime stands for taking dual of the space in question. The

249


dual (momentum) space X’ is identified with

X’

k

RvNI k

0. with the inner product (k, u)

m-mk" u.

(2.6)

Thus Ikl 2 is the symbol of -A and -A IPl 2. We use extensively the natural bilinear form on X x X’: (x, k) x. k. Given a generic vector k X’, its projections on X and (xa) will be denoted by k, and k a, correspondingly. Accordingly, the momenta canonically conjugate to Xa and x and corresponding to k and k will be denoted by Pa and pa, respectively. Thus Ta Ipl 2. Using the bilinear form above, we define the generator of dilations as

A

1/2((p, x5 + (x, p))

7

1/2((p, 2) + (2, p)),

and the selfadjoint operator

associated with the angle between the velocity and coordinate. Again, for decomposed systems A splits into the operator

A

}((pa, Xa) nt" (X a, pa))

corresponding to the internal motion of the clusters, and the operator

Aa 1/2((Pa, Xa) -1- (Xa, Pa)) corresponding to the motion of the centers of mass of the clusters. Finally, we mention some notation. We denote Ea F(H A) for an interval A c R and set Ha HEa. P will stand for the orthogonal projection on the pure point spectrum subspace of H. 3. Asymptotic completeness. In this section we cast the intuitive notion of asymptotic completeness into precise mathematical terms and discuss different aspects of this notion. To this end we need the notion of channel. The channel is a scenario according to which the scattering process develops. Different scenarios are labeled by an asymptotic state of the system in question at Thus a channel is described by a pair: (a, m), where a is a decomposition of the system into subsystems and m specifies a stable motion (i.e. a bound state) within each subsystem of this decomposition. We introduce the conditions on the potentials which are used in this paper. (A) lyl I=110=vj(y)l are At-compact for Il 0, 1, 2. (B) There is R > 0 such that for lYl > R, Vj C 2 and obey IO=V(y)l with I1-- 1, 2 and/ > 0.

250

SIGAL AND SOFFER

For long-range potentials, even as clusters move away from each other, the interduster interation cannot be entirely ignored. Its effect remains in the form of time-dependent modification of the internal potentials of the clusters. We proceed to the definition of this modification. For any cluster decomposition a, we write x (x x), the pair of orthogonal projections of x onto the subspaces X and X, respectively. We introduce the following cutoff function

,

F,;,(x)

I-I F[(x’) > 2]

(3.1)

14a

with the standard convention for F[(xt),> 2] given in Section 1. Note that this function is supported in the region of the configuration space in which the distance between clusters in a is at least 2. Now we are prepared to define the modified Hamiltonian of independent clusters,

H(t)

Ha + Wa(t),

(3.2)

where W(t) is the family of pseudodifferential operators written in detail as

Wa(t

Wa,t(X a, Vat

(3.3)

(note here that x and/)a commute). Here 13 (O/Opa)lPal 2 is the velocity operator, i.e. the differential operator with the symbol (O/Oka)Ika[ z, and (note the identification X (X a, Xa)

Wa, t(X)

W(x)F

( )

>

and

Wa(x

Ia(x)Fa,,

(3.4)

where 6 > 0 is a geometrical constant depending only on N and the m’s and 2 > 0 is sufficiently small and is specified in appropriate places. Due to condition (B), the effective potentials W,, obey the estimates

ItgtgflW,,(x)l

C,t(1 + Ixl + Itl)

(3.5)

for Icl + fl 2. These inequalities hold strictly speaking only for sufficiently large so that 2t > R. However, in order not to complicate the notation, we ignore this qualification. Since Ha is selfadjoint and W(t) is uniformly (in t) bounded, one concludes that Ha(t) generates the evolution Ua(t) (see e.g. [RSII-I). More precisely, there is a family Ua(t) of unitary operators (with a certain group property) which solves uniquely the Schr6dinger-Cauchy problem:

u(t)

d----T-= Ha(t)Ua(t),

Ua(O)

id.

(3.6)

251


We introduce Ua,(t) F(lval 2)Ua(t), where 2 is the same as in (3.4). Of course, F(lpa] > 2)commutes with Ua(t). Next we define the channel Hamiltonians for the long-range scattering

H(t)

e

+ [pal 2 + W,(0, vat)

L2(Xa). Note that H,(t) is independent of 2. The crucial difference between this operator and Ha(t) is that the former commutes with H while the latter does not. Denote now by U,(t) the evolution on L2(Xa) generated by H,(t): on

fft U,(t) H,(t) U,(t)

and

U,(0)

id

(3.7)

where U,(t) are the channel evolution operators. We can now formulate the property of asymptotic completeness. We say that a system obeying (B) with # > 1/2 is asymptotically complete if for any L2(X)

(3.8)

0

u.

"

+ for some Here are the eigenfunctions of H corresponding to eigenvalues e ", and the sum extends over all channels including a trivial one with la()l 1. This definition is equivalent to a standard one in terms of the channel wave operators (provided the latter exist). The smoothness restriction in condition (B) is immaterial; however, the restriction on the decay is essential. For # < 1/2 one has to modify the channel evolution operators further (see [Sigl]). Of course, one can use the same channel evolutions for all/ > 0; then for # > 1/2, the difference between those evolutions and the ones we use will vanish as tl In this paper we use the simplest evolutions rather than the most general ones. We formulafe the main result of this paper. as

.

THEOREM 3.1. Assume conditions (A)-(B) hold with # and assume H >/0 whenever al > 4. (This holds, for example, if the number of particles N < 4.) Then asymptotic completeness holds. 4. Asymptotic energy cutoffs and sharp energy localization. As the statement of the problem already indicates, we have to control evolutions generated by timedependent Hamiltonians of the form

H(t)

H

+ W(t),

(4.1)

where H is a selfadjoint N-particle Schr6dinger operator satisfying restrictions (A) and (B) (or (B) alone) of Section 3 and W(t) is the multiplication operator by a real

252

SIGAL AND SOFFER

and smooth function W(x, t) obeying

dOW(x, t)= 0((1 + ]tl) --Il-a)

(4.2)

with # > 0 the same as in condition (B). Note that # in (4.2) can be taken to be different from that of condition (B). Let U(t) stand for the evolution generated by

n(t). Since now we are dealing with time-dependent Hamiltonians, the energy is not conserved and cutoffs in H are not sufficient anymore since they do not commute with the evolution U(t). In [SigSof4] the notion of asymptotic energy cutoffs was introduced and used in the study of evolutions generated by time-dependent Hamiltonians. This notion plays an important role in the present paper. It is shown in [SigSof4] that the norm limits

E

(4.3)

lim U(t)*EnU(t)

exist for any bounded interval f, provided conditions (A) and (B) hold with # > 0. The definition of implies that

Et

Eo,

E

L

(4.4)

Similarly to [SigSof4, Thm. 4.1(ii)], we have the following theorem.

THEOREM 4.1. Let v IR. Then the strong limits

Q{

=s- lim

E

(4.5)

exist. Here the limit is taken over a sequence of open intervals containing v and shrinking to it. The limits are independent of the sequence taken.

Proof.

Since

E > 0 and are monotonically decreasing as f shrinks, there are

Q{ such that E Q{ weakly, as f shrinks to {v}. Next, for any f and f’ such that En(2)En,(2) En(2), we have by (4.5) that

II(E- E,)ull 2= IIEtull 2 + IIE,ull 2- 2(Eau, u). Since

(4.6)

E < 1, we derive from this inequality II(E- E,)ull 2

II(E)X/2ul12 + I[(E,)X/2ull2-2(Eu, u)

u5

(E u, uS.

Hence [[(E E,)ull 0 as f and f’ shrink to v in such a way that En(2)En,(2) En(2) holds. The latter restriction can clearly be replaced by the restriction f c f’.

253


Thus

E with f shrinking to v form a strong Cauchy sequence. This implies that

Ea Q strongly, as fl shrinks to v.

(4.5) implies that Q are projection,operators. We will need later the following simple consequences of this fact and the definition of Q"

O .L Ran Q = EO

0

as f

{v}

(4.7)

and

Ran Q = Et

0

(4.8)

if v e f.

The results above hold if H is replaced by Ha. The corresponding threshold projections will be denoted by Qv and

Q,v. thresh, of

(4.9)

H

In this paper the projections Q and Qa are always considered on the entire L2(X). In what follows, whenever a threshold v is fixed and whenever it does not cause a confusion, we drop the subindex v from the notation. Finally, note that for any smooth function f with limits

C

derivatives the norm

f+(H)= lim U(t)-lf(H)U(t)

,

exist (see [SigSof4]). Now we show that the energy of an orbit U(t)O starting at _+m, namely, that for any fl > 0 squeezed around v as

qt,

F(talH- vl < 1)q6 + O(t-u+a).

THEOREM 4.2 (see [SigSof4, Thm. 4.1(i)). Let

I1,- eao,

e Ran

Q{ gets (4.10)

Ram Q+. Then

ClAl-Xt -"

(4.11)

with C independent of and A.

Proof. Denote Ek

U(t)*EAU(t)andletO Ran Q+.SinceqJ

---II(e

ek)qJll.

EO, wehave

(4.12)

254

SIGAL AND SOFFER

Using that d

U(s)*[W(x, s), Ea] U(s) ds,

(4.13)

we obtain that

II(E

E*)ll

(4.14)

II[W(x, s), Elll ds.

Due to estimates (4.2) on W and due to (6.22) with R

III-W(x, s), E-III

and tr

IAI, we obtain that

ClAl-ls -x-

(4.15)

with C independent of s and A. The last two relations together with (4.12) yield (4.11). m -a, v + -a] in (4.11), one arrives at (4.10). Taking A Iv

5. Proof of asymptotic completeness. In this section we combine the results of the following sections into a proof of Theorem 3.1. Another purpose of this section is to serve as a guide for the rest of the paper. Let H be an N-body Schr6dinger operator satisfying conditions (A)-(B) of Section 3 with # 1. In what follows we assume is considered +. The case in a similar fashion. Let A be a compact interval away from the thresholds and eigenvalues of H and let ff Ran EA(H). Then by Theorem 8.3(i) there are ffa L2(X) and 2 > 0 such that

-

e-im

U.,z(t)J.

(5.1)

0

as +o. Here the sum extends over all a with lal 2, and the evolutions Ua.x(t) are defined in (3.6) and in the sentence after this equation. We analyze now the second term on the left-hand side of (5.1), i.e., the orbits Ua, x(t)qk with b L We begin with the study of the simplest situation: the standing waves. Let P, be the orthogonal projection on qJ’, an eigenfunction of H" (corresponding to an eigenvalue e’). We have the following statement.

.

THEOREM 5.1. Assume 0 we can pick a neighbourhood f of the threshold set of H so that lim

IlE(n)btlll

IIE.(n)Ua(t)btXll

0 there is a set of ’s with a() a and there are 21 > 0 and functions Z and fb such that

lit-*

/9

+

U,(t)tl)

U(t)Pz b 0 there are gb and 21 > 0 such that

F ast

([A-I > P) Ua’(t)t2)

b pt} for any p > 0. Hence the evolution localized to such a region is asymptotically clustering even for threshold energies. Now we consider the most difficult terms F((IAal/t)< P)Ua(t)2) with p > 0 arbitrarily small and lal > 2. Of course, it suffices to consider Ran Qa+.v, where v is

257


v

a fixed threshold of Ha. (Remember that Qa Qa, v, where the sum is taken over the threshold set of Ha.) The first key step is showing that, for any 0 Ran Qa+,v, Ua(t)O is very sharply localized in energy:

F(talH

U(t)O

-

vl < 1)Ua(t)O + O(t -+)

> 0 (Theorem 4.2). Using this result, we show that x F((lZl/t) < p)U(t) 2). Namely we prove (Theorem 13.1) that

for any fl

2

and (Theorem 13.3) that, as e

dt

(5.11)

is well localized on

< CIIqll 2

(5.12)

0,

(5.13)

--}0

for any

e Ran

Qa+,v, provided lal > 2, p is sufficiently small and -p2 -N-1
0

whenever

Ibl

(5.15)

4.

All the hard work in this paper goes into proving estimates (5.12) and (5.13) (Sections 9-13). Note that the results of Sections 9-11 hold for general many-particle systems while restriction (5.15) is needed only in Section 13. Estimate (5.12) allows us to pass from the evolution Ua, z(t) to the evolution Ua, o(t) e-m’e o wo.so,vo), which differs from e -too’ only by a phase factor. To this end, we use the fact that the difference between the generators of Ua, z(t) and of Ua, o(t), W,t(x vat) W,,(O, vat), is O(t -1-+) in the region Ixal < Ct and therefore is integrable for a (1 + 2--x) and assume + exist on Ran Qa+ in (5.16) obey (5.14). Then I4(.a,0

estimate (5.12) holds. Let

258

SIGAL AND SOFFER

Proof. Let > 0. Fix a threshold v of Ha. Denote b F(((xa)/z ) < 1) and W(x a, t) Wa, t(x a, vat) and omit the indices a and 2 from the notation. Pick up fl obeying/2 > fl > 2(1 #). Using that fl < and estimating multiple commutators in a standard way (see Lemma 6.2 of this paper and Lemma A.2 of [SigSof3]), we derive

F(talH- vl > 2)bF(talH- vl < 1)= O(t-2).

(5.17)

This implies

rfiF(talH- vl

< 1)-- F(taIH vl < 2)rfiF(talH vl < 1) + O(t-2).

Using this relation and Theorem 4.2, we obtain for 9

Wo+=s-

Ran Q+

lim W(t)d/,

where

F(taIH- vl < 2)Uo(t)*rkU(t).

W(t)

Next, following the Cook argument and using Theorem 4.2 and that, since v] < 2) is of the form F(ttl H vl 2),

F’(ttlH

F’(taIH- vl < 2)bF(tal H provided

vl < 1)= O(t-2),

> fl, by (5.17), we obtain for q Ran Q+,

W(T)

A

blt=oq

O(t --+) dt

+B+

,

(5.18)

0

where, with the obvious abbreviation,

A and since W(x, vt)U(t)

fro

FUo(t)*DrpU(t) dt,

W(x)U(t) (remember W(x)

FUo(t)*4(W,(x)

(5.19) W(x, vt)),

W,(0))U(t) dt.

(5.20)

Now we compute

(5.21)

259


.

b’= F’((x)/z < p). This together with the fact that 0 < -((x)/z’)b < and estimate (5.12) implies j’]o iiiDlX/zU(t)ll=(dt/t) < Cilll and a similar -Cb’ inequality with Uo(t) replacing U(t). Hence, we conclude that integral (5.19) converges as T +oz on Ran Q/. Next the integral in (5.20) converges in norm as T oz due to the relation where

(W(x)

W(0))F(- < 1)=O(z-l-u+’),

valid by virtue of the mean value theorem and estimate (3.5) on W. Thus W(T) converges as T oz strongly on Ran Q/. A similar statement holds for T -oz and Ran Q-. 121

The above theorem shows that for any 0 e Ran

Ua, o(t)FO +

IIqU,(t)O as

Q,+,

there is 0 + such that

0

+. Here b is the same as in the proof of Theorem 5.2. Using that O(t -x+a)

[Fa, b] by standard estimates, where

Ft F(ttlHa- vl < 1) and using equation (4.10), we obtain

rVa,(t)O Va,(t)O + O(t -+)

+,

for any 0 Ran Qa

v.

Next, since Ft commutes with Ua, o(t) and since

Ftf where a(tr)

a and e

(5.23)

P,f,

v, we have

IlFaU,o(t)f U(t)Pfll

(5.24)

0

oz. Here U,(t) is the evolution generated by e" + Ipal 2 4- Wa(0, Vat) as in (3.7) as but acting on (R) L2(Xa). Combining (5.22)-(5.24), we find

"

as

IlqUa, z(O0

U,(t)P,FO +

0

+oz. This together with (5.13) yields that for any 0 Ran Qa+,v there is 0 +

260

SIGAL AND SOFFER

such that

-o

lim

+

(5.25)

"

v. as p 0, where a obey a(a) a and Picking a finite-dimensional part of Ran Q+ as the dimension increases and using equations (5.4)-(5.5), (5.8), (5.9), and (5.25), we conclude that for any a with lal > 2, any q L 2, and any > 0 there is a finite set ofa’s with a(a) a and functions Zat

and b such that

(5.26) t- +

b
0, there is a finite set B of ’s with a() < a and functions qat such that

li’t-*

+O

Ua, z(t)q-

Uat(t)Patqat B

Equations (5.1) and (5.27) imply that for any set C of a’s and functions at., such that

0 there is a finite

e.

(5.27)

Cr

t--

On the other hand, by a standard result we have that, under conditions (A) and (B) 1/2 of Section 3, the wave operators

with # >

f+

s

lim U(t)*P U(t)

exist. Equation (5.27) with e < 1/2 and the last fact imply that in turn implies (3.8), i.e. asymptotic completeness. El

(5.28)

at Ran f+. This

We emphasize again that the condition/ (in (B)) was used only in the proof of asymptotic clustering (equations (5.1) and (5.9)) while condition (5.15) in the proof of estimates (5.12) and (5.13). 6. Time-dependent observables. In this section we outline some technical ideas used in this paper. Let H be a selfadjoint Schr6dinger operator and, as before, Ea Ea(H). By the time-dependent observable we usually mean a normdifferentiable family q(t) of selfadjoint bounded operators, which map D(H) into itself. In those few cases when we use unbounded operators it is clear from the


261

construction how to manipulate them. Let H(t) be a time-dependent selfadjoint Schr6dinger operator with the domain D(H) and generating the evolution U(t). Denote if, U(t)k, an orbit of H(t). We study the means

((t)),

((t)t/J,, ,)

with the ’s localized in some appropriate energy interval, say ff Ran Ea. Given small A, we choose 4(t)so that (ff(t))t essentially increases. This will imply that the propagation t moves away from the boundary of the phase-space support of (t) which in turn will lead to propagation estimates. To put this in more technical terms, define

D)(t)

d---T- + i[H, (t)-I.

We call this operation the Heisenberg derivative. (It is similar to the Lagrange or Liouville derivative in classical mechanics.) In applications below we use observables of the form Ear/)Ea. A key relation used there is d

d (Ea(t)Ea), (EaD)(t)Ea), + O(t--),

(6.1)

which is valid, provided

[H(t), Ea]

O(t --)

(6.2)

and

[H(t)- H, (t)-I

O(t--’).

(6.3)

The last two estimates for some e > 0 and for all A’s are assumed in the rest of this section and are easily proven in the applications (equations (13.2) and (3.5) are used in the latter case). To show that ((t)) increases we demonstrate that D(t) is positive modulo inessential terms. We will seek estimates of the form

ED(t)E > EF(t)E-

F,(t)

(6.4)

where F and Fi are time-dependent observables with Fi satisfying for all/

(6.5)

Ran E a. Then the desired propagation estimate is given in the and for all following lemma:

262

SIGAL AND SOFFER

LEMMA 6.1. Let (6.2)-(6.5) hold. Let b(t) be H-bounded uniformly in t. Then

.

for all Proof.

2

dt

< CIIq,

2

(6.6)

Since b(t) is uniformly (in t) bounded, we have

E(t)EA), dt

IEA(T)E)r- (EA(O)E)ol

< const II011

.

This inequality together with (6.1), (6.4), and (6.5) yields (6.6).

Remark. This lemma descends from the Putnam-Kato theorem (see RSII), and in particular it extends the latter to time-dependent observables ((t)). However, the main difference is that while the Putnam-Kato theorem requires the commutator to be positive which reduces dramatically its applications, Lemma 6.1 allows the Heisenberg derivative (which plays the role of commutator in the time-dependent case) not to be sign definite and establishes the propagation estimate only for the positive part of the commutator. This allows iterative application of this estimate and establishing control of different parts of the phase-space piece by piece. The most difficult task in this enterprise is to choose appropriate observables (t) and to prove for them estimates of the form (6.4)-(6.5). In this task we are aided by the following two technical devices borrowed from [SigSof4] (see also [SigSof3]). We consider smooth functions f obeying

(6.7)

ds
--

+ O(t-),

(7.23)

where di depends on the sharpness of the cutoff functions involved and --, 0 as the latter increases (i.e. supp F’I 0). Taking 6 and tr 2 sufficiently small and using (7.6)-(7.7) and an appropriate symmetrization of the left-hand side, we obtain that there are A around E and K such that

EaF’

( > p) >OEaF’

F

\ t2-

>p

O, and rp(t)=F’(A/t>.. 0( ’, f). Then the integral on the right-hand side 2

dt

finite

finite

Next, there are 0 < 2: < A: < o such that ]k ’(x)l 2

dx dt

tl

Collecting the estimates above, we obtain

Il

dt

CIIqll 2

(7.24c)

with C independent of the @’s. Taking into account that

(7.25)

273


where t 2 is the measure of supp F’(2 that

EaDqEa >

> p), we derive from (7.19) and (7.24)-(7.25) EaF’FEa + EaREd,,

(7.26)

where R R1 + R2 -1- R3 with R1, R2, and R 3 are given by (7.11b), (7.20), and (7.24b), respectively. By (7.7), 0 > p. Taking 2 < 19 0 and combining (7.26) with (7.13), (7.21), and (7.24c) and using Lemma 6.1, we arrive at (7.8). To prove (7.9) we use the maximal velocity bound of [SigSof3, Theorem 4.3] to estimate j’ f((x)/t > V/)a, tll (dr E!

Using Theorem 7.1, one can extend Theorem 6.1 of [SigSof3] to the threshold energies without changing the proof of the latter:

THEOREM 7.4. Assume conditions (A)-(B) with l > O. Let E e IR and let e > 0 if E is either a threshold or an eigenvalue of H and e 0 otherwise. Then for any time-dependent phase-space operator qt supported in {Ix" kl PSE. there is > 0 such that

> et} (ll < ,t) for

some 2 > 0 and vanishing on

2

dt

for any / L2(X), for any A E such that IAI result holds also for < O. For/3 2

2p, Lemma 7.3 implies 2

dt

tr

> O. A similar

< CIIII 2

(7.28)

for all e L2(X) and for any o > 2 > 0. This together with either Theorem 7.1 or Theorem 7.4 yields the following result.

THEOREM 7.5.

Theorems 7.1 and 7.4 remain true if F(IAI/t > p) and { Ix[/> et}, respectively.

are replaced by F((x)/t

> p) and {Ix" k[ > et}

Instead of working with the operator A/t, we could work with the operator V, as it is done in [SigSofl, 3]. Both approaches are similar and, in most of the cases, are equivalent. A simpler and more elegant proof of Theorems 7.1 and 7.4 can be deduced from [Gr] (see also [Kit, Tam]). 8. Asymptotic clustering revisited. The asymptotic clustering states roughly that t--, + a system starting in a given energy interval, disintegrates into noninteracting subsystems. This property was proven in [SigSof3] for pair potentials obeying (A)-(B) with # 1 and for nonthreshold energies. In this section as

274

SIGAL AND SOFFER

we reexamine this proof in order to extend the result to time-dependent Hamiltonians H(t) and to threshold energies. In the latter case we have to microlocalize this notion. Here H(t) is the time-dependent Schrtidinger operator defined at the beginning of Section 4. Let Ua,(t) be the evolution defined in equations (3.2)-(3.6) (see also the sentence after equation (3.6)), but with Wa(x) replaced by Wa(x) + W(x). Below we use the notation a,t introduced in (7.4). Let P 1 P, where, recall, P is the projection onto the pure point subspace for H. First, we recall the definition of asymptotic clustering [SigSofl, 3-1, slightly modifying it:

Definition 8.1. Assume condition (B) with/ > 1/2 holds. A system is said to be asymptotically clustering at an energy E if there is an interval A around E such that for any p L2(X) as

-

_+o

for some @f and for some 2 > 0. Here the sum extends over all a with (8.1) holds, we say that g’a,t is asymptotically clustering.

(8.1)

lal

2. If

Now we microlocalize this notion: Definition 8.2. Assume condition (B) with/ > 1/2 holds. A system is said to be asymptotically clustering in a region f x A X x IR x X’ x IR, if for any timedependent phase-space operator b supported in t2 and any 6 L2(X)

-

OlA’t-- Ela[>2 Ua’2(t)l/la+-

"-+0

(8.2)

_+o for some qa+- L2(X) and for some 2 > 0. Correspondingly, if(8.2) holds, as we will say that bqa,, is asymptotically clustering (or q6 is asymptotically clustering in t2 x A). The latter term we will also use for general bounded operators b. As we mentioned in the introducton for threshold energies some of the channels have nonzero intercluster velocities (propagating channels), while the others have zero ones (diffusing channels). The propagation estimates on which the asymptotic clustering is based hold only for the propagating channels. Thus we expect the asymptotic clustering for threshold energies to hold only in a part of the extended phase-space inaccessible to the threshold channels. In particular, the region Ix" kl > eltl (or Ixl > eltl) supports only channels whose intercluster velocity is > const e. Therefore, we expect that the system is asymptotically clustering in this region. The main result of this section is the following one: are satisfied. Then (i) THEOREM 8.3. Assume conditions ( A ) and (B) with # P@a, is asymptotically clusterin# in any compact enerly interval A disjoint from the thresholds of H, provided 2 enterin# (8.1) belongs to (0, 6 mine 0 depends on N and the mi’s only, and (ii) @a, is asymptotically

275


clustering in {Ix’kl eltl} A for any > 0 and for any A sufficiently small, provided 2 entering (8.2) belongs to (0, 6e), where 6 > 0 depends on N, the re{s, and A only.

Proof. (i) The proof is a straightforward modification of the proof for U(t) exp(-iHt) given in [SigSof3]. One uses estimates (4,2) to show that the additional contributions are integrable. (ii) The proof proceeds in the same way as the proof of (i) with the exception that, instead of the propagation result of [SigSofl] (see Theorem 5.2 of [SigSof3]), one uses Theorem 7.1 of the previous section. In any case, it follows from Theorem 7,4 of the previous section along the lines of [SigSof3] (see Section 8 of the latter paper). 1:21 Remark 8.4. In Supplement we show that for 2 > 0, F((x)/t > 2)a,t is asymptotically clustering for sufficiently small A. However, we do not use the latter result in this paper. Since Vb commutes with x b, the analysis above is applicable to the evolution Ub,o(t)- F(lvb[ > p)Ub(t) generated by nb(t)= Hb + W,t(x", vbt) and yields the

following result.

THEOREM 8.5. Theorem 8.3 still holds if U(t) is replaced by Ub,,(t) with

p

> 0, provided the sums on the ri#ht-hand side of (8.1) and (8.2) are restricted by

9. Many scales of configuration space. In this section we analyze the geometry of the configuration space X with a view at describing various breakup processes developing according to different time scales. In particular, we construct a multiscale partition of unity on L2(X), which is used in the subsequent analysis, and derive some geometric relations used in proving the propagation estimates of the next section. The geometrical analysis of X and construction of partition of unity originates in [SigSofl, Sect. 5] (except that we have no use of the momentum dependence and we drop it) and is inspired by that in [Gr], Let M be defined by

(9.1)

a, b, where a v b is defined in the standard way. Pick up numbers t,

t < (20M)-Xt. Let

if b < a.

>

obeying

(9,2)

(t, all a’s). We introduce the subsets of X:

F.,, {x Xllxl. > const a, (Xa) ’ 2t.},

(9.3)

276

SIGAL AND SOFFER

where the constant is independent of x and and

?a

min b>a

(9.4)

t.

These domains describe clustering of the system at different time scales. The constant in (9.3) is not fixed but is adjusted according to a relation at hand. In particular, it could be different for different F.,’s involved. It is shown in [SigSofl] that

{ (x) >_. t,,,,} m

F,.,.

(9.5)

lal2

Consequently, one can associate a smooth partition with this open covering. Explicitly, it is constructed as follows [SigSofl, Sect. 5, equations (5.3)-(5.4) with the momentum dependence dropped from equation (5.3a)]’

Note that we have inserted here an extra factor, n 1, not present in [SigSofl, equation (5,4)], Consequently, (see [SigSofl, Thm. 5.1(ii)-(iii)])

F((x) > 1} \t,,..

Ja,

(9.7)

/

and moreover,

suppja,, =

F.,,.

We demonstrate here the last inclusion. Let (i, j) (9.1)

Ix xl

(9.8) a and let b

(ij) v a. Then by

>- lx bl -I xa I,

which implies that on supp ja,,

(9.9) Thus x e Fb,,; so (9.8) holds. Note that for lal 2

J"t

F(X t.

1)F(- > 1).

(9.10)


277

The next lemma elaborates Theorem 5.1(iv)of [SigSofl]:

LEMMA 9.1. With the constants in the Fb.t’s chosen appropriately, (9.11)

Proof. Let x be such that Ixl t. Then relation (9.5) adapted to the configuration space X b states that there is c < b such that min lc

Ixl

min

I(x)l

l,gc

l const ?,

Ixl

Let now x left-hand side of (9.11). Since other hand, since x e F.t,

Ixl Ixl min ,gb Since

,

for some c < b.

(9.12)

tb (9.12) holds for this x. On the const

?b.

?b > ?c, we have

minb Ixl’ minb Ixl)

xl min

t

> const ? i.e., x

Fc.,. E!

Note that since

tb/? as

"+

0

, Ixlb in the definition of F., can be replaced, for sufficiently large, by Ixlb’ {Ixl 2tb, IXblb C ’b} = {Ixl 2tb, IXblb

(9.13)

for appropriate 0 < C 2 < C ,< Let 0 b

fib).

(9.15)

278

SIGAL AND SOFFER

We set (20M) 1/$,

o

(9.16)

and in what follows we assume that

>

o

In this case

(20M)-t

(9.17)

if b < a.

In order not to cram expressions below, we choose the scales as

.

t. Then o obey (9.2), provided {t, o with time.

(9.18)

> to. In notations (9.3) and (9.6) we identify

10. Localization of momentum. In this section we show that the channel momentum for threshold (-= diffusive) channels can be very sharply localized around zero. Let H be an N-body Schr6dinger operator obeying (A) and (B). Recall that P, is the eigenprojection corresponding to the channel tr (and the eigenvalue e), i.e., the orthogonal projection on the eigenfunction if’. Introduce the intercluster distance

Ixl.

min

(x’),

14;a

i.e., the minimal distance between particles from different clusters of a. We will use the symbol F(Ixl, > 6) to denote a smooth cutoff function supported in Ixl > which can be realized for instance using (3.1).

THEOREM 10.1. Let fl < lae and let either a a(cr), where r is a threshold channel with e" = v or inf cont spec H" v. Then (for > O)

F(ttlH-- vl )F([@" > 1)Qapa where Q, id

P.

P. if a

v, and a(a), e N lal.) Proof. First, we show that

P. for Irl

O(t-t/2),

Pa if inf cont spec H"

Q.F(taIH.- vl < 1)p.

O(t-11/2).

(10.1)

v. (Remember that

(10.2)

279


In the case Qa

Pr this follows from P,F(tlIH,- vl < 1)= P,F(tJp2 < 1).

In the case Qa

(10.3)

Pa we observe that, due to Ha=Hatl d- l(p2a

(10.4)

and

Po(H

v) > O,

we have that

F(talH

vl < )

F(talH- vl < -)F(tap2 < 1),

(10.5)

which in turn implies (10.2). Now equation (10.2) yields that

F(talH- vl < )F(I > 1)QaPa

O(t-1/2) + R,

(10.6)

where

R

F(talH

vI
I)F(taIHa- vI > )Q.P

Using that p commutes with Q (and H) and using the simple estimate

I(H-i)-IF( > 1)-F(I> 1)(Ha-i)-llp. O(t-.), we conclude that it suffices to show that for fl )

O(t-"t’"-a)), (10.7)

where n is the degree of differentiability of the potentials in a neighbourhood of c. This is done in Lemma 10.3 below.

280

SIGAL AND SOFFER

-

Remark 10.2. A slightly worse, but sufficient for our purposes, estimate

ttJlH- v]

F

o’ ]Ha

F

v]

>

O(t -nmin(au-#’#-#’))

has an elementary proof. Indeed, it is shown by using repeatedly the relation

(t(H-v)

i)-lF(- > 1)-F(I> 1)(ta(H,-

and observing that (tt(H

II(ta(n

v)

i)"F(talH

v)- i)-"F(t

’ IH

v[

vl

v)- i)-

O(t -’u+t)

< 1/2)is bounded for any n, while )11

(1 + 1/4ta-a’) -".

Observe that (10.2), (10.3), (10.5), and (10.7) imply that

F

taln-v[

Qa

F(talH- vl < )F(I > 1)QaF(ttp2a
O. Then (10.7) holds.

1)

+ O(t-n(’u-a)),

(10.8)

is the degree of differentiability of

of differentiability of the potentials at

Proof. In order to simplify notation we prove this lemma in the two body case. In other words we replace Ha by Ho and F(Ixla/t > 1) by F((x)/t > 1) and set v 0. Denote F (2) F(2 < 1). Using Lemma A.1 of the appendix (this lemma is a straightforward extension of Lemma 5.2), we obtain

F( > 1)F(taH) F(taHo)F( > 1) =,Xk=0 (k’)-ItkaFk)(taH) adk(F(- > l)) + where (V

H

Ho) ad(B)

HoB- BH

[Ho, B]

BV

R"(F)’

281


and

IIR(Fx)F(H

1)11

c

f2oo

Expanding ad"(F), we get for some numerical coefficients Ck, k

II

adn(F)

ck

(k,

k,),

ado(F"V"),

where the sum is taken over nonnegative integers ki’s,/{s, and mi’s, 1, n, n that for 1. s sufficiently large obeying Using and ki + l F((x)/t > 1)V (9(t-r), using that

m

ado and using that for k

ado

(F (- 1)VI)F()

< (degree of differentiability of V in a neighbourhood of o)

(F (- > 1)"Vt)F(> ),2

+ rn + k and rt + r 2

for n

=0,

1, r:, r 2

(Ho+ 1) -k/2 =O(t-n’’)

0, 1, we obtain

and therefore

R.(F )F(H

< 1) O(t-"’-tJ)).

Since

F(tlJHo 2)Fk(ttJHo) O, we obtain that

2)F( > 1) Ft (taH) =F(taHo> 2)IF( > 1)F(taH) F(ttJHo)F(> 1)]F(H

F(taHo >

for n specified in the lemma. El

1)

282

$IGAL

AND SOFFER

11. Evolution of threshold channels. In this section we prove maximal velocity estimates for the threshold channels. The Hamiltonians under consideration are of the form

H(t)

H + W(x, t),

where H obeys (A) and (B) of Section 3 and W obeys (4.2). Recall that U(t) denotes the evolution, generated by H(t), and Ot U(t),. We begin with definitions. Given an operator-valued function At and a subspace R of L(X), we say

(11.1) if and only if for all $ e Ran Q+ the estimates

(11.2) hold, with the constant C the same for all $ Ran Q+. In an obvious way we define also At 0. Similar definitions are valid for < 0. Here and in the remainder of the paper, v is a fixed threshold of H. Next, for a > b we denote by xg the projection of x onto Xb Xa, i.e.,

X

Xb

(11.3)

X

We also use the notation xb Ib

min (xg),

(11.4)

a>b

which is just the distance Ixl for the projection xb. Again, F(Ixala/t > 1) will stand for a smooth realization of the corresponding cutoff function, though we write

symbolically

>

t---g--

(11.5)

It is clear from (3.1) how to understand such a gradient. Remember that # is the rate of decay at o of the pair potentials. THEOREM 11.1. Let Qa be the same as in Theorem 10.1. In the case when Qa P, we assume that either (x)P is bounded for some rl > 0 or that H > 0 for Ibl > we assume that H > 0 for Ib[ > lal / 1. Let lal + 1. In the case when Qa (1 + 2 -N-I)-1 # < 1. Let 1/2 < a, O, < obey 6 < 0 and

,

tr=

min

[+

min(a,5),

+

#2-N-11

> 1.

(11.6)


Then

F

t2_

>

F

t.

Q,

_

283

(11.7)

0

and

t2_

F

F t 0 for Ibl

>1 Q.-" 0.

(11.8)

lal + 1. The remaining case

is simpler. We use the propagation observable

(/)

-FI\

F2

t2

> F3(talp, 2 < 1),

(11.9)

where 2(1 i) < fl < 5#. We have labeled the cutoff functions here to make it easier to keep track of them when we omit their arguments. Our purpose is to compute D. First we figure out the contribution of each of the factors on the right-hand side. Using that

that

tnF2 O(t-), and that PaF3

O(t-#/2), we obtain [H, f2]f3

O1/2(t-min((#/2)+0’20)).

(11.10)

Next we compute

1

=-1

G,

(11.12)

284

SIGAL AND SOFFER

Ct -I1 and

where 7.b(X, t) are smooth functions obeying [9Zb(X, t)l

6> 2-sV. The latter is possible because of(ll.6). Next, by (4.10) and (10.7)

V

1

,= V(taln- vl < l)V

for any $ Ran Q:, provided P 1, we have that

62V > ft.

F(talHb-vl< 1)=0if v a,

,

62 > fl and

+ O(t a-u)

Ibl > lal,

(11.13)

and therefore

=F(talpbl 2 0, for any $ A through zF((x)/t

A"

(11.15)

O(t-*),

Ran Q. This together with (11.12) and after commuting 1) yields

((x,)

)( )

Taking this into account, we return to equation (11.11) to conclude that

(DF)F2F3’ where a2

min(26,

+ al) >

23 (xa) 2 2

and $

FF2F$’

Ran Q.

+ O(t-*2)’

(11.17)

285


Finally, we note that F1F2 with 6 < 0 is supported in Ixl 1/2t (provided is sufficiently large). This together with condition (B) of Section 3 and a simplified version of [SigSofl, equation (A.31)] yields

Fi F2[F3(talPal 2

< 1), la]- O(t-{x+u)+ta/2)).

(11.18)

Similarly, (4.2) and [SigSof3, equation (A.13)] implies

[F3(talpl z

< 1), W]--- O(t-x-u+a/u)).

(11.19)

Combining estimates (11.10) and (11.17)-(11.19) and computing the remaining time derivatives of the second and third factors, we obtain

Dq

26

(xa) 2 t2 +1 FF2F3 +

OFxF’2F3

4-

-ffl-lPal2FxF2(-F’3) 4- O(t-O’3)

where 0 3 > 1. The first three terms on the right-hand side are positive. This, together with the inequalities (11.16) and (x")Z/tZ)F > FI and with Lemma 6.1, implies t2_

F

> F(ttlPal u < 1)--" 0

F

(11.21)

Next we consider the propagation observable

t2 >1 ) b, b=Fo\{(x)

(11.22)

where b is given by (11.9). As in (11.10), we have

i[H, Fo]F3

2 A t2_

F’oF3 + O(t -2) (11.23)

O(t -’) with a4 > 1. This yields

Db

2 (Xa) 2

2F’q + FDb + O(t-’s)’

(11.24)

where a5 > 1. Since the first two terms on the right-hand side can be symmetrized to positive plus integrable parts, this relation together with Lemma 6.1 and equations (11.9) and (11.20)implies t2_

F

(

F\

t6

F(t alpl

> 1)

0. (11.25)

286

SIGAL AND SOFFER

Now since 6 < 0, equations (11.21) and (11.25) remain valid if IXala there is replaced by IXla. Moreover, equations (4.10) and (10.9) imply that for any e Ran Q+ and for 0/ > fl

QaF(> 1)O,=F(talpal2< 1)Q,F(- > 1)O,+O(t-"+a).

(11.26)

This relation and the conclusions of the previous sentence yield (11.7) and (11.8).

El

Remark 11.2. One can strengthen (11.7)-(11.8) by sacrificing the uniformity as follows:

\ ta

F

CIl(x)X/2011,

Qat

(11.27)

where p e [0, tr- 1]. In order to prove this estimate, we use the propagation observable

b2

IXala < 0,

(11.28)

.---_

where is defined in (11.9), and then proceed as above but using, in addition, that due to p < min(0, ), Dx > 0. As a result we arrive at (11.27) with Qa replaced by F(t IPal 2 < 1). This together with (11.26) implies (11.27). 12. Estimates for free evolution. In this section we obtain estimates for free motion at any energy. Recall Ho -A.

THEOREM 12.1. Let 6 (0, 1) and a > O. Then

(tL / F(-=

2

dt

< Cllll 2

(12.1)

for any d/ L2(X). Proof. First, we rescale tr into 1. Second, we say Bt "-" 0 if dt

e-m’O T < CllOll 2

for any L2(X). Note the difference between this definition and (11.1)-(11.2). Naturally, B, 0 will mean that B, can be written as a sum of a nonnegative operator and of an operator which is "-" 0. Denote

DoK

OK

+ i[Ho, K].

(12.2)

287


We prove that

(12.3) by using four propagation observables K

F( > 1)F(tl-Olp[
-x, and using that

F’(t-lp[

O)

(12.5)

0,

we obtain

DoKx < 20-6+6 F’F- F(- F’). (1-6)0

(12.6)

where 6 is a small constant related to the sharpness of the cutoff functions involved and which vanishes as that sharpness increases. Recalling that 20- 6 < 0 and taking the cutoff functions so sharp that 6x < 6 20, we derive from (12.6)

F(= 1)F(tX-alPl
1)F(t-lpl

O)

O,

(12.8)

and

provided 0 2 f(tX-alp[ > O)

(12.10)

288

SIGAL AND SOFFER

with 2 > 0 and 0 > 0 obeying

02 > 2(26- 1)/2.

(12.11)

Using that 0 F(tX_lPl dt

> O) > 0,

we find

DoK2 > (2 Pick

61

IPl 2

Isupp F’(s > 2)1 to obey 0 < 1 < 202

t2a_-----i-

(2

> 2 < (2 + 6I)F’

t2a_

1)2. Using that

t2a_

>2

(12.12)

and using that

t2tx-)lpl2F(tX-lp]

> O) > 02F(t-lpl > 0),

we arrive at

DoK 2 >-F’F with a

202

(26

(12.14)

61 > 0. This yields that

1)2

2 F(t 1- Ipl

F t2_

> 0)

(12.15)

0,

provided (12.11) holds. Next we define, for 2 > 6/2 and 0 > 0, obeying (12.11),

F

K3

k

t2

>

F t2a_

>2

F(tl-nlpl

Using that ((x)2/t2O)F’((x)2/t 2 > 1) < (1 + 61)F’((x)2/t 2 sharpness of the cutoff function involved, we find

DK3 >

4A

26(1

61)

)

F’FF +

(21p12

\-{-_--f

> 0).

(12.16)

> 1), where 6

-1

is the

289


As above (see (12.11)-(12.13)), the last term is positive modulo O(t -) with tr > 1. Since 22 > 6, 61 could be chosen so small that

where 6 2 > 0. Using this, we obtain after estimating simple commutators

DoK3 >-[

F’FF

t2O_

(12.17)

for some a > 0. This yields

F

,t2a.

F

t2O. > 2 F(tl-Olp[ > O)

0

(12.18)

provided 2 > 6/2 and 0 > 0 obey (12.11). Combining (12.15) with (12.18), we conclude that (12.18) holds for any 2 and 0 > 0 obeying (12.11). Finally, we consider the propagation observable

>

K F.

F

t2=i

(1/M)lxl- IXa[, where M is defined in (9.1), for lal 2, KG is supported in IXla > const 2"-1. This and equations (13.6) and (13.7) imply that

[Ia, K]G=O(t-)

(13.11)

with a (1 + t)(2 1). Note that a > by condition (13.4). This together with (13.9) and with the relations [IPal 2, G-I [Ia, G] 0 yields

Dq,

(DK,)G,

(13.12)

where

c3Ki + DK --SiiI-IPI2’ K].

(13.13)

292

SIGAL AND SOFFER

Applying inequalities (12.6), (12.14), (12.17), and (12.22) to (13.12), we arrive in a familiar fashion (see e.g. Section 6) at

This relation, the spectral theorem, the equality A

\ ta < and the condition 2a

6>

AaF\

t
1, and therefore (Ab/t 2"-l)jb,, 0, and using that AbJb,,k, O(t -/2)) for ff Ran Q+ by (4.10) and Theorem 10.1, and therefore (Ab/t 2"- )jb, O, we conclude that (A/t 2- )jb,t O. Hence it suffices to show that F((x)/t 1)jb,, O. The latter follows from property (9.8) Ofjb,,, the condition H b > 0 and equation (11.8) with a b. (In fact, we have covered one part of the estimate twice.) El

THEOREM 13.2. Assume (13.2)-(13.3) hold. Let obey (13.4). Then as e $ 0

(13.17)

for any Ran Q+. Proof. Property (9.8) of the partition of unityja.t together with equations (9.3), (9.18), (11.27) and (13.3)imply that if lal

> 3.

Hence, due to (9.7), it suffices to consider the vector standing under the sign of norm on the left-hand side of (13.17) multiplied by ja,, with lal 2. The result is of the

293


form kF(IAI/t

< e), where t2 >1 t. / 1) and using (13.5) and (13.8), we find that Db 0. Using this relation and following the lines of the proof of Theorem 5.2, we obtain the following lemma.

LEMMA 13.3. Let and

obey

wave operators

Ua+ exist on

S-

#2

--

< # and lal

3e)b,

F(IH

-

vl > 3e)bF(lpol 2 > 2e), + O(t-). (13.26) G where G*AaG < tG*G, that

A a + Aa, 1)F(H < M) for some M < A

that

F((xa)/t < Lemma 6.6 of [SigSofl]

, and that 6 < 1, we obtain similarly

to

(13.27) Thus it suffices to show that (we change here 2e to e)

\ to

F

2e)F

"-0.

(13.28)

In order to prove this relation, we use the propagation observable

Ckx

F\

>1 f(lpa[ 2

f t2 o e so that

F’(2)

0 for 2

>

2e)F(-A-),

< -e,

2/e for -e/2

-1F([2[ < e).

(13.29)

< 2 < e/2 (13.30)

In computing Dbx we control the term involving DF((xa)2/t 20 < 1) by equation (13.8) and the term involving DF((x)2/t 2 > 1) by Theorem 13.1, and we use that

>F

DF

< e -/(IPal 2

due to the localization property of F(IAal/t 0>0

Db which yields (13.23).

e)- Ct -2

(13.31)

< e). As a result, we obtain for some

0

(13.32)

}- (left-hand side of (13.28)),

E!

Equation (13.23) and a standard argument involving wave operators implies that

bF(lH"-vl> 6e)F as

(- < e),,

0

(13.33)

+oe for any p s Ran Q+. This together with (13.22) yields (13.17). El

295


APPENDIX Expansion of commutators.

LEMMA A.1. Let A, Ao, and B be selfadjoint operators on a Hilbert space such that A and B, and Ao and B obey (6.8)-(6.9). Let g C(IR) be bounded and such that for some n

lsO(s)l ds
it. In this supplement we prove asymptotic clustering in the region (x> > et, e > 0, for time-dependent Hamiltonians defined in Section 4 and for arbitrary energies.

LEMMA S.1. Let E e IR and 8 2 > 26 > O. Then there

is an interval

A around E

such that

-

+ for any k e L2(X), provided the sharp. A similar result holds for -. as

Proof.

cutoff functions above are sufficiently

Pick up 2 > 0 so large that

(S.2) as --, oz and 2

dt

< CllOll

(S.3)


297

for all L2(X) by the maximal velocity estimate (see [SigSof3, Thm. 4.3 and equation (9.6)]). Due to (7.28), for any L2(X) there is a sequence {t,} such that and t,

as n

.

Moreover, (S.2)-(S.3), (7.28), and the Cook argument imply that

converges strongly as

.

This together with (S.4) yields (S.1).

TIEOREM S.2, Let E IR and e > O. There is an interval A around E such that for L2(X), F((x)/t > e)@a,, is asymptotically clusterin9 at the eneroy E. Proof. Pick up 6 < e2/2 and pick up 2 > 0 such that (S.2) holds and write

any

Applying (S.2) together with a simple commutator estimate to the first term on the right-hand side, Lemma S.1 to the second one, and Theorem 8.3 to the third term, we obtain the statement of Theorem S.2.

REFERENCES [CFKS] [Com] [En]

[Gr] [H6r]

[Kal]

H. CYCON, R. FROESE, W. Kmscn, AND B. SIMOr, Schr6dinoer Operators, with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin, 1987. J.-M. COMBES, Relatively compact interactions in many particle systems, Comm. Math. Phys. 12 (1969), 283-295. V. ENSS, "Quantum scattering theory for two- and three-body systems with potentials of short- and long-range" in Schr6dinoer Operators, Lectures, Second Session, C.I.M.E., Como, 1984, ed. by S. Graffi, Lecture Notes in Math. 1159, Springer-Verlag, Berlin, 1985, 39-176. G.-M. GRAF, Asymptotic completeness for N-body short-range quantum systems: a new proof, Comm. Math. Phys. 132 (1990), 73-101. L. Ht)RMANDER, The Analysis of Linear Partial Differential Operators IV, Grundlehren Math. Wiss. 275, Springer-Verlag, Berlin, 1985. T. KATO, Fundamental properties of Hamiltonian operators of Schr6dinger type, Trans. Amer. Math. Soc. 70(1951), 195-211.

298 [Ka2]

SIGAL AND SOFFER

, Wave operators and similarity for some non-self-adjoint operators, Maths. Ann. 162

[SigSof4] [SigSofS]

(1966), 258-279. H. KITADA, Asymptotical completeness of N-body wave operators I. Short-range quantum systems, Rev. Math. Phys. 3 (1991), 101-124. E. MOURRE, Op.rateurs conjugus et proprit.s de propagation, Comm, Math. Phys. 91 (1983), 279-300. P. PERRY, Propagation of states in dilation analytic potentials and asymptotic completeness, Comm. Math. Phys. $1 (1981), 243-259. M. REED AND B. SIMON, Methods of Modern Mathematical Physics II, III, Academic Press, New York, 1979. I.M. SIGAL, "Mathematical questions of quantum many-body theory" in S.minaire Equations aux Derivers Partielles 1986-1987, lcole Polytechnique, Paris, expos6 XXIII. On long-range scattering, Duke Math. J. 60 (1990), 473-496. I.M. SIGAL AND A. SOFFER, The N-particle scattering problem: asymptotic completeness for short-range systems, Ann. of Math. (2) 126 (1987), 35-108. "Asymptotic completeness of multiparticle scattering" in Differential Equations and Mathematical Physics, Proceedings, International Conference, Birmingham, 1986, ed. by I. W. Knowles and Y. Saito, Lecture Notes in Math. 1285, Springer-Verlag, Berlin, 1987, 435-472. Long-range many-body scattering. Asymptotic clustering for Coulomb-type potentials, Invent. Math. 99 (1990), 115-143. , Local decay and velocity bounds, preprint. , Asymptotic completeness for Coulomb-type 3-body systems, preprint, Princeton

Ill]

A.G. SIGALOV

[Kit]

[Mo] [Pe]

[RS] [Sigl] [Sig2] [SigSofl

[SigSof2]

[SigSofT]

,

Univ., 1991.

[SinMuth]

[Tam]

AND I. M. SIGAL, Description of the spectrum of the energy operator of quantum mechanical systems, Theoret. and Math. Phys. 5 (1970), 990-1005. K. SINHA AND PL. MUTHURAMALINGAM, Asymptotic evolution of certain observables and completeness in Coulomb scattering I, J. Funct. Anal. 55 (1984), 323-343. H. TAMURA, Propagation estimate for N-body quantum systems, Bull. Fac. Sci. Ibaraki Univ.

Ser. A 22 (1990), 29-48. SIGAL: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, TORONTO, ONTARIO, MSS 1A1, CANADA SOFFER: DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY 08544, USA

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