C O L L O Q U I U M M A T H E M A T I C U M VOL. 79 1999 NO. 1

C O L L O Q U I U M M A T H E M A T I C U M

VOL. 79 1999 NO. 1

CHARGE TRANSFER SCATTERING IN A CONSTANT ELECTRIC FIELD

BY

LECH Z I E L I ´ N S K I (PARIS)

We prove the asymptotic completeness of the quantum scattering for a Stark Hamiltonian with a time dependent interaction potential, created by N classical particles moving in a constant electric field.

1. Introduction. We consider a model describing the quantum dy- namics of a light particle (such as an electron) in collisions with some heavy particles (such as some ions) obeying the laws of classical dynamics. Thus only the light particle is considered a quantum particle, while the heavy par- ticles follow some classical trajectories R ∋ t 7→ χ _k (t) ∈ R ^d . If V _k denotes the quantum interaction potential between the quantum particle and the kth classical particle, the total quantum time-dependent interaction V (t) is the operator of multiplication by

(1.1) V (t, x) = X

1≤k≤N

V k (x − χ k (t)),

and the total time-dependent Hamiltonian H(t) is a self-adjoint operator in L ² (R ^d ),

(1.2) H(t) = H 0 + V (t, x),

where H 0 denotes the free motion Hamiltonian. The subject of scattering theory is to describe the large time behaviour of the evolution propagator {U (t, t 0 )} t≥t 0 of H(t), that is, the family of unitary operators in L ² (R ^d ) satisfying

(1.3) i d

dt U (t, t 0 )ϕ = H(t)U (t, t 0 )ϕ, U (t 0 , t 0 )ϕ = ϕ, for ϕ from the domain of H ₀ .

The first papers describing such a model considered the case of linear classical trajectories and H 0 the Laplace operator [10, 25, 26]. The papers

1991 Mathematics Subject Classification: 81U10, 47A40, 47N50.

[37]

[7, 29] deal with classical trajectories which are only asymptotically lin- ear and the papers [30, 31, 32] deal with the dispersive case when H 0 is a more general elliptic operator. We note that all these papers consider the hypothesis that different classical trajectories have different asymp- totic velocities lim t→∞ χ ^′ _k (t), which implies the separation of trajectories:

|χ _k (t) − χ _k ^′ (t)| ≥ ct with c > 0 if k 6= k ^′ .

The aim of this paper is to consider the situation arising in the presence of a constant electric field E ∈ R ^d \ {0}, when the free motion Hamiltonian for a particle of mass m > 0 and charge q 6= 0 has the form

h 0 (x, p) = p ²

2m − qE · x

and the Hamilton equations ˙p(t) = qE, ˙x(t) = p(t)/m have the solutions of the form

p(t) = qEt + mυ, x(t) = qE

2m t ² + υt + ω,

where υ = p(0)/m ∈ R ^d and ω = x(0) ∈ R ^d . Thus the above solutions of the Hamilton equations describe the motion that is free in the directions orthogonal to the constant field E and uniformly accelerated in the direction parallel to E.

We shall consider only the simplest situation when different classical trajectories have different asymptotic accelerations lim t→∞ χ ^′′ _k (t). More pre- cisely we begin by assuming the following separation condition: there exist constants T 0 , c > 0, such that for t ≥ T 0 ,

(1.4) |χ _k (t) − χ _k ^′ (t)| ≥ ct ² if 1 ≤ k < k ^′ ≤ N.

Let m k , q k be the mass and the charge of the kth classical particle and assume that χ(t) = (χ ₁ (t), . . . , χ _N (t)) is a solution of the Newton equations (1.5) m k χ ^′′ _k (t) = q k E − X

k ^′ ∈{1,...,N }\{k}

∇V k,k ^′ (χ k (t) − χ k ^′ (t)),

where the classical interaction potentials V k,k ^′ satisfy the decay condition (1.6) |∇V _k,k ^′ (x)| ≤ C ₀ |x| ^{−1−µ 0} for |x| ≥ C ₀

with C 0 , µ 0 > 0.

It is clear that (1.4)–(1.6) imply

(1.7) χ ^′′ _k (t) = z k + O(t ^{−2(1+µ 0 )} ) with z k = q k

m k

E

as t → ∞, i.e. z k = (q k /m k )E = lim t→∞ χ ^′′ _k (t) is the asymptotic acceleration of the trajectory χ k . Since (1.7) means that _dt ^d (χ ^′ _k (t) − z k t) = O(t ^{−2−2µ 0} ), the limit

υ k = lim

t→∞ (χ ^′ _k (t) − z k t)

exists and introducing e χ k by the relation

(1.8) χ _k (t) = ¹ ₂ z _k t ² + υ _k t + e χ _k (t), we have

(1.9) χ e ^′′ _k (t) = O(t ^{−2−2µ 0} ), χ e ^′ _k (t) = O(t ^{−1−2µ 0} ) as t → ∞.

The Hamiltonian of the free motion for a quantum particle of mass m 0 >

0 and charge q 0 6= 0 has the form

(1.10) H 0 = p ²

2m 0

− q 0 E · x, where p = (p 1 , . . . , p d ) = (−i∂ x 1 , . . . , −i∂ x d ).

For quantum interactions V k we assume that for some constants C, b C, ε 0

> 0,

V _k (x)(1 + p ² ) ^{−1+ε 0} is a compact operator in L ² (R ^d ), (1.11a)

|∂ _x ^α V k (x)| ≤ C for |x · E| ≥ b C and |α| ≤ 2, (1.11b)

and V k = V _k ^l + V _k ^s with real valued functions V _k ^l , V _k ^s , such that for some µ > 0 we have

|∂ _x ^α V _k ^l (x)| ≤ C(1 + |x|) ^−µ−|α| for x ∈ R ^d and |α| ≤ 1, (1.11c)

|∂ _x ^α V _k ^s (x)| ≤ C(1 + |x|) −µ+(|α|−1)/2 for |x · E| ≥ b C (1.11d)

and |α| ≤ 1.

Theorem 1. Let U (t, t 0 ) be defined by (1.3) with H(t) given by (1.1), (1.2), (1.10). For k = 0, 1, . . . , N , let z k = q k E/m k be such that z k 6= z k ^′ if 0 ≤ k < k ^′ ≤ N . Assume that the trajectories χ _k (t) have the form (1.8) with e

χ k (t) satisfying (1.9) for some µ 0 > 0. If V k = V _k ^l + V _k ^s satisfy (1.11a–d) for some µ > 0, ε 0 > 0, then the limit

(1.12)

Ω(t 0 )ψ = lim

t→∞ U (t, t 0 ) ^∗ e ^{−itH 0 −iS(t)} ψ with S(t) =

t

\

1

dτ X

1≤k≤N

V _k ^{l 1} ₂ z 0 τ ² − χ k (τ )
,

exists in the norm of L ² (R ^d ) for every ψ ∈ L ² (R ^d ). Moreover , the asymp- totic completeness holds, i.e. the wave operator Ω(t 0 ) defined by (1.12) is unitary.

We recall the result of I. M. Sigal [20] (cf. also [3, 4, 5]) which guarantees the absence of eigenvalues for 2-body Stark Hamiltonians H k = H 0 + V k (x).

This allows us to neglect bound states and the asymptotic completeness

formulated in Theorem 1 implies that for every ϕ ∈ L ² (R ^d ) there exists ψ ∈

L ² (R ^d ) such that ϕ = Ω(t 0 )ψ. Thus U (t, t 0 )ϕ−e ^{−itH 0 −iS(t)} ψ → 0 as t → ∞,

which means that the asymptotic behaviour of U (t, t 0 )ϕ is asymptotically the same as for the free evolution (modulo a phase factor e ^−iS(t) ).

We note that the approach used in the proof below comes from recent developments of scattering theory of N -body systems ([6, 8, 21]). We also mention the references [9, 12, 15–17, 19, 23, 24, 27, 28, 33] concerning Stark scattering in the 2-body case and [1, 2, 13, 14, 18, 22] in the N -body case.

In Section 2 we begin by describing in Lemma 2.1 asymptotic concentra- tion of the free evolution trajectories e ^{−itH 0} ϕ on classical Stark trajectories.

Then it is easy to prove the existence of the wave operator Ω(t ₀ ) given by (1.12). Clearly Ω(t 0 ) is an isometric injection and in order to prove the asymptotic completeness it suffices to prove the existence of the limit (1.12 ^′ ) Ω(t 0 ) ^∗ ϕ = lim

t→∞ e ^{itH 0 +iS(t)} U (t, t 0 )ϕ

for every ϕ ∈ L ² (R ^d ). Indeed, if Ω(t 0 ) ^∗ given by (1.12 ^′ ) exists, then applying the chain rule we get Ω(t 0 )Ω(t 0 ) ^∗ ϕ = ϕ, that is, Ω(t 0 ) is surjective and hence unitary.

To begin the proof of the existence of (1.12 ^′ ) we assume for simplicity V _k ^s = 0 and introduce the auxiliary observable η t . This observable is used in Proposition 3.2 to introduce an energy cut-off, similarly to the “boosted Hamiltonian” of Graf [7]. However, instead of Enss approach used in Graf [7], our next step is based on the existence of the wave operators Ω k (t) of Proposition 3.7 (similar to the Deift–Simon operators of the N -body theory developed in Graf [8]). Then Proposition 3.7 allows us to localize and “distinguish” interactions of different classical charges, reducing the problem to the 2-body problem when the number of classical charges is N = 1.

The situation N = 1 is studied in Section 4 using the ideas of the Mourre estimate. More precisely, knowing that z 0 ·p is the conjugate operator for H 0

(i.e. we have the positive commutator [iH 0 , z 0 · p] = z ₀ ² I), we find the propa- gation estimate of Proposition 4.3 using a suitable cut-off g 1 (z 0 ·p/t) instead of z ₀ · p. Finally, in Section 5 we sketch the idea allowing one to modify the observable η t in order to recover all the previous results in the case of inter- action potentials with singularities, V _k ^s 6= 0.

2. Preliminary estimates. For U ⊂ R ^d , C ₀ ^∞ (U ) is the set of smooth

functions with compact support in U . We write a t = O(f (t)) if there is a

constant C > 0 such that ka _t k ≤ Cf (t), where k · k is the norm of L ² (R ^d )

or the norm of bounded operators B(L ² (R ^d )). The analogous notation will

be used when a t = (a ¹ _t , . . . , a ^d _t ) assuming ka t k = (ka ¹ _t k ² + . . . + ka ^d _t k ² ) ^1/2 .

Moreover, a _t = b _t +O(f (t)) means a _t −b _t = O(f (t)). For Z ⊂ R, 1 _Z denotes

the characteristic function of Z on R.

Assume that V 0 is a real function satisfying

(2.1) |∂ _t ⁿ ∂ _x ^α V 0 (t, x)| ≤ Ct ^{−2µ−2|α|−n} for |α| + n ≤ 1, and denote by U 0 (t, t 0 ) the evolution propagator of the Hamiltonian (2.2) H 0 (t) = H 0 + V 0 (t, x),

where H 0 is given by (1.10). By rescaling we may assume further on that m ₀ = 1.

Let y t = (y ¹ _t , . . . , y _t ^d ), w t = (w ¹ _t , . . . , w ^d _t ) be systems of d commuting self-adjoint operators,

(2.3) y _t = 2x

t ² − z ₀ , w _t = p t − z ₀ .

Lemma 2.1. Let U 0 (t, t 0 ), y t , w t be as above and ϕ ∈ C ₀ ^∞ (R ^d ). Then (2.4) w t U 0 (t, t 0 )ϕ = O(t ⁻¹ ), y t U 0 (t, t 0 )ϕ = O(t ⁻¹ )

and for every κ > 0 and j = 1, . . . , d one has 1 _[κ;∞[ (|y ^j _t |)U ₀ (t, t ₀ )ϕ = O(t ⁻¹ ).

P r o o f. Define U 0 (t, t 0 ) = U _t ⁰ , f (t) = U _t ^0∗ pU _t ⁰ ϕ and g(t) = U _t ^0∗ xU _t ⁰ ϕ.

Then

f ^′ (t) = U _t ^0∗ [iH 0 (t), p]U _t ⁰ ϕ = z 0 ϕ + O(t ^−2(1+µ) ), g ^′ (t) = U _t ^0∗ [iH 0 (t), x]U _t ⁰ ϕ = f (t)

= f (t 0 ) +

t

\

t 0

f ^′ (τ ) dτ = tz 0 ϕ + O(1), hence w t U _t ⁰ ϕ = t ⁻¹ U _t ⁰ (f (t) − z 0 tϕ) = O(t ⁻¹ ). Moreover,

g(t) = g(t 0 ) +

t

\

t 0

g ^′ (τ ) dτ = ¹ ₂ z 0 t ² ϕ + O(t), and x− ¹ ₂ z 0 t ²

U _t ⁰ ϕ = U _t ⁰ g(t)− ¹ ₂ z 0 t ² ϕ

= O(t) implies the second estimate (2.4). Finally, using κ ² 1 _[κ;∞[ (|λ|) ≤ λ ² and the second estimate (2.4) we obtain

(κ ² 1 _[κ;∞[ (|y ^j _t |)U _t ⁰ ϕ, U _t ⁰ ϕ) ≤ ((y _t ^j ) ² U _t ⁰ ϕ, U _t ⁰ ϕ) = ky _t ^j U _t ⁰ ϕk ² = O(t ⁻² ).

Note that (1.9) implies the existence of

(2.5) lim

t→∞ χ e k (t) = ω k with χ e k (t) = ω k + O(t ^{−2µ 0} ), hence

(2.5 ^′ ) χ ^′ _k (t) = z k t + υ k + O(t ^{−1−2µ 0} ), χ k (t) = ¹ ₂ z k t ² + υ k t + O(1).

By rotation of the coordinate system we may assume further on that E =

(E 1 , 0, . . . , 0) with E 1 ∈ R \ {0}, hence z k = (z ¹ _k , 0, . . . , 0) with z _k ¹ =

E 1 q k /m k . Further, we set

(2.6) τ = ₁₆ ¹ min{|z _k ¹ − z _k ^{1 ′} | : 0 ≤ k < k ^′ ≤ N }.

Fix J ⁰ ∈ C ₀ ^∞ (] − 4τ ; 4τ [) such that 0 ≤ J ⁰ ≤ 1, J ⁰ = 1 on [−2τ ; 2τ ], define J ⁰ = 1 − J ⁰ and let

V _0k (t, x) = J ⁰ (4x ₁ /t ² − 2z ¹ _k )V _k ^l (x − χ _k (t)) (2.7)

= J ⁰ (2y ¹ _t − 2e z k )V _k ^l (x − χ k (t)) where we have set e z k = z ¹ _k − z ₀ ¹ . Then we have

Proposition 2.2. Let V 0 = P

1≤k≤N V 0k , where V 0k is given by (2.7).

Then (2.1) holds and for every ϕ ∈ L ² (R ^d ) the following limits exist:

(2.8)

Ω(t e 0 ) ^∗ ϕ = lim

t→∞ e ^{itH 0 +iS(t)} U 0 (t, t 0 )ϕ, Ω(t e 0 )ϕ = lim

t→∞ U 0 (t, t 0 ) ^∗ e ^{−itH 0 −iS(t)} ϕ.

P r o o f. Since χ k (t) = ¹ ₂ z k t ² + O(t) there is T 0 such that for t ≥ T 0 we have

J ⁰ (4x 1 /t ² − 2z _k ¹ ) 6= 0 ⇒ |4x 1 /t ² − 2z ¹ _k | ≥ 2τ

⇒ |x − χ _k (t)| ≥ 

x 1 − ¹ ₂ z _k ¹ t ² 

 −   ¹ ₂ z _k t ² − χ _k (t)  

≥ ¹ ₂ τ t ² − C ^′ t ≥ ¹ ₄ τ t ² and applying (1.11) we find

(2.9) |x − χ _k (t)| ≥ ¹ ₄ τ t ² ⇒ |(∂ ^α V _k ^l )(x − χ _k (t))| ≤ Ct ^{−2(µ+|α|)} if |α| ≤ 1.

We conclude that V 0 satisfies (2.1) noting that

∂

∂x 1

(J ⁰ (4x 1 /t ² − 2z _k ¹ )) = O(t ⁻² ), ∂

∂t (J ⁰ (4x 1 /t ² − 2z _k ¹ )) = O(t ⁻¹ ).

Since C ₀ ^∞ (R ^d ) is dense in L ² (R ^d ), to obtain the existence of e Ω(t 0 ) ^∗ ϕ it suffices to consider ϕ ∈ C ₀ ^∞ (R ^d ) and to check that

(2.10) d

dt (e ^{itH 0 +iS(t)} U 0 (t, t 0 )ϕ)

= e ^{itH 0 +iS(t)} i(S ^′ (t) − V 0 (t, x))U 0 (t, t 0 )ϕ = O(t ^−1−2µ ).

However, for 1 ≤ k ≤ N we have |z _k ¹ − z ₀ ¹ | ≥ 16τ , hence J ⁰ (2z ₀ ¹ − 2z ¹ _k ) = 1 and

(2.11) V ₀ t, ¹ ₂ z ₀ t ²

= X

1≤k≤N

J ⁰ (2z ₀ ¹ − 2z _k ¹ )V _k ^{l 1} ₂ z ₀ t ² − χ _k (t)

= S ^′ (t).

Thus we may write

V 0 (t, x) − S ^′ (t) = V 0 (t, x) − V 0 t, ¹ ₂ z 0 t ²

= γ t · x − ¹ ₂ z 0 t ²

= ¹ ₂ γ t · t ² y t

with

γ t =

1

\

0

dθ ∇ x V 0 t, (1 − θ)x + ¹ ₂ θz 0 t ²
and (2.1) implies t ² γ t = O(t ^−2µ ). Therefore

k(S ^′ (t) − V 0 (t, x))U 0 (t, t 0 )ϕk = ¹ ₂ t ² γ t · y t U 0 (t, t 0 )ϕ (2.12)

≤ Ct ^−2µ ky _t U ₀ (t, t ₀ )ϕk

and by (2.4) the right hand side of (2.12) is O(t ^−1−2µ ), i.e. (2.10) follows.

We may use V 0 (t, x) = 0 in Lemma 2.1, hence it is clear that e ^{−itH 0} satisfies the same estimates as U 0 (t, t 0 ), and we obtain the existence of the second limit (2.8) as above with e ^{−itH 0} and U 0 (t, t 0 ) interchanged.

Proof of the existence of Ω(t 0 ). Using the chain rule and the existence of (2.8), we note that it suffices to prove the existence of lim t→∞ U (t, t 0 ) ^∗

× U 0 (t, t 0 )ϕ, where as before we may assume ϕ ∈ C ₀ ^∞ (R ^d ). Let J ∈ C ₀ ^∞ (R ^d ) be such that J(x) = 1 for |x| ≤ τ , J(x) = 0 for |x| ≥ 2τ , 0 ≤ J ≤ 1. Then Lemma 2.1 implies

k(1 − J)(y t )U 0 (t, t 0 )ϕk ≤ k1 _{[τ ;∞[} (|y t |)U 0 (t, t 0 )ϕk = O(t ⁻¹ ), i.e.

t→∞ lim U (t, t 0 ) ^∗ J(y t )U 0 (t, t 0 )ϕ = lim

t→∞ U (t, t 0 ) ^∗ U 0 (t, t 0 )ϕ and it suffices to show that

(2.13) d

dt (U (t, t 0 ) ^∗ J(y t )U 0 (t, t 0 )ϕ)

= U (t, t ₀ ) ^∗ (D _H 0 J(y _t ) + iJ(y _t )(V (t, x) − V ₀ (t, x)))U ₀ (t, t ₀ )ϕ

= O(t ^−1−2µ ) + O(t ⁻² ),

where D _a t b _t = [ia _t , b _t ] + _dt ^d b _t denotes the Heisenberg derivative.

However, a simple calculation gives

(2.14) D _H

0 J(y t ) = 2 t

X

1≤j≤d

∂ j J(y t )(w ^j _t − y _t ^j ) + O(t ⁻³ ) and using (2.4) we obtain (D _H 0 J(y _t ))U ₀ (t, t ₀ )ϕ = O(t ⁻² ).

Next for 1 ≤ k ≤ N we have J ⁰ (2y ¹ _t − 2e z k ) 6= 0 ⇒ |y _t ¹ − e z k | < 2τ

⇒ |y _t ¹ | ≥ |e z k | − 2τ = |z _k ¹ − z ₀ ¹ | − 2τ ≥ 14τ ⇒ J(y t ) = 0, hence J(y t )J ⁰ (2y _t ¹ − 2e z k ) = J(y t ) and

J(y t )(V − V 0 )(t, x) = X

1≤k≤N

J(y t )J ⁰ (2y _t ¹ − 2e z k )V _k ^s (x − χ k (t)).

If T 0 is as at the beginning of the proof of Proposition 2.2, then for t ≥ T 0

we have

J ⁰ (2y _t ¹ − 2e z k ) 6= 0 ⇒ |x − χ k (t)| ≥ ¹ ₄ τ t ² ⇒ |V _k ^s (x − χ k (t))| ≤ Ct ^−1−2µ . Until the end of Section 4 we assume that V _k ^s = 0, that is, V k = V _k ^l . We now introduce

η ⁰ _t = 1 2

 p ₁ t − 2x ₁

t ²

 2

+ 1 4

 2x ₁ t ² − z ₀ ¹

 2

+ 1 2

X

2≤j≤d

p ² _j t ² + I, (2.15)

η t = η ⁰ _t + V (t, x) t ² . (2.16)

Lemma 2.3. If η _t ⁰ , η t are given by (2.15)–(2.16) and D is defined as below (2.13), then D H(t) η t = D H 0 η ⁰ _t + r t with

(2.17) r t = d dt

 V (t, x) t ²



−



iV (t, x), x 1 p 1 + p 1 x 1

t ³

 . P r o o f. A simple transformation of the expression (2.15) gives

η _t ⁰ = 1 2

 p ² ₁

t ² − 2 x 1 p 1 + p 1 x 1

t ³ + 4x ² ₁ t ⁴



+ 1 4

 4 x ² ₁

t ⁴ − 4 z ¹ ₀ x 1

t ² + (z ₀ ¹ ) ²

 + 1

2 X

2≤j≤d

p ² _j t ² + I

= 1 2

p ² ₁

t ² − x 1 p 1 + p 1 x 1

t ³ +

 1

2 · 4 + 1 4 · 4

 x ² ₁ t ⁴

− z ₀ ¹ x ₁

t ² + (z ₀ ¹ ) ² 4 + 1

2 X

2≤j≤d

p ² _j t ² + I

= 1 t ²

 1

2 p ² − z ₀ ¹ x 1



− x 1 p 1 + p 1 x 1

t ³ + 3x ² ₁

t ⁴ + (z ₀ ¹ ) ² 4 + I.

Therefore we may express η ⁰ _t in the following way:

(2.15 ^′ ) η ⁰ _t = H ₀

t ² − x ₁ p ₁ + p ₁ x ₁ t ³ + 3x ² ₁

t ⁴ + (z ₀ ¹ ) ² 4 + I and compute

D _H(t) η t = D H(t)



η _t ⁰ + V (t, x) t ²



= D H(t)

 H(t)

t ² − x 1 p 1 + p 1 x 1

t ³ + 3x ² ₁ t ⁴



= D H(t)

 H(t) t ²



−



iV (t, x), x ₁ p ₁ + p ₁ x ₁ t ³



+ D H 0



− x 1 p 1 + p 1 x 1

t ³ + 3x ² ₁ t ⁴



= d dt

 H ₀

t ² + V (t, x) t ²



−



iV (t, x), x ₁ p ₁ + p ₁ x ₁ t ³



+ D H 0



− x 1 p 1 + p 1 x 1

t ³ + 3x ² ₁ t ⁴



= D H 0

 H 0

t ²

 + D H 0



− x 1 p 1 + p 1 x 1

t ³ + 3x ² ₁ t ⁴



+ r t = D H 0 η _t ⁰ + r t . Lemma 2.4. If r _t is given by (2.17) then r _t = O(t ⁻² ).

P r o o f. First note that d

dt (t ⁻² V (t, x)) = t ⁻² ∂ t V (t, x) − 2t ⁻³ V (t, x) = t ⁻² ∂ t V (t, x) + O(t ⁻³ ).

Thus setting χ ^′ _k (t) = ( ˙ χ ¹ _k (t), ˙ χ ^⊥ _k (t)) ∈ R × R ^d−1 and using ˙ χ ^⊥ _k (t) = O(1), we have

t ² r _t = ∂ _t V (t, x) −



iV (t, x), x ₁ p ₁ + p ₁ x ₁ t



+ O(t ⁻¹ )

= X

1≤k≤N

∂ x 1 V k (x − χ k (t))

 2x 1

t − ˙χ ¹ _k (t)



+ O(1).

But 2x 1 /t − ˙ χ ¹ _k (t) = (2/t)(x 1 − χ ¹ _k (t)) + O(1) by (2.5 ^′ ) and we complete the proof noting that ∂ x 1 V k (x − χ k (t))(x 1 − χ ¹ _k (t)) = O(1).

Proposition 2.5. If η t is given by (2.16) and D as below (2.13), then (2.18) D _H(t) η t = − 3

t

 p ₁ t − 2x ₁

t ²

 2

− X

2≤j≤d

p ² _j

t ³ + O(t ⁻² ).

P r o o f. By Lemmas 2.3 and 2.4 it suffices to check that

(2.19) D _H

0 η _t ⁰ = − 3 t

 p 1

t − 2x 1

t ²

 2

− X

2≤j≤d

p ² _j t ³ . Now we note that formally

(2.20) D _a

t (b _t eb t ) = (D _a t b _t )eb _t + (b _t D _a

t eb t ).

If a t and b t are self-adjoint, then (2.20 ^′ ) D _a

t (b t ) ² = b t (D a t b t ) + (D a t b t )b t = 2b t (D a t b t ) + hc,

where m t + hc = ¹ ₂ (m t + m ^∗ _t ) denotes the Hermitian symmetrization of the operator m t . In particular, using

(2.21) D _H

0 w _t = − w t

t , D _H

0 y ¹ _t = 2

t (w ¹ _t − y _t ¹ )

[where w t , y t are given by (2.3)], we obtain 1

4 D _H

0 (y _t ¹ ) ² = 1 2 y ¹ _t D _H

0 y _t ¹ + hc = 1

t y ¹ _t (w ¹ _t − y ¹ _t ) + hc, 1

2 D _H

0 (w _t ¹ − y _t ¹ ) ² = (w ¹ _t − y ¹ _t )D H 0 (w _t ¹ − y _t ¹ ) + hc

= 1

t (w ¹ _t − y ¹ _t )(2y _t ¹ − 3w ¹ _t ) + hc.

Introducing w _t ^⊥ = (w _t ² , . . . , w ^d _t ) = (p 2 /t, . . . , p d /t) we may express (2.15) in the form

η _t ⁰ = ¹ ₂ (w _t ¹ − y _t ¹ ) ² + ¹ ₄ (y ¹ _t ) ² + ¹ ₂ |w ^⊥ _t | ² + I and it is clear that ¹ ₂ D _H

0 |w ^⊥ _t | ² = − ¹ _t |w ^⊥ _t | ² . To complete the proof we com- pute

1 2 D _H

0 (w ¹ _t − y ¹ _t ) ² + 1 4 D _H

0 (y _t ¹ ) ²

= 1

t (w ¹ _t − y _t ¹ )(2y ¹ _t − 3w ¹ _t ) + 1

t (w ¹ _t − y ¹ _t )y _t ¹ + hc

= 1

t (w ¹ _t − y _t ¹ )(3y ¹ _t − 3w ¹ _t ) + hc = − 3

t (w ¹ _t − y _t ¹ ) ² . 3. Propagation estimates. We denote by G(H) the set of operator- valued functions t 7→ M (t) ∈ B(L ² (R ^d )) satisfying

(3.1)

T

1

dt Re(M (t)U (t, t 0 )ϕ, U (t, t 0 )ϕ) ≤ Ckϕk ² for all ϕ ∈ L ² (R ^d ), all T ≥ 1 and for some constant C > 0.

Sometimes we write M (t) ∈ G(H(t)) instead of M ∈ G(H). We note that if M (t) = O(t ^−1−ε ) with ε > 0, then M ∈ G(H),

(3.2)

if ( f M ∈ G(H) and M (t) ≤ f M (t) for all t ≥ 1), then M ∈ G(H).

(3.3)

If D H(t) M (t) is well defined, then writing U (t, t 0 )ϕ = ϕ t we have

(3.4)

T

\

1

dt ((D _H(t) M (t))ϕ t , ϕ t ) =

T

\

1

dt d

dt (M (t)ϕ t , ϕ t ) = [(M (t)ϕ t , ϕ t )] ^T ₁ and if M (t) = O(1), then D _H(t) M (t) ∈ G(H(t)).

Note that η _t ⁰ ≥ I and η t = η ⁰ _t + O(t ⁻² ), hence for n ≥ 1, t ≥ T 0 , e η n,t = (1 + η t /n) ⁻¹ is well defined and satisfies 0 ≤ e η n,t ≤ I. Introducing

(3.5) M ₀ (t) = 1

t η e _n,t (3(w ¹ _t − y ¹ _t ) ² + |w _t ^⊥ | ² )e η _n,t ,

we find that Proposition 2.5 gives

(3.6) nD _H(t) e η n,t = −e η n,t (D _H(t) η t )e η n,t = M 0 (t) + O(t ⁻² ).

It is clear that (3.4), (3.2) and (3.6) give

Corollary 3.1. If M 0 is given by (3.5), then M 0 ∈ G(H).

Proposition 3.2. For every ϕ ∈ L ² (R ^d ) we have

n→∞ lim sup

t≥T 0

k(I − e η ² _n,t )U (t, t 0 )ϕk = 0.

P r o o f. First we set U (t, t ₀ )ϕ = ϕ _t and note that 0 ≤ λ ≤ 1 ⇒ (1 − λ ² ) ² ≤ 4(1 − λ), hence

k(I − e η _n,t ² )ϕ t k ² = ((I − e η ² _n,t ) ² ϕ t , ϕ t ) ≤ 4((I − e η n,t )ϕ t , ϕ t ).

It remains to note that e η _n,T 0 ϕ _T 0 → ϕ _T 0 as n → ∞, and −nD _H(t) η e _n,t ≤

−M 0 (t) + Ct ⁻² ≤ Ct ⁻² allows us to estimate [((I − e η n,t )ϕ t , ϕ t )] ^T _{T 0} = −

T

\

T 0

dt ((D _H(t) η e n,t )ϕ t , ϕ t ) ≤

T

\

T 0

dt Ct ⁻² /n ≤ C/n.

Further on in this section we assume n ≥ 1 fixed and write simply e

η t = e η n,t . As below (2.20 ^′ ), M (t)+hc denotes the symmetrization ¹ ₂ (M (t)+

M (t) ^∗ ).

Lemma 3.3. Let J 0 ∈ C ₀ ^∞ (R). Then M 1 ∈ G(H) if

(3.7) M 1 (t) = 1

t η e t (y ¹ _t − w _t ¹ )J 0 (y _t ¹ )e η t + hc.

P r o o f. Let J ∈ C ^∞ (R) be such that the derivative J ^′ = −J 0 , and set M 1,0 (t) = e η t J(y ¹ _t )e η t .

Then D _H(t) M 1,0 = M 1,1 + M 1,2 with

M 1,1 (t) = e η t (D H(t) J(y ¹ _t ))e η t = 2M 1 (t) + O(t ⁻³ ), M 1,2 (t) = 2e η t J(y _t ¹ )D _H(t) η e t + hc.

From (3.4) we have D H(t) M 1,0 ∈ G(H) and it is clear that in order to show M ₁ ∈ G(H) it suffices to check that −M _1,2 ∈ G(H).

Noting that

w ^⊥ _t η e t = O(1), y _t ¹ η e t = O(1), (w ¹ _t − y ¹ _t )e η t = O(1), it is easy to estimate the commutators

n[e η _t , w ^⊥ _t ] = −e η _t [η ⁰ _t + O(t ⁻² ), w ^⊥ _t ]e η _t = O(t ⁻² ), n[e η t , w _t ¹ − y _t ¹ ] = e η t [η _t ⁰ + O(t ⁻² ), y ¹ _t − w _t ¹ ]e η t

= e η t  ₁

4 (y _t ¹ ) ² , y _t ¹ − w ¹ _t  e

η t + O(t ⁻² ) = O(t ⁻² ),

n[e η t , J(y ¹ _t )] = −e η t [η ⁰ _t , J(y ¹ _t )]e η t = O(t ⁻² ).

Using (2.18) to express D H(t) η e t in M 1,2 (t) it is easy to see that the above commutator estimates allow us to write

−M 1,2 (t) = 2

t η e t (3(w ¹ _t − y ¹ _t )a t (w _t ¹ − y _t ¹ ) + w _t ^⊥ a t w ^⊥ _t )e η t + O(t ⁻² ) with a t = −n ⁻¹ J(y _t ¹ )e η t + hc, and it is clear that the inequality a t ≤ CI implies

(3.8) −M _1,2 (t) ≤ 2CM ₀ (t) + Ct ⁻²

where M 0 is given by (3.5). By Lemma 3.3 the right hand side of (3.8) belongs to G(H) and consequently −M 1,2 ∈ G(H).

Proposition 3.4. Let J ₀ ∈ C ₀ ^∞ (R \ {e z ₁ , . . . , e z _N }) where e z _k = z _k ¹ − z ₀ ¹ . Then M 2 ∈ G(H) if

(3.9) M ₂ (t) = 1

t η e _t J ₀ (y ¹ _t )y ¹ _t η e _t .

P r o o f. If M 1 is given by (3.7), then M 1 ∈ G(H) and M 2 = 3M 1 + M 3

with

M 3 (t) = 1

t η e t (3w _t ¹ − 2y _t ¹ )J 0 (y ¹ _t )e η t + hc.

Thus it remains to show that M 3 ∈ G(H). But for 1 ≤ k ≤ N , e z k 6∈ supp J 0

and

J ₀ (y _t ¹ ) 6= 0 ⇒ |y ¹ _t − e z _k | = |2x ₁ /t ² − z _k ¹ | ≥ c > 0

⇒ |x − χ k (t)| ≥ 

x 1 − ¹ ₂ z _k ¹ t ² 

 − C ^′ t ≥ ¹ ₂ ct ² − C ^′ t implies

[iV (t, x), w ¹ _t ]J 0 (y t ) = −∂ x V (t, x)J 0 (y t )t ⁻¹ = O(t ⁻³ ).

Therefore introducing

M 3,0 (t) = e η t (y _t ¹ − w ¹ _t )J 0 (y t )e η t + hc, we find that D _H(t) M 3,0 = M 3,1 + M 3,2 + M 3,3 with

M 3,1 (t) = e η t (D H(t) (y _t ¹ − w _t ¹ ))J 0 (y _t ¹ )e η t = M 3 (t) + O(t ⁻³ ), M 3,2 (t) = e η t (y ¹ _t − w _t ¹ )(D _H(t) J 0 (y _t ¹ ))e η t + hc,

M 3,3 (t) = 2e η t (y ¹ _t − w ¹ _t )J 0 (y ¹ _t )D H(t) η e t + hc.

As before, (3.4) gives D _H(t) M _3,0 ∈ G(H) and M ₃ ∈ G(H) follows if we know that −M 3,2 , −M 3,3 ∈ G(H). To show −M 3,3 ∈ G(H) we note that we may replace M 1,2 by M 3,3 in (3.8) using a t = n ⁻¹ J 0 (y ¹ _t )(w ¹ _t − y ¹ _t )e η t + hc ≤ CI to express −M _3,3 similarly to −M _1,2 . Also

−M _3,2 (t) = − 2

t η e _t (y _t ¹ − w ¹ _t )J ₀ ^′ (y ¹ _t )(y _t ¹ − w ¹ _t )e η _t + O(t ⁻³ )

≤ CM 0 (t) + Ct ⁻³ ∈ G(H(t)).

We keep the notations J ⁰ , e z k , V 0k , V 0 , H 0 (t), U 0 (t, t 0 ) introduced in Sec- tion 2. Moreover, for 1 ≤ k ≤ N we denote by U k (t, t 0 ) the evolution propagator of the Hamiltonian

(3.10)

H _k (t) = H ₀ + V ^k (t, x) with V ^k (t, x) = V k (x − χ k (t)) + X

k ^′ ∈{1,...,N }\{k}

V 0k ^′ (t, x).

Corollary 3.5. If M 0 , M 2 , H k are as above, then M 0 , M 2 ∈ G(H k ).

P r o o f. Define η _t ^k by using V ^k (t, x) instead of V (t, x) in (2.16). As before we obtain

M ₀ ^k (t) = 1

t η e _t ^k (3(w _t ¹ − y _t ¹ ) ² + |w ^⊥ _t | ² )e η ^k _t ∈ G(H k (t))

with e η _t ^k = (1 + η ^k _t /n) ⁻¹ . We recall that |∂ _t ⁿ ∂ _x ^α V 0k ^′ (t, x)| ≤ Ct ^{−2µ−2|α|−n} for

|α| + n ≤ 1, and reasoning as in the proof of Proposition 3.4 we find M ₂ ^k (t) = 1

t e η ^k _t J 0 (y _t ¹ )y ¹ _t η e ^k _t ∈ G(H k (t))

for J 0 ∈ C ₀ ^∞ (R \ {e z 1 , . . . , e z N }). However, η t = η _t ^k + O(t ⁻² ) implies

((w _t ¹ −y _t ¹ ) ² +|w ^⊥ _t | ² )(e η t − e η ^k _t ) = ((w ¹ _t −y ¹ _t ) ² +|w _t ^⊥ | ² )e η _t ^k (η t −η ^k _t )e η t /n = O(t ⁻² ), hence

M j (t) = M _j ^k (t) + O(t ⁻² ) ∈ G(H k (t)), j = 0, 2.

The following well known lemma is the basic tool allowing us to obtain the existence of wave operators (we give its proof in the Appendix):

Lemma 3.6. Let U (t, t ₀ ) and e U (t, t ₀ ) be the evolution propagators of H(t) = H 0 + V (t) and e H(t) = H 0 + e V (t) respectively. Assume that for M (t) ∈ B(L ² (R ^d )) we may define D H 0 M (t) as bounded operators with (3.11) ( e V (t) − V (t))M (t) = O(t ^−1−ε ) and

D _H

0 M (t) = f M (t) + O(t ^−1−ε )

where ε > 0, and that there exists f M 0 ∈ G(H)∩G( e H) satisfying the estimates (3.11 ^′ ) − f M 0 (t) ≤ f M (t) ≤ f M 0 (t) and M f 0 (t) ≥ 0 for all t ≥ 1.

If ϕ ∈ L ² (R ^d ) and Ω _t = e U (t, t ₀ ) ^∗ M (t)U (t, t ₀ ), then the limit lim _t→∞ Ω _t ϕ exists.

Proposition 3.7. Set J(y _t ¹ ) = 1 − P

1≤k≤N J ⁰ (y ¹ _t − e z k ) ² and define (3.12) Ω ₀ (t, t ₀ ) = U ₀ (t, t ₀ ) ^∗ J(y _t ¹ )U (t, t ₀ ),

Ω k (t, t 0 ) = U k (t, t 0 ) ^∗ J ⁰ (y _t ¹ − e z k )U (t, t 0 ) for k = 1, . . . , N.

Then for every ϕ ∈ L ² (R ^d ), k = 0, 1, . . . , N , the following limits exist:

(3.12 ^′ ) Ω k (t 0 )ϕ = lim

t→∞ Ω k (t, t 0 )ϕ.

P r o o f. Consider first the case k = 0. By Proposition 3.2 it suffices to show that

t→∞ lim U 0 (t, t 0 ) ^∗ J(y _t ¹ )e η ² _n,t U (t, t 0 )ϕ

exists for every n ≥ 1. Further on n is fixed, we write e η _t = e η _n,t and we apply Lemma 3.6 with e H(t) = H 0 (t) and M (t) = J(y ¹ _t )e η ² _t .

We begin by noting that the first condition of (3.11) follows from (3.13) (H(t) − H 0 (t))J(y ¹ _t ) = X

1≤k≤N

J ⁰ (2y _t ¹ − 2e z k )V _k ^l (x − χ k (t))J(y _t ¹ ) = 0.

To check (3.13) we note that J ⁰ (y _t ¹ − e z _k ) 6= 0 ⇒ |y _t ¹ − e z _k | < 4τ and for k ^′ 6= k we have |e z k −e z k ^′ | = |z _k ¹ −z ¹ _k ^′ | ≥ 16τ , hence J ⁰ (y _t ¹ −e z k ) 6= 0 ⇒ J ⁰ (y ¹ _t −e z k ^′ ) = 0 for k ^′ 6= k. Thus it is clear that J ⁰ (2y ¹ _t − 2e z k ) 6= 0 ⇒ |y ¹ _t − e z k | < 2τ ⇒ J ⁰ (y _t ¹ − e z k ) = 1 ⇒ J(y ¹ _t ) = 1 − J ⁰ (y ¹ _t − e z k ) ² = 0.

Next we find that D H 0 M = f M 1 + f M 2 with M f ₁ (t) = (D _H 0 J(y ¹ _t ))e η _t ² = 2

t η e _t (w ¹ _t − y _t ¹ )J ^′ (y _t ¹ )e η _t + hc + O(t ⁻² ), (3.14)

M f 2 (t) = 2e η t J(y ¹ _t )D H(t) η e t + hc + O(t ⁻² ).

(3.15)

Next for k = 1, . . . , N , we have |y _t ¹ | ≤ 2τ ⇒ |y ¹ _t − e z k | ≥ 14τ ⇒ J ⁰ (2y _t ¹ − 2e z _k ) = 0. Therefore J = 1 on [−2τ ; 2τ ] and 0 6∈ supp J ^′ allows us to define J 0 ∈ C ₀ ^∞ (R \ {e z 1 , . . . , e z N }) satisfying J 0 (λ)λ = J ^′ (λ) ² and to estimate (3.16) ± (w _t ¹ − y _t ¹ )J ^′ (y _t ¹ ) + hc ≤ 2(w ¹ _t − y ¹ _t ) ² + 2J 0 (y _t ¹ )y ¹ _t

⇒ ± f M 1 ≤ 4M 0 + 4M 2

with M 0 , M 2 given by (3.5), (3.9). Then similarly to the proof of Lemma 3.3 we find ± f M 2 (t) ≤ CM 0 (t) + Ct ⁻² , hence it is clear that the hypotheses of Lemma 3.6 hold with f M 0 = C 0 M 0 + 4M 2 ∈ G(H) ∩ G(H k ) by Corollary 3.1, 3.5 and Proposition 3.4.

In the case k = 1, . . . , N , we apply Lemma 3.6 with e H(t) = H k (t) and M (t) = e J(y _t ¹ )e η _t ² , where e J(λ) = J ⁰ (λ − e z k ). As before we have

(3.17) (H(t) − H k (t)) e J (y _t ¹ )

= X

k ^′ ∈{1,...,N }\{k}

J ⁰ (2y _t ¹ − 2e z k ^′ )V _k ^{l ′} (x − χ k ^′ (t)) e J (y ¹ _t ) = 0.

Indeed, e J(y _t ¹ ) 6= 0 ⇒ |y _t ¹ − e z k | < 4τ ⇒ |y ¹ _t − e z k ^′ | ≥ 2τ for k ^′ 6= k ⇒

J ⁰ (2y _t ¹ − 2e z k ^′ ) = 0 for k ^′ 6= k. We complete the proof noting that e J = 0 on

[−2τ ; 2τ ] and (3.14)–(3.16) still hold if J is replaced by e J.

4. Asymptotic completeness. In order to obtain the asymptotic completeness it remains to prove

Proposition 4.1. If k = 1, . . . , N and ϕ ∈ L ² (R ^d ), then

t→∞ lim J ⁰ (y _t ¹ − e z _k )U _k (t, t ₀ )ϕ = 0.

Indeed, using Propositions 2.2, 3.7 and 4.1, we can see that via the chain rule,

e ^{itH 0 +iS(t)} U (t, t 0 )ϕ = e ^{itH 0 +iS(t)} 

J(y ¹ _t ) + X

1≤k≤N

J ⁰ (y ¹ _t − e z k ) ² 

U (t, t 0 )ϕ

= e ^{itH 0 +iS(t)} U 0 (t, t 0 )Ω 0 (t, t 0 )ϕ

+ X

1≤k≤N

e ^{itH 0 +iS(t)} J ⁰ (y ¹ _t − e z k )U k (t, t 0 )Ω k (t, t 0 )ϕ converges to e Ω 0 (t 0 ) ^∗ Ω 0 (t 0 )ϕ, i.e. the limit (1.12 ^′ ) exists.

Before starting the proof of Proposition 4.1 we introduce more notation.

We set

(4.1) H 0k = ¹ ₂ p ² + e z k x 1 , H e k = H 0k + V k (x − ω k ), where k = 1, . . . , N and ω k is as in (2.5). We define

χ ⁰ _k (t) = ¹ ₂ z k t ² + υ k t, χ ˙ ⁰ _k (t) = z k t + υ k , (4.2)

H e k (t) = H 0k + e V k (t, x) (4.3)

with

V e k (t, x) = V ^k (t, x + χ ⁰ _k (t))

= V _k (x − e χ _k (t)) + X

k ^′ ∈{1,...,N }\{k}

V _0k ^′ t, x + ¹ ₂ z _k t ² + υ _k t
. It is easy to see that V _0k ^′ t, x+ ¹ ₂ z _k t ² +υ _k t

satisfies estimates (2.1) similarly to V 0k ^′ . The following lemma allows us to compare e H k and e H k (t).

Lemma 4.2. (a) We have V k (x − e χ k (t)) = V k (x − ω k ) + O(t ^{−2µ 0} ) and d

dt V k (x − e χ k (t)) = −e χ ^′ _k (t) · ∇V k (x − e χ k (t)) = O(t ^{−1−2µ 0} ).

(b) If h ∈ C ₀ ^∞ (R) then h( e H k (t)) = h( e H k ) + O(t ^{−2µ 0} ) + O(t ^−2µ ) and D e ^{H k (t)} h( e H k (t)) = d

dt h( e H k (t)) = O(t ^{−1−2µ 0} ) + O(t ^−1−2µ ).

(c) If g, e h ∈ C ₀ ^∞ (R) then [e h( e H k ), g( e w t )] = O(t ⁻¹ ).

We note that our assumptions ∇V _k = ∇V _k ^l = O(1) and (1.9) give imme-

diately the indicated estimate of _dt ^d V k (x − e χ k (t)), while the first estimate of

Lemma 4.2(a) follows by integration. The proof of estimates in (b) and (c) is given in the Appendix.

Proposition 4.3. Let g ∈ C ₀ ^∞ 

− ³ ₄ |e z k |; ³ ₄ |e z k | 

and h ∈ C ₀ ^∞ (R). Then (4.4) M f h (t) = 1

t h( e H k (t))g( e w t ) ² h( e H k (t)) ∈ G( e H k (t)), where we have set e w t = p 1 /t.

P r o o f. Let n ∈ N be such that h ∈ C ₀ ^∞ (]−n; n[). Since ( f M h (t)ϕ, ϕ) = t ⁻¹ kg( e w t )h( e H k (t))ϕk ² , it is clear that f M h 1 +h 2 (t) ≤ 2 f M h 1 (t) + 2 f M h 2 (t).

Thus it suffices to show that for every λ ∈ [−n; n] there is δ > 0 such that M f h (t) ∈ G( e H k (t)) with h ∈ C ₀ ^∞ (]λ − δ; λ + δ[), |h| ≤ 1.

Let g 1 ∈ C ^∞ (R) satisfy g ₁ ^′ = −g ² and set

M 0 (t) = e z k h( e H k (t))g 1 ( e w t )h( e H k (t)).

Let ε = min{1, 2µ ₀ , 2µ}. Then Lemma 4.2 allows us to write

D e ^{H k (t)} M 0 (t) = e z k h( e H k (t))(D e ^{H k (t)} g 1 ( e w t ))h( e H k (t)) + O(t ^−1−ε )

= e z k h( e H k )(D e ^{H k} g 1 ( e w t ))h( e H k ) + O(t ^−1−ε ).

We now show that choosing δ > 0 small enough we have (4.5) e z _k h( e H _k )[iV _k (x−ω _k ), g ₁ ( e w _t )]h( e H _k ) ≥ − z e ² _k

8t h( e H _k )g( e w _t ) ² h( e H _k )−Ct ⁻² . Using (1.11b) and the standard pseudo-differential expansion [(A.1) of Ap- pendix with n = 2 and then with n = 1] we find the following expression of the commutator:

[iV k (x − ω k ), g 1 ( e w t )] = − 1

t ∂ x 1 V k (x − ω k )g ₁ ^′ ( e w t ) + O(t ⁻² ) (4.6)

= 1

t g( e w t )∂ x 1 V k (x − ω k )g( e w t ) + O(t ⁻² ),

and since e H k has no eigenvalues (cf. [20]), 1 [λ−2δ;λ+2δ] ( e H k ) → 0 strongly as δ → 0. As ∂ x 1 V k (x − ω k )1 [−n;n] ( e H k ) is compact, for δ > 0 small enough we have

(4.7) z e _k eh( e H _k )∂ _x 1 V _k (x − ω _k )e h( e H _k ) ≥ − ¹ ₈ e z _k ²

if e h ∈ C ₀ ^∞ (]λ−2δ; λ+2δ[), 0 ≤ e h ≤ 1. Using e h such that h = he h and Lemma 4.2(c) we obtain (4.5) from (4.6)–(4.7). Next we note that D H 0k g 1 ( e w t ) =

−t ⁻¹ (e z _k + e w _t )g ₁ ^′ ( e w _t ) = t ⁻¹ (e z _k + e w _t )g( e w _t ) ² and since λ ∈ supp g ⇒ |λ| ≤

3

4 |e z k | ⇒ e z k (e z k + λ) ≥ ¹ ₄ z e _k ² , it is clear that (4.8) z e _k h( e H _k )(D _H 0k g ₁ ( e w _t ))h( e H _k ) ≥ 1

4t z e _k ² h( e H _k )g( e w _t ) ² h( e H _k ).

Let M 1 , M 2 denote the left hand sides of (4.5) and (4.8). Then (4.4) follows from

1

8 e z _k ² M f h (t) ≤ (M 1 + M 2 )(t) + Ct ^−1−ε

= D e ^{H k (t)} M ₀ (t) + O(t ^−1−ε ) ∈ G( e H _k (t)).

Proof of Proposition 4.1 . Step 1. Introduce

(4.9)

G _k (t) = e ^{−iΦ k (t)} e ^{−ix· ˙ χ 0 k (t)} e ^{ip·χ 0 k (t)} where Φ k (t) =

t

\

1

dτ z 0 · χ ⁰ _k (τ ) + ¹ ₂ χ ˙ ⁰ _k (τ ) ²
.

Since e ^{−ix· ˙ χ 0 k (t)} p = (p + ˙ χ ⁰ _k (t))e ^{−ix·χ 0 k (t)} and e ^{ip·χ 0 k (t)} x = (x + χ ⁰ _k (t))e ^{ip·χ 0 k (t)} , we compute

G ^′ _k (t) = − z ₀ · χ ⁰ _k (t) − ¹ ₂ χ ˙ ⁰ _k (t) ² − x · z _k + (p + ˙ χ ⁰ _k (t)) · ˙ χ ⁰ _k (t)
iG _k (t), iG k (t)H k (t) = ¹ ₂ (p + ˙ χ ⁰ _k (t)) ² − z 0 · (x + χ ⁰ _k (t)) + V ^k (t, x + χ ⁰ _k (t))

iG k (t)

= e H k (t) + p · ˙ χ ⁰ _k (t) + ¹ ₂ χ ˙ ⁰ _k (t) ² − z k · x − z 0 · χ ⁰ _k (t)
iG k (t)

= i e H k (t)G k (t) + G ^′ _k (t).

Thus we have d

dt ( e U k (t, t 0 ) ^∗ G k (t)U k (t, t 0 )ϕ)

= e U k (t, t 0 ) ^∗ (G ^′ _k (t) + i e H k (t)G k (t) − iG k (t)H k (t))U k (t, t 0 )ϕ = 0, which implies

(4.10) U e k (t, t 0 ) = G k (t)U k (t, t 0 )G k (t 0 ) ⁻¹ . We write e y t = 2x 1 /t ² . Then

G k (t)J ⁰ (y ¹ _t − e z k ) = J ⁰ (e y t + 2υ _k ¹ /t)G k (t) = J ⁰ (e y t )G k (t) + O(t ⁻¹ ) and using (4.10) we obtain

(4.11) lim

t→∞ kJ ⁰ (y ¹ _t − e z k )U k (t, t 0 )ϕk = lim

t→∞ kJ ⁰ (e y t ) e U k (t, t 0 )G k (t 0 )ϕk.

Step 2 . It suffices to show that for every h ∈ C ₀ ^∞ (R) we have

(4.12) lim inf

t→∞ kJ ⁰ (e y t )h( e H k (t)) e ϕ t k = 0 where we have set e ϕ t = e U k (t, t 0 )G k (t 0 )ϕ.

Indeed, note first that (4.11) is the limit of the norms of ϕ(t) = U k (t, t 0 ) ^∗

× J ⁰ (y _t ¹ − e z k )U k (t, t 0 )ϕ and that ϕ(t) converges in L ² (R ^d ), by a reasoning

analogous to the proof of Proposition 3.7. Thus the limits (4.11) exist and

we may replace them by lim inf.

However, taking h 0 ∈ C ₀ ^∞ (R) such that h 0 = 1 in a neighbourhood of 0, 0 ≤ h 0 ≤ 1, we have h 0 ( e H k (T 0 )/n)ψ → ψ as n → ∞ and by Lemma 4.2(b),

[((I − h 0 ( e H k (t)/n)) ² ϕ e t , e ϕ t )] ^T _{T 0} =

T

\

T 0

dt (D e ^{H k (t)} (I − h 0 ( e H k (t)/n)) ² ϕ e t , e ϕ t )

≤

T

\

T 0

dt C ϕ t −1−2 min{µ,µ 0 } /n ≤ e C ϕ /n, which implies

n→∞ lim sup

t≥T 0

k(I − h 0 ( e H k (t)/n)) e ϕ t k = 0.

Step 3 . Instead of (4.12) it suffices to show that M (t) ∈ G( e H k (t)) with

(4.13) M (t) = 1

t h( e H _k (t))J ⁰ (e y _t ) ² h( e H _k (t)).

Indeed, (M (t) e ϕ _t , e ϕ _t ) ∈ L ¹ ([t ₀ ; ∞[, dt) implies 0 = lim inf

t→∞ t(M (t) e ϕ t , e ϕ t ) = lim inf

t→∞ kJ ⁰ (e y t )h( e H k (t)) e ϕ t k ² .

Step 4 . To complete the proof of Proposition 4.1 it suffices to prove Lemma 4.4. Let g ∈ C ₀ ^∞ 

− ³ ₄ |e z k |; ³ ₄ |e z k | 

be such that g = 1 on

 − ₃ ² |e z k |; ² ₃ |e z k |  . Then

(4.14) (1 − g)( e w _t )J ⁰ (e y _t )h( e H _k (t)) = O(t ⁻¹ ).

Indeed, if g, M , f M h are as before, then Lemma 4.4 and Proposition 4.3 give

M (t) = 1

t h( e H k (t))g( e w t )J ⁰ (e y t ) ² g( e w t )h( e H k (t)) + O(t ⁻² )

≤ f M h (t) + Ct ⁻² ∈ G( e H k (t)).

P r o o f (of Lemma 4.4). We set J = J ⁰ and g = 1 − g. Then (4.14) follows if we show

(4.14 ^′ ) (−i + e H k ) ⁻¹ J(e y t )g( w e t ) ² J(e y t )(i + e H k ) ⁻¹ ≤ Ct ⁻² . Writing g(λ) ² = eg(λ) λ ² − ¹ ₄ e z _k ²

eg(λ) we have eg ∈ S 1 ⁻¹ (R) ⇒ [J(e y t ), e g( e w t )]

× (1 + | e w t |) = O(t ⁻³ ) (cf. Appendix), and J(e y t ) 6= 0 ⇒ |e y t | ≤ 4τ ≤ ¹ ₄ |e z k | ⇒

−e z _k y e _t ≤ ¹ ₄ e z _k ² allows us to estimate J(e y t )eg( e w t ) e w ² _t − ¹ ₄ z e ² _k

eg( e w t )J(e y t )

= eg( e w t )J(e y t ) e w ² _t − ¹ ₄ z e ² _k

J(e y t )eg( e w t ) + O(t ⁻³ )

≤ eg( e w _t )J(e y _t )( e w ² _t + e z _k y e _t )J(e y _t )eg( e w _t ) + Ct ⁻³

≤ eg( e w t )J(e y t )2t ⁻² H 0k J(e y t )eg( e w t ) + Ct ⁻³ .

Since [e y t , eg( e w t )] and [ e w ² _t , J(e y t )]eg( e w t ) are O(t ⁻³ ), we obtain (4.14 ^′ ) noting that H 0k J(e y t )eg( e w t )(i + e H k ) ⁻¹ = J(e y t )eg( e w t )H 0k (i + e H k ) ⁻¹ + O(t ⁻¹ ) = O(1).

5. Interaction potentials with singularities. Let b C be as in (1.11b) and θ ∈ C ₀ ^∞ (R) be such that θ(x 1 ) = 1 for |x 1 | ≤ b C/|E|. Then

(5.1) kV _k (x)θ(x ₁ )ϕk ≤ ¹ ₅ kp ² θ(x ₁ )ϕk + Ckϕk ≤ ¹ ₂ kH ₀ ϕk + C ^′ kϕk and V k (x)(1 − θ)(x 1 ) is bounded. Therefore H 0 + V k (x) is well defined as a self-adjoint operator on the domain of H 0 and the operators H 0 (H 0 + V k (x) +i) ⁻¹ , (H 0 +V k (x))(H 0 +i) ⁻¹ are bounded. The analogous assertion clearly holds if V k (x) is replaced by V k (x − χ k (t)) or by V (t, x) (using constants locally bounded with respect to t).

Further on β > 0 is fixed small enough. Following [7] or [9] we may state Lemma 5.1. There exist functions u ^j _t ∈ C ₀ ^∞ (R), j = 1, . . . , d, such that for t ≥ 1 one has

u ¹ _t (λ) = ˙ χ ¹ _k (t)/t − z ¹ ₀ for λ ∈ [e z k − t ^−β ; e z k + t ^−β ], (5.2a)

u ¹ _t (λ) = λ for λ 6∈ [

1≤k≤N

[e z k − 2t ^−β ; e z k + 2t ^−β ], (5.2b)

u ^j _t (λ) = ˙ χ ^j _k (t) for λ ∈ [ ˙ χ ^j _k (t) − t ^−β ; ˙ χ ^j _k (t) + t ^−β ], j ≥ 2, (5.2c)

u ^j _t (λ) = λ for λ ∈ [−C + t ^−β ; C − t ^−β ] (5.2d)

\ [

1≤k≤N

[ ˙ χ ^j _k (t) − 2t ^−β ; ˙ χ ^j _k (t) + 2t ^−β ], j ≥ 2,

(u ^j _t ) ^′ (λ) = d

dλ u ^j _t (λ) = 0 for |λ| ≥ C, j ≥ 2, (5.2e)

d dt u ^j _t (λ)

  ≤ Ct ^−1−β , (u ^j _t ) ^′ (λ) ≥ 0,

|(u ^j _t ) ⁽ⁿ⁾ (λ)| =    

d ⁿ d ⁿ λ u ^j _t (λ)

  ≤ C ⁿ t ^(n−1)β for λ ∈ R, n ≥ 1, (5.2f)

where e χ ^′ _k (t) = ( ˙ χ ¹ _k (t), . . . , ˙ χ ^d _k (t)) and C is fixed large enough.

We write a t = O(b t ) if b t ≥ I for t ≥ T 0 and b ^−1/2 _t a t b ^−1/2 _t = O(1). Note that a t = O(b t ) holds if we have a t b ⁻¹ _t = O(1) and b ⁻¹ _t a t = O(1). Further, we denote x ⊥ = (x 2 , . . . , x d ), e y ^⊥ _t = x ⊥ /t, u ^⊥ _t (e y _t ^⊥ ) = (u ² _t (x 2 /t), . . . , u ^d _t (x d /t)), u ^⊥ _t ^′ (e y ^⊥ _t ) = ((u ² _t ) ^′ (x 2 /t), . . . , (u ^d _t ) ^′ (x d /t)),

(5.3) η ^⊥ _t = ¹ ₂ |w ^⊥ _t | ² − u ^⊥ _t (e y ^⊥ _t ) · w ^⊥ _t /t + hc + C ⊥ I with C ⊥ > 0 large enough and

(5.3 ^′ ) η ⁰ _t = ¹ ₂ (w ¹ _t − u ¹ _t (y ¹ _t )) ² + ¹ ₄ (y ¹ _t ) ² .

Proposition 5.2. Let η t = η ⁰ _t + η ^⊥ _t + V (t)/t ² . If ε > 0 is small enough, then

(w _t ) ^2θ ≤ C(η _t ⁰ + η ^⊥ _t ) ^θ for 0 ≤ θ ≤ 1, (5.4a)

η t − (η _t ⁰ + η ^⊥ _t ) = t ⁻² V (t) = O(t ^−2ε (η ⁰ _t + η ^⊥ _t )), (5.4b)

D _H(t) η _t = D _H 0 (η ⁰ _t + η ^⊥ _t ) + O(t ^−1−ε η ^1−ε _t ), (5.4c)

D _H

0 η _t ^⊥ = − 1 t

X

2≤j≤d

w _t ^j (1 + (u ^j _t ) ^′ (x _j /t))w ^j _t + O(t ^−1−ε η ^1/2 _t ), (5.4d)

D _H

0 η _t ⁰ = − 1

t (w ¹ _t − y ¹ _t )(1 + 2(u ¹ _t ) ^′ (y _t ¹ ))(w ¹ _t − y ¹ _t ) + O(t ^−1−ε η _t ^1/2 ).

(5.4e)

P r o o f. By interpolation it suffices to prove (5.4a) for θ = 1. As u ^⊥ _t (e y _t ^⊥ ) is bounded, we have |w ^⊥ _t | ² ≤ Cη _t ^⊥ . Then using u t (y ¹ _t ) ² = (y ¹ _t ) ² + O(t ^−β ) we may estimate

(w ¹ _t ) ² = (w _t ¹ − u ¹ _t (y _t ¹ )) ² + u ¹ _t (y ¹ _t ) ² + 2(w _t ¹ − u ¹ _t (y _t ¹ ))u t (y _t ¹ ) + hc (5.5)

≤ 2(w ¹ _t − u ¹ _t (y ¹ _t )) ² + 2u ¹ _t (y ¹ _t ) ² + 1 ≤ 12η ⁰ _t + Ct ^−β . Thus (5.4b) follows from (5.4a) by the estimate

t ⁻² e ^{−iχ k (t)·p} V _k (x)e ^{iχ k (t)·p} ≤ Ct ⁻² (1 + p ² ) ^1−ε ≤ C ^′ t ^−2ε (1 + |w _t | ² ) ^1−ε . Next we note that

u ¹ _t (y ¹ _t ) + z ¹ ₀ − ˙ χ ¹ _k (t)/t 6= 0 ⇒ |x 1 − χ ¹ _k (t)| ≥ ¹ ₂ t ^2−β

⇒ ∇V _k (x − χ _k (t)) = O(t ^{−µ(2−β)} ), u ^⊥ _t (e y _t ^⊥ ) − ˙ χ ^⊥ _k (t) 6= 0 ⇒ |x ⊥ − χ ^⊥ _k (t)| ≥ ¹ ₂ t ^1−β

⇒ ∇V k (x − χ k (t)) = O(t ^{−µ(1−β)} ),

hence using the fact that ˙ χ ¹ _k (t)/t, ˙ χ ^⊥ _k (t), u ¹ _t (y ¹ _t )η ^−1/2 _t , u ^⊥ _t (e y _t ^⊥ ) are O(1), we obtain

∂ _x 1 V _k (x − χ _k (t))(u ¹ _t (y ¹ _t ) + z ¹ ₀ − ˙ χ ¹ _t (t)/t) = O(t ^−ε η ^1/2 _t ), (5.6)

∂ x ⊥ V k (x − χ k (t))(u ^⊥ _t (e y _t ^⊥ ) − ˙ χ ^⊥ _k (t)) = O(t ^−ε ).

(5.6 ^′ )

Then reasoning as in the proof of Lemma 2.4 we can see that (5.6)–(5.6 ^′ ) imply (5.4c).

Finally, we obtain (5.4d, e) calculating D _H

0 u ^⊥ _t (e y ^⊥ _t ) = 1

t u ^⊥′ _t (e y _t ^⊥ )(p _⊥ − e y _t ^⊥ ) + O(t ^−2+β ),

−tD _H 0 η _t ⁰ + O(t ^−ε ) = (w ¹ _t ) ² − 2(y ¹ _t − w _t ¹ )u ^1′ _t (y _t ¹ )w _t ¹ − u ¹ _t (y _t ¹ )w _t ¹ + 2(y _t ¹ − w ¹ _t )(u ^1′ _t u ¹ _t )(y _t ¹ ) + (y _t ¹ − w ¹ _t )y ¹ _t

= (w ¹ _t − y ¹ _t ) ²

+ 2(w ¹ _t − y ¹ _t )(1 + 2u ^1′ _t )(y _t ¹ )(w _t ¹ − y _t ¹ ) + O(t ^−ε ).