Inverse scattering at high energies for the multidimensional Newton equation in a long range potential

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Inverse scattering at high energies for the

multidimensional Newton equation in a long range potential

Alexandre Jollivet

To cite this version:

Alexandre Jollivet. Inverse scattering at high energies for the multidimensional Newton equation in a long range potential. 2013. �hal-00834476�

Inverse scattering at high energies for the multidimensional Newton equation in a long

range potential

Alexandre Jollivet June 15, 2013

Abstract

We define scattering data for the Newton equation in a potential V ∈ C²(Rⁿ,R), n ≥2, that decays at infinity like r^−α for some α ∈ (0,1]. We provide estimates on the scattering solutions and scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of the potential up to the knowledge of the long range tail of the potential. The Born approximation at fixed energy of the scattering data is also considered.

We then change the definition of the scattering data to study inverse scattering in other asymptotic regimes. These results were obtained by developing the inverse scattering approach of [Novikov, 1999].

1 Introduction

Consider the multidimensional Newton equation in an external static force F deriving from a scalar potential V:

¨

x(t) =F(x(t)) =−∇V(x(t)), (1.1) where x(t)∈Rⁿ, x(t) =˙ ^dx_dt(t),n ≥2.

When n = 3 then equation (1.1) is the equation of motion of a nonrel- ativistic particle of mass m = 1 and charge e = 1 in an external and static electric (or gravitational) field described by V (see [8]) where x denotes the position of the particle, ˙x denotes its velocity and ¨xdenotes its acceleration and t denotes the time.

We also assume throughout this paper thatV satisfies the following con- ditions

F =F^l+F^s, (1.2)

where F^l := −∇V^l, F^s := −∇V^s and (V^l, V^s) ∈ (C²(Rⁿ,R))², and where V^l satisfies the following long range assumptions

|∂_x^jV^l(x)| ≤β_|^l_j|(1 +|x|)^−(α+|j|), (1.3) and V^s satisfies the following short range assumptions

|∂_x^jV^s(x)| ≤β_|^s_j|+1(1 +|x|)^{−(α+1+|j|)}, (1.4) for x ∈ Rⁿ and |j| ≤ 2 and for some α ∈ (0,1] (here j is the multiindex j ∈ (N∪ {0})ⁿ,|j| = Pn

m=1jm, and β_m^l and β_m^s′ are positive real constants for m = 0,1,2 and for m^′ = 1,2,3). Note that the assumption 0 < α ≤ 1 includes the decay rate of a Coulombian potential at infinity. Indeed for a Coulombian potential V(x) = _|¹_x|, estimates (1.3) are satisfied uniformly for

|x|> εandα = 1 for anyε >0. Although our potentials (V^l, V^s) are assumed to be C² on the entire space Rⁿ, our present work may provide interesting results even in presence of singularities for the potentials (V^l, V^s).

For equation (1.1) the energy E = 1

2|x(t)˙ |² +V(x(t)) (1.5) is an integral of motion.

For σ >0 we denote by B(0, σ) the Euclidean open ball of center 0 and radius σ, B(0, σ) = {y ∈ Rⁿ | |y| < σ}, and we denote by B(0, σ) = {y ∈ Rⁿ | |y| ≤σ} its closure. We setµ:=

q2⁵nmax(β₁^l,β₂^l)

α . Under conditions (1.3) the following is valid (see Lemma 2.1 given in the next Section): for any v ∈Rⁿ\B(0, µ), there exists a unique solutionz_±(v, .) of the equation

¨

z(t) =F^l(z(t)), t∈R, (1.6)

so that

˙

z_±(v, t)−v =o(1), ast → ±∞, z_±(v,0) = 0, and

|z_±(v, t)−tv| ≤ 2⁵²n¹²β₁^l

α|v| |t| fort ∈R.

When F^l≡0 thenβ₁^l,β₂^l and µcan be arbitrary close to 0, and we have that z_±(v, t) = tv for (t, v)∈R×Rⁿ, v 6= 0.

Then under conditions (1.3) and (1.4), the following is valid: for any (v₋, x₋) ∈ Rⁿ\B(0, µ)×Rⁿ, the equation (1.1) has a unique solution x ∈ C²(R,Rⁿ) such that

x(t) =z₋(v₋, t) +x₋+y₋(t), (1.7)

where|y˙₋(t)|+|y₋(t)| →0,ast→ −∞; in addition for almost any (v₋, x₋)∈ Rⁿ\B(0, µ)×Rⁿ,

x(t) =z+(v+, t) +x++y+(t), (1.8) for a unique (v+, x+) ∈ Rⁿ×Rⁿ, where |v+| = |v₋| ≥ µ by conservation of the energy (1.5), and where v+ =: a(v₋, x₋), x+ =:b(v₋, x₋), and |y˙+(t)|+

|y+(t)| →0, as t → +∞. A solution x of (1.1) that satisfies (1.7) and (1.8) for some (v₋, x₋),v₋6= 0, is called a scattering solution.

We call the mapS : (Rⁿ\B(0, µ))×Rⁿ→(Rⁿ\B(0, µ))×Rⁿgiven by the formulas

v+=a(v₋, x₋), x+ =b(v₋, x₋), (1.9) the scattering map for the equation (1.1). In addition, a(v₋, x₋), b(v₋, x₋) are called the scattering data for the equation (1.1), and we define

asc(v₋, x₋) = a(v₋, x₋)−v₋, bsc(v₋, x₋) =b(v₋, x₋)−x₋. (1.10) Our definition of the scattering map is derived from constructions given in [6, 1]. We refer the reader to [6, 1] and references therein for the forward classical scattering theory.

By D(S) we denote the set of definition of S. Under the conditions (1.3) and (1.4) the map S : D(S) → (Rⁿ\B(0, µ))×Rⁿ is continuous, and Mes(((Rⁿ\B(0, µ))×Rⁿ)\D(S)) = 0 for the Lebesgue measure on Rⁿ×Rⁿ. In addition the map S is uniquely determined by its restriction to M(S) = D(S)∩Mand byF^l, where M={(v₋, x₋)∈Rⁿ×Rⁿ|v₋ 6= 0, v₋·x₋ = 0}. (Indeed if x(t) is a solution of (1.1) then x(t+t0) is also a solution of (1.1) for any t₀ ∈R.)

One can imagine the following experimental setting that allows to mea- sure the scattering data without knowing the potential V inside a (a priori bounded) region of interest. First find a potential V^lthat generates the same long range effects asV does. Then compute the solutions z_±(v, .) of equation (1.6). Then for a fixed (v₋, x₋)∈(Rⁿ\B(0, µ))×Rⁿ send a particle far away from the region of interest with a trajectory asymptotic to x₋+z₋(v₋, .) at large and negative times. When the particle escapes any bounded region of the space at finite time, then detect the particle and findS(v₋, x₋) = (v+, x+) so that the trajectory of the particle is asymptotic to x₊+z₊(v₊, .) at large and positive times far away from the bounded region of interest.

In this paper we consider the following inverse scattering problem for equation (1.1):

Given S and given the long range tail F^l of the force F, find F^s. (1.11) The main results of the present work consist in estimates and asymptotics for the scattering data (asc, bsc) and scattering solutions for the equation (1.1)

and in application of these asymptotics and estimates to the inverse scatter- ing problem (1.11) at high energies. Our main results include, in particular, Theorem 1.1 given below that provides the high energies asymptotics of the scattering data and the Born approximation of the scattering data at fixed energy.

Consider

TSⁿ⁻¹ :={(θ, x)∈Sⁿ⁻¹ ×Rⁿ | θ·x= 0}, and for any m ∈N consider the x-ray transformP defined by

P f(θ, x) :=

Z +∞

−∞

f(tθ+x)dt

for any function f ∈ C(Rⁿ,R^m) so that |f(x)| =O(|x|^−β˜) as |x| →+∞ for some ˜β >1. For (σ,β, r,˜ α)˜ ∈(0,+∞)²×(0,1)×(0,1], lets0 =s0(σ, r,β,˜ α)˜ be defined as the root of the equation

1 = 4 ˜βn(σ+ 1)

˜ αr(^s0

2³² −r)(1−r)^α+2˜ 1 + 1

s0

2³² −r 2

, s0 >2³²r. (1.12) Then we have the following results.

Theorem 1.1. Let (θ, x) ∈ TSⁿ⁻¹. Under conditions (1.3) and (1.4) the following limits are valid

s→lim+∞sasc(sθ, x) = P F^l(θ, x) +P F^s(θ, x), (1.13)

s→lim+∞s²θ·

bsc(sθ, x)−W(sθ, x)

= −P V^s(θ, x), (1.14) where

W(v, x) :=

Z 0

−∞

Z σ

−∞

F^l(z₋(v, τ) +x)−F^l(z₋(v, τ))

dτ dσ (1.15)

− Z +∞

0

Z +∞ σ

F^l(z+(a(v, x), τ) +x)−F^l(z+(a(v, x), τ)) dσdτ

,

for (v, x)∈ D(S).

In addition,

asc(sθ, x)− Z +∞

−∞

F(τ sθ+x)dτ ≤

4n²(3|x|+ 5)β² 1 + ¹

2⁻³²s−r

2

α²(1−r)^2α+3(2⁻³²s−r)² , (1.16)

bsc(sθ, x)−W(sθ, x)− Z 0

−∞

Z σ

−∞

F^s(τ sθ+x)dτ dσ

+ Z +∞

0

Z +∞ σ

F^s(τ sθ+x)dτ dσ ≤

4n²(3|x|+ 5)β² 1 + ¹

2⁻³²s−r

2

α²(1−r)^2α+2(2⁻³²s−r)³ , (1.17) for (r,(θ, x))∈(0,1)×TSⁿ⁻¹ and for s > s0(|x|, r, β, α),

where β = max(β₁^l, β₂^l, β₂^s, β₃^s).

Note that the vector W defined by (1.15) is known from the scattering data and fromF^l. Then from (1.13) (resp. (1.14)) and inversion formulas for the X-ray transform for n ≥ 2 (see [12, 4, 9, 10]) it follows that F^s can be reconstructed from asc (resp. bsc).

Note that (1.16) and (1.17) also give the asymptotics of asc, bsc, when the parameters α, n, s, θ and x are fixed and β decreases to 0 (where β = max(β₁^l, β₂^l, β₂^s, β₃^s)). In that regime the leading term of sasc(sθ, x) and s²θ· (bsc(sθ, x)−W(sθ, x)) for (θ, x)∈TSⁿ⁻¹ and for s > s0(|x|, r, β, α) is given by the right hand sides of (1.13) and (1.14) respectively. Therefore Theorem 1.1 gives the Born approximation for the scattering data at fixed energy when the potential is sufficiently weak, and it proves that F^s can be reconstructed from the Born approximation of the scattering map at fixed energy.

Theorem 1.1 is a generalization of [10, formulas (4.8a), (4.8b), (4.9a) and (4.9b)] where inverse scattering for the classical multidimensional New- ton equation was studied in the short range case (F^l ≡ 0). We develop Novikov’s framework [10] to obtain our results. Note that results [10, formu- las (4.8b) and (4.9b)] also provide the approximation of the scattering data (asc(v₋, x₋), bsc(v₋, x₋)) for the short range case (F^l ≡0) when the param- eters α, n, v₋ and β are fixed and |x₋| →+∞. Such an asymptotic regime is not covered by Theorem 1.1. Therefore we shall modify in Section 3 the definition of the scattering map to study these modified scattering data in the following three asymptotic regimes: at high energies, Born approxima- tion at fixed energy, and when the parameters α, n, v₋ and β are fixed and

|x₋| →+∞.

For inverse scattering at fixed energy for the multidimensional Newton equation, see for example [7] and references therein.

For the inverse scattering problem in quantum mechanics for the Schr¨odinger equation, see for example [3], [2], [11] and references given in [11].

Our paper is organized as follows. In Section 2 we transform the differen- tial equation (1.1) with initial conditions (1.7) in an integral equation which takes the form y₋ = A(y₋). Then we study the operator A on a suitable space (Lemma 2.2) and we give estimates for the deflection y₋(t) in (1.7) and for the scattering data asc(v₋, x₋), bsc(v₋, x₋) (Theorem 2.4). We prove

Theorem 1.1. Note that we work with small angle scattering compared to the dynamics generated by F^l through the “free” solutionsz₋(v₋, t): In par- ticular, the angle between the vectors ˙x(t) = ˙z₋(v₋, t) + ˙y₋(t) and ˙z₋(v₋, t) goes to zero when the parameters β, α, n, v₋/|v₋|, x₋ are fixed and |v₋| increases. We also provide similar results when one replaces the “free” so- lutions z₋(v, .) by some other functions “z₋,N(v, .)” that may be easier to compute in practise (Formulas (2.47) and (2.48)). In Section 3 we change the definition of the scattering map so that one can obtain for the modified scat- tering data (˜asc(v₋, x₋),˜bsc(v₋, x₋)) their approximation at high energies, or their Born approximation at fixed energy, or their approximation when the parameters α, n, v₋ and β are fixed and |x₋| →+∞(Theorems 3.3, 3.4 and formulas (3.40) and (3.41)). Sections 4, 5, 6 and 7 are devoted to proofs of our Theorems and Lemmas.

2 Scattering solutions

2.1 Integral equation

First we need the following Lemma 2.1 that generalizes the statements given in the Introduction on the existence of peculiar solutions z_± of the equation (1.6).

Lemma 2.1. Assume conditions (1.3). Let(v, x, w, h)∈(Rⁿ)⁴ so thatv·x= 0 and

|w−v| ≤ |v| 4√

2 and |h|<1 + |x|

√2, (2.1)

and assume

2⁵nmax(β₁^l, β₂^l)

α|v|²(1 + ^√|x₂^| − |h|)^α ≤1. (2.2) Then there exists a unique solution z_±(w, x+h, .) of the equation (1.6) so that

˙

z_±(w, x+h, t)−w=o(1), as t → ±∞, z_±(w, x+h,0) =x+h, (2.3) and

|z_±(w, x+h, t)−x−h−tw| ≤ 2⁵²n¹²β₁^l

α|v|(1 + ^√|x₂^| − |h|)^α|t|, (2.4) for t ∈R.

A proof of Lemma 2.1 is given in Section 4.

For the rest of this Section we set µ :=

r2⁵nmax(β₁^l, β₂^l)

α , (2.5)

z_±(v, t) = z_±(v,0, t) for t ∈R, when µ≤ |v|, (2.6)

β₂ := max(β₂^l, β₂^s). (2.7)

Let (v₋, x₋)∈Rⁿ×Rⁿ, v₋·x₋ = 0 and |v₋| ≥µ. Then the function y₋ in (1.7) satisfies the integral equation y₋=A(y₋) where

A(f)(t) = Z t

−∞

Z σ

−∞

F(z₋(v₋, τ) +x₋+f(τ))−F^l(z₋(v₋, τ))

dτ dσ (2.8) fort ∈Rand forf ∈C(R,Rⁿ), sup_(−∞,0]|f|<∞. Under conditions (1.3) and (1.4) we have A(f)∈C²(R,Rⁿ) forf ∈C(R,Rⁿ) so that sup_(−∞,0]|f|<∞. For r > 0 we introduce the following complete metric space Mr defined by

Mr ={f ∈C(R,Rⁿ) | sup

(−∞,0]|f|+ sup

t∈[0,+∞)

|f(t)| 1 +|t|

≤r}, (2.9) and endowed with the normk.kwherekfk= sup_(−∞,0]|f|+sup_t∈[0,+∞)

|f(t)| 1+|t|

. Then we have the following estimate and contraction estimate for the map A restricted to Mr.

Lemma 2.2. Let (v₋, x₋)∈ (Rⁿ\B(0, µ))×Rⁿ, v₋·x₋ = 0, and let r > 0, r <max(^|v−|

2³² ,1 + ^|√x−₂^|). Then the following estimates are valid:

kA(f)k ≤ ρ(n, α, β₂,|x₋|,|v₋|, r) (2.10) := β2(n(3|x₋|+ 2r) + 2√

n) (^|v−|

2³² −r)(1−r)^α

2 α(^|v−|

2³² −r)+ 1 (α+ 1)(1−r)

,

and

kA(f1)−A(f2)k ≤λ(n, α, β2, β₃^s,|x₋|,|v₋|, r)kf1−f2k, (2.11) λ(n, α, β₂, β₃^s,|x₋|,|v₋|, r) := 2n

α(^|v−|

2³² −r)(1−r+ ^|x√−₂^|)^α β₂+ β₃^s

1−r+ ^|x√−₂^| + β₃^s

|v₋| 2³² −r

× 1

1−r+ ^|x√−₂^| + 1

|v₋| 2³² −r

,

for (f, f1, f2)∈M_r³.

A proof of Lemma 2.2 is given in Section 5.

We also need the following result.

Lemma 2.3. Let (v₋, x₋)∈ (Rⁿ\B(0, µ))×Rⁿ, v₋·x₋ = 0, and let r > 0, r < max(^|v−|

2³² ,1 + ^|x√−₂^|). When y₋ ∈ Mr is a fixed point for the map A then z₋(v₋, .) +x₋+y₋(.) is a scattering solution for equation (1.1) and

z₋(v₋, t) +x₋+y₋(t) =z₊(a(v₋, x₋), t) +b(v₋, x₋) +y₊(t), (2.12) for t ≥0, where

a(v₋, x₋) := v₋+ Z +∞

−∞

F(z₋(v₋, τ) +x₋+y₋(τ))dτ, (2.13) b(v₋, x₋) := x₋+l(v₋, x₋, y₋) +l1(v₋, x₋) +l2(v₋, x₋, y₋), (2.14) y+(t) :=

Z +∞ t

Z +∞ σ

F(z₋(v₋, τ) +x₋+y₋(τ))−F^l(z+(a(v₋, x₋), τ)) dτ dσ, (2.15) for t ≥0, and where

l(v₋, x₋, y₋) :=

Z 0

−∞

Z σ

−∞

F z₋(v₋, τ) +x₋+y₋(τ)

−F^l z₋(v₋, τ) dτ dσ

− Z +∞

0

Z +∞ σ

F^s z₋(v₋, τ) +x₋+y₋(τ)

dτ dσ, (2.16)

l1(v₋, x₋) :=− Z +∞

0

Z +∞ σ

F^l(z+(a(v₋, x₋), τ)+x₋)−F^l(z+(a(v₋, x₋), τ)) dτ dσ, (2.17) l₂(v₋, x₋, y₋) := −

Z +∞ 0

Z +∞ σ

F^l(z₋(v₋, τ)+x₋+y₋(τ))−F^l(z₊(a(v₋, x₋), τ)+x₋) dτ dσ, (2.18)

for t ≥0.

Lemma 2.3 is proved in Section 4. Note that l1 is known from the scat- tering data and the knowledge of F^l.

2.2 Estimates on the scattering solutions

In this Section our main results consist in estimates and asymptotics for the scattering data (asc, bsc) and scattering solutions for the equation (1.1).

Theorem 2.4. Under the assumptions of Lemma 2.3 the following estimates are valid

|y˙₋(t)| ≤ β2 n(|x₋|+r) +√ n (α+ 1) ^|₂^v√−₂^| −r

1−r+|t| ^|₂^v√−₂^| −rα+1, (2.19)

|y₋(t)| ≤ β2(n(|x₋|+r) +√ n) α(α+ 1) ^|₂^v√−₂^| −r2

1−r+|t| ₂^|v√−₂^|−rα, (2.20) for t ≤0. In addition

|asc(v₋, x₋)| ≤ 2n¹²

|v₋| 2√

2 −r

(1 + ^|√x−₂^|−r)^α β₁^l

α + β2

(α+ 1)(1 + ^|x√−₂^| −r) .

(2.21)

|l(v₋, x₋, y₋)| ≤ β2n¹²(n¹²(|x₋|+r) + 2)

α(α+ 1)(1−r)^α ₂^|v√−₂^| −r2, (2.22) and

asc(v₋, x₋)− Z +∞

−∞

F(z₋(v₋, τ) +x₋)dτ

≤ 4 max(β2, β₃^s)²n³²(n¹²(3|x₋|+ 2r) + 2) α²(^|v−|

2³² −r)²(1−r)^2α+3

1

|v₋|

2³² −r + 12

, (2.23)

|l(v₋, x₋, y₋)−l(v₋, x₋,0)| ≤ 4 max(β2, β₃^s)²n³²(n¹²(3|x₋|+ 2r) + 2) α²(α+ 1)(^|v−|

2³² −r)³(1−r)^2α+2

× 1

|v₋|

2³² −r + 12

. (2.24)

In addition when

8nmax(β₁^l, β2) α ^|v−|

2³² −r2

(1−r)^α+1 ≤1, (2.25)

then

|l₁(v₋, x₋)| ≤ 8β2n|x₋|

α(α+ 1)|v₋|², (2.26)

|l2(v₋, x₋, y₋)| ≤ 2n³²β₂²(n¹²(2|x₋|+r) + 3) α²(α+ 1)²(1−r)^2α(^|v−|

2³² −r)⁴, (2.27)

|y+(t)| ≤ 2n¹²β2

α(α+ 1)(^|v−|

2³² −r)²(1−r+^|√x−₂^|+t(^|v−|

2³² −r))^α

× 1 + 2nβ2(n¹²(2|x₋|+r) + 3) α(α+ 1)(^|v−|

2³² −r)²(1−r)^α

, (2.28)

for t ≥0.

A proof of Theorem 2.4 is given in Section 6. We now prove Theorem 1.1 combining Theorem 2.4, Lemma 2.2 and estimate (2.4).

2.3 Proof of Theorem 1.1

Let (v₋, x₋)∈ Rⁿ×Rⁿ, v₋·x₋ = 0 and |v₋| ≥µ. We first prove estimates (2.32) and (2.34) given below. We use the following estimate (2.29)

|x₋+ηv₋τ+ (1−η)z₋(v₋, τ)| ≥ |x₋+τ v₋| − |z₋(v₋, τ)−τ v₋|

≥ |√x₋|

2 + (|√v₋|

2 − 2⁵²n¹²β₁^l

α|v₋| )|τ| ≥ |√x₋|

2 +|τ||v₋|

2³² , (2.29)

for η ∈ (0,1) and τ ∈ R (we used (2.4) and (2.2)). Then from (1.3), (1.4), (2.29) and (2.4) it follows that

|F^s(z₋(v₋, τ) +x₋)−F^s(τ v₋+x₋)| ≤ sup

η∈(0,1)

nβ₃^s|z₋(v₋, τ)−τ v₋|

(1 +|x₋+ηv₋+ (1−η)z₋(v₋, τ)|)^α+3

≤ 2⁵²n³²β₃^sβ₁^l|τ| α|v₋|(1 + ^|x√−₂^|+|τ|^|v−|

2³² )^α+3, (2.30) for τ ∈R. Similarly

|F^l(z₋(v₋, τ) +x₋)−F^l(τ v₋+x₋)| ≤ 2⁵²n³²|τ|β₂β₁^l α|v₋|(1 + ^|x√−₂^| +|τ|^|v−|

2³² )^α+2, (2.31) for τ ∈R. Then using (2.30) and (2.31) we have

Z +∞

−∞

F(z₋(v₋, τ) +x₋)dτ− Z +∞

−∞

F(τ v₋+x₋)dτ

≤ 2⁷²n³² max(β2, β₃^s)β₁^l α|v₋|

Z +∞

−∞

|τ|dτ (1 + ^|x√−₂^|+ ^|v−|

2³² |τ|)^α+2 ≤ 2¹⁵² n³²max(β2, β₃^s)β₁^l α²|v₋|³(1 + ^|√x−₂^|)^α .

(2.32) Set

∆1(v₋, x₋) = Z 0

−∞

Z σ

−∞

F^s z₋(v₋, τ)+x₋

dτ dσ− Z +∞

0

Z +∞ σ

F^s z₋(v₋, τ)+x₋ dτ dσ.

(2.33) Then using (2.30) we have

∆1(v₋, x₋)− Z 0

−∞

Z σ

−∞

F^s τ v₋+x₋

dτ dσ+ Z +∞

0

Z +∞ σ

F^s τ v₋+x₋ dτ dσ

≤ 2⁷²n³²β₁^lβ₃^s α|v₋|

Z 0

−∞

Z σ

−∞

|τ|dτ dσ

(1 + ^|√x−₂^|+ ^|₂^v√−₂^||τ|)^α+3 ≤ 2¹³² n³²β₁^lβ₃^s

α²(α+ 1)|v₋|³(1 + ^|√x−₂^|)^α. (2.34) Let r >0,r <max(^|v−|

2³² ,1). Note that max ρ

r, λ, 8nmax(β₁^l, β2) α ^|v−|

2³² −r2

(1−r)^α+1

≤ 4βn(|x₋|+ 1) αr(^|v−|

2³² −r)(1−r)^α+2 1 + 1

|v₋| 2³² −r

2

,

(2.35) where ρ and λ are defined by (2.10) and (2.11) respectively. Assume that

|v₋|> s0(|x₋|, r, β, α) wheres0 is the root of the equation (1.12). Then from (1.12) and Lemma 2.2 it follows thatAhas a unique fixed point inMrdenoted by y₋. Then adding (2.23) and (2.32) we obtain (1.16). Note also that

l(v₋, x₋,0) = Z 0

−∞

Z σ

−∞

F^l z₋(v₋, τ) +x₋

−F^l z₋(v₋, τ) dτ dσ +∆1(v₋, x₋).

Hence adding (2.34), (2.27) and (2.24) we obtain (1.17). Theorem 1.1 is

proved.

2.4 Motivations for changing the definition of the scat- tering map

For a solution x at a nonzero energy for equation (1.1) we say that it is a scattering solution when there existsε >0 so that 1+|x(t)| ≥ε(1+|t|) fort∈ R (see [1]). In the Introduction and in the previous subsections we choose to parametrize the scattering solutions of equation (1.1) by the solutionsz_±(v, .)

of the equation (1.6) (see the asymptotic behaviors (1.7) and (1.8)), and then to formulate the inverse scattering problem (1.11) using this parametrization.

To compute the ”free” solutions z_±(v, .) one has to integrate equation (1.6).

For some cases solving (1.6) leads to simple exact formulas (see [8, Section 15]

when F^l is a Coulombian force). In general one may choose to approximate the solutions z_±(v, .) by the functions z_±,N+1(v, .) defined below. In general the functionsz_±,N+1(v, .) are easier to compute, and in this Subsection we use these approximations to obtain an other formulation of the inverse scattering problem and to mention results similar to Theorem 1.1 and to those given in the previous subsections.

Assume without loss of generality that α 6∈ {m¹ | m ∈ N, m > 0} and set N =⌊α⁻¹⌋ the integer part of α⁻¹. Then let (x, v, w, h)∈ (Rⁿ)⁴ so that v ·x = 0 and (2.1) and (2.2) are satisfied. We define by induction (see also [6])

z_±,0(w, x+h, t) =x+h+tw,(2.36) z₋,m+1(w, x+h, t) =x+h+tw+

Z t 0

Z σ

−∞

F^l(z₋,m(w, x+h, τ))dτ,(2.37) z+,m+1(w, x+h, t) =x+h+tw−

Z t 0

Z +∞ σ

F^l(z+,m(w, x+h, τ))dτ,(2.38) for t ∈ R and for m = 0. . . N. Then one can prove the following estimates by induction (see the proof of Lemma 2.1 and see also [6])

|z_±,m(w, x+h, t)−wt−x−h| ≤ 2⁵²n¹²β₁^l

α|v|(1 + ^√|x₂^| − |h|)^α|t|, t∈R, m= 1. . . N+1, (2.39)

|z_±,m+1(w, x+h, t)−z_±,m(w, x+h, t)| ≤ 2^3(m+1)n^m+12(β₂^l)^mβ₁^l α^m+1|v|^2m+2Π^m+1_j=1 (1−jα)j

×

1 + |x|

√2− |h|+|t||v| 2³²

1−(m+1)α

− 1 + |x|

√2− |h|1−(m+1)α

, (2.40) for m= 0. . . N −1 and for ±t≥0,

|z_±,N+1(w, x+h, t)−z_±,N(w, x+h, t)| ≤ 2^3(N+1)n^N+12(β₂^l)^Nβ₁^l

α^N+1|v|^2N+2Π^N+1_j=1 j|1−jα| 1 + ^√|x₂^| − |h|(N+1)α−1, (2.41)

for ±t≥0, and

|z_±,m+1(w, x+h, t)−z_±,m(w, x+h, t)| ≤ 2^4m+52n^m+12(β₂^l)^mβ₁^l

α^m+1|v|^2m+1(1 + ^√|x₂^| − |h|)^(m+1)α|t|, (2.42)

for t ∈ R and for m = 1. . . N. We set z_±,m(v, .) := z_±,m(v,0, .) for m = 1. . . N + 1 and |v| ≥ µ. For (v₋, x₋) ∈ Rⁿ ×Rⁿ, |v₋| ≥ µ, there exists a unique solution x(t) of equation (1.1) so that

x(t) =x₋+z₋,N+1(v₋, t) +y₋(t), t∈R and lim

t→−∞(|y₋(t)|+|y˙₋(t)|) = 0.

(2.43) In addition when the solution x in (2.43) is a scattering solution then there exists a unique (v+, x+)∈Rⁿ×Rⁿ, |v+|=|v₋| so that

x(t) =x₊+z_+,N+1(v₊, t) +y₊(t), t∈R and lim

t→+∞(|y₊(t)|+|y˙₊(t)|) = 0.

(2.44) In that case we define the scattering data (aN(v₋, x₋), bN(v₋, x₋)) := (v+, x+), and we consider the inverse scattering problem

Given (aN, bN) and given the long range tail F^l of the force F, find F^s. (2.45) The function y₋ in (2.43) satisfies the following integral equation y₋ = AN(y₋) where

AN(f)(t) = Z t

−∞

Z σ

−∞

F(z₋,N+1(v₋, τ) +x₋+f(τ))−F^l(z₋,N(v₋, τ)) dτ dσ (2.46) for t ∈ R and for f ∈ C(R,Rⁿ), sup_(−∞,0]|f| < ∞. Then with appropriate changes in the proof of Lemma 2.2 we can study the operator AN restricted toMr and we can obtain estimate and contraction estimate similar to (2.10) and (2.11). We also obtain the analog of Lemma 2.3 by appropriate change in its proof, and the decomposition (2.12) remains valid by replacing a,b and z₋(v₋, τ) +x₋+y₋(τ) by aN, bN and z_−,N+1(v₋, τ) +x₋+y₋(τ) in (2.12)–

(2.18), and by replacingz+ andF^l z₋(v₋, τ)

byz+,N+1 andF^l z₋,N(v₋, τ) in (2.12) and (2.16), and by replacing z+ by z+,N in (2.15), (2.17) and (2.18). An analog of Theorem 2.4 can be proved for the scattering solu- tions and scattering data (aN, bN). Set asc,N(v₋, x₋) := aN(v₋, x₋)−v₋ and bsc,N(v₋, x₋) := bN(v₋, x₋)−x₋. Finally the following high energies limits are valid. Let (θ, x)∈TSⁿ⁻¹, then

s→lim+∞sasc,N(sθ, x) = P F^l(θ, x) +P F^s(θ, x),(2.47)

s→lim+∞s²θ·

bsc,N(sθ, x)−WN(sθ, x)

= −P V^s(θ, x), (2.48)

where

WN(v, x) :=

Z 0

−∞

Z σ

−∞

F^l(z₋,N(v, τ) +x)−F^l(z₋,N(v, τ))

dτ dσ (2.49)

− Z +∞

0

Z +∞ σ

F^l(z+,N(aN(v, x), τ) +x)−F^l(z+,N(aN(v, x), τ)) dσdτ,

for (v, x) ∈ Rⁿ ×Rⁿ, v ·x = 0 and |v| > C for some constant C. The vector WN defined by (2.49) is known from the scattering data and fromF^l. For the Problem (2.45), from (2.47) (resp. (2.48)) and inversion formulas for the X-ray transform for n ≥ 2 (see [12, 4, 9, 10]) it follows that F^s can be reconstructed from asc,N (resp. bsc,N).

The limits (2.47) and (2.48) follow from estimates similar to (1.16) and (1.17) that also give the Born approximation of asc,N, bsc,N at fixed energy.

However these similar estimates also do not provide the asymptotics of the scattering data (asc,N, bsc,N) when the parameters α, n, v₋ and β are fixed and |x₋| → +∞. Motivated by this disadvantage, in the next section we modify the definition of the scattering map given in the Introduction so that one can obtain a result on this asymptotic regime.

3 A modified scattering map

3.1 Changing the parametrization of the scattering so- lutions

We set

µ(σ) :=

s2⁵nmax(β₁^l, β₂^l)

α(1 + ^√σ₂)^α , for σ ≥0. (3.1) Under conditions (1.3) and (1.4), the following is valid: for any (v₋, x₋) ∈ Rⁿ\{0} ×Rⁿ so that |v₋| ≥µ(|x₋|) andv₋·x₋ = 0, then the equation (1.1) has a unique solution x∈C²(R,Rⁿ) such that

x(t) =z₋(v₋, x₋, t) +y₋(t), (3.2) where |y˙₋(t)|+|y₋(t)| → 0, as t → −∞, and where z₋(v₋, x₋, .) is defined in Lemma 2.1 (for ”(w, x, v, h) = (v₋, x₋, v₋,0)”).

In addition the function y₋ in (3.2) satisfies the integral equation y₋ = A(y₋) where

A(f)(t) = Z t

−∞

Z σ

−∞

F(z₋(v₋, x₋, τ) +f(τ))−F^l(z₋(v₋, x₋, τ)) dτ dσ

(3.3)

fort ∈Rand forf ∈C(R,Rⁿ), sup_(−∞,0]|f|<∞. Under conditions (1.3) and (1.4) we have A(f)∈C²(R,Rⁿ) forf ∈C(R,Rⁿ) so that sup_(−∞,0]|f|<∞. We study the map A defined by (3.3) on the metric space Mr defined by (2.9). Set

˜k(v₋, x₋, f) =v₋+ Z +∞

−∞

F z₋(v₋, x₋, τ) +f(τ)

dτ, (3.4)

for f ∈Mr. For the rest of the section we set β2 = max(β₂^l, β₂^s).

The following Lemma 3.1 is the analog of Lemma 2.2.

Lemma 3.1. Let (v₋, x₋) ∈ Rⁿ×Rⁿ, v₋·x₋ = 0, |v₋| ≥ µ(|x₋|), and let r >0, r <max(^|v−|

2³² ,1 + ^|x√−₂^|). Then the following estimates are valid:

kA(f)k ≤ ρ(n, α, β˜ 2,|x₋|,|v₋|, r) (3.5)

= 2β2n¹²(n¹²r+ 1) (^|v−|

2³² −r)(1−r+ ^|x√−₂^|)^α

1

(α+ 1)(1−r+ ^|x√−₂^|) + 2 α(^|v−|

2³² −r) ,

kA(f1)− A(f2)k ≤λ(n, α, β2, β₃^s,|x₋|,|v₋|, r)kf1−f2k, (3.6) and

|k(v˜ ₋, x₋, f)−v₋| ≤ 2n¹²

|v₋| 2√

2 −r

(1 + ^|√x−₂^|−r)^α β₁^l

α + β₂

(α+ 1)(1 + ^|x√−₂^| −r)

,

(3.7) for (f, f1, f2)∈M_r³, where λ is defined in (2.11).

Proof of Lemma 3.1 is given in Section 5.

Let (v₋, x₋)∈ Rⁿ×Rⁿ,v₋·x₋= 0, and letr∈(0,max ¹₂+^|x−|

2³² ,2⁻³²|v₋| ).

Assume that

20nmax(β₁^l, β2) α ^|₂^v√−₂^| −r_{2 1}

2 +^|x−|

2³² −rα ≤1. (3.8) Then using Lemma 2.1 and (3.7) one can consider the free solutionz+(˜k(v₋, x₋, f), x₋, .) (for ”(w, v, x, h) = (˜k(v₋, x₋, f), v₋, x₋,0)”) for f ∈ Mr. In addi- tion, with appropriate changes in the proof of Lemma 2.3 one can prove that when y₋ ∈Mr is a fixed point of the operator Athen z₋(v₋, x₋, .) +y₋(.) is a scattering solution of (1.1) (in the sense given in Section 2.4), and one can prove that the following decomposition is valid

z₋(v₋, x₋, t)+y₋(t) =z+(˜a(v₋, x₋), x₋+h, t)+ G^v−,x₋(h)−h

+H(v₋, x₋, y₋, h)(t), (3.9)