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

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

range potential

Alexandre Jollivet

July 12, 2014

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 and consider also 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

∗Laboratoire de Math´ematiques Paul Painlev´e, CNRS UMR 8524/Univer- sit´e Lille 1 Sciences et Technologies, 59655 Villeneuve d’Ascq Cedex, France, [email protected]

electric (or gravitational) field described byV (see [11]) wherex 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_|j|^l (1 +|x|)^−(α+|j|), (1.3) and V^s satisfies the following short range assumptions

|∂_x^jV^s(x)| ≤β_|j|+1^s (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=1j_m, and β_m^l and β_m^s0 are positive real constants for m = 0,1,2 and for m⁰ = 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) = _|x|¹ , 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:=

q2⁴nmax(β₁^l,β₂^l)

α . Under conditions (1.3) the following is valid (see Section 4 for a proof): for any v ∈ Rⁿ\B(0, µ), there exists a unique solution z±(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, (1.7) and

|z±(v, t)−tv| ≤ 2³²n¹²β₁^l

α|v| |t| fort ∈R. (1.8)

When F^l ≡0 then z±(v, t) =tv for any (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), (1.9) 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), (1.10) 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, ast →+∞. A solution x of (1.1) that satisfies (1.9) and (1.10) 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))×Rⁿ given by the formulas

v₊=a(v−, x−), x₊ =b(v−, x−), (1.11) 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

a_sc(v−, x−) = a(v−, x−)−v−, b_sc(v−, x−) =b(v−, x−)−x−. (1.12) 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)∩ M, where M={(v−, x−)∈Rⁿ×Rⁿ |v−6= 0, v−·x− = 0}. Indeed if x(t) is a solution of (1.1) thenx(t+t0) is also a solution of (1.1) for anyt0 ∈R, and we also have z±(v, t+t₀)−z±(v, t)−t₀v = Rt0+t

t

Rσ

±∞F^l(z±(v, τ))dτ dσ which gives |z±(v, t+t₀)−z±(v, t)−t₀v|+|z˙±(v, t+t₀)−z˙±(v, t)| → 0 as t → ±∞for any (v, t0)∈(Rⁿ\B(0, µ))×R. We then obtain

a(v−, x−+t₀v−) =a(v−, x−), b(v−, x−+t₀v−) = b(v−, x−) +t₀a(v−, x−), (1.13) for any t₀ ∈Rand for (v₋, x₋)∈ D(S).

For the forward classical scattering theory we refer the reader to [1, 7, 17, 8, 16, 2, 3] and references therein. Our definition of the scattering map is derived from constructions given in [8, 3].

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 choose a potential V^l that generates the

same long range effects as V does. Then compute the solutions z±(v, .) of equation (1.6). Then for a fixed (v₋, x₋) ∈ (Rⁿ\B(0, µ))×Rⁿ send a par- ticle 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 find S(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. (1.14) 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 (1.14) at high energies. Our main results include, in particular, Theorem 1.1 and Corollary 1.2 given below that provide the high energies asymptotics of the scattering data and the Born approximation of the scat- tering data at fixed energy respectively.

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], let ρ₀ =ρ₀(σ, r,β,˜ α)˜ be defined as the root of the equation

1 = 6 ˜βn(σ+ 1)

˜ αr(^ρ0

2³² −r)(1−r)^α+2˜ 1 + 1

ρ0

2³² −r

, ρ₀ >2³²r. (1.15) 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

ρ→+∞lim ρa_sc(ρθ, x) = P F^l(θ, x) +P F^s(θ, x), (1.16)

ρ→+∞lim ρ²θ·

bsc(ρθ, x)−W(ρθ, x)

= −P V^s(θ, x), (1.17)

where W(v, x) :=

Z 0

−∞

Z σ

−∞

F^l(z₋(v, τ) +x)−F^l(z₋(v, τ))

dτ dσ (1.18)

− Z +∞

0

Z +∞

σ

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

for (v, x)∈ D(S).

In addition,

ρa_sc(ρθ, x)− Z +∞

−∞

F(τ θ+x)dτ ≤

8n²(|x|+ 2)β²ρ 1 + ¹

2⁻³²ρ−r

2

α²(1−r)^2α+3(2⁻³²ρ−r)² , (1.19) ρ²θ· b_sc(ρθ, x)−W(ρθ, x)

+P V^s(θ, x) ≤

11n²(|x|+ 2)β²ρ² 1 + ¹

2⁻³²ρ−r

2

α²(α+ 1)(1−r)^2α+2(2⁻³²ρ−r)³, (1.20) for (r,(θ, x))∈(0,1)×TSⁿ⁻¹ and for ρ > ρ₀(|x|, r, β, α),

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

Note that the vector W defined by (1.18) is known from the scattering data and fromF^l. Then from (1.16) (resp. (1.17)) and inversion formulas for the X-ray transform for n ≥ 2 (see [15, 6, 12, 13]) it follows that F^s can be reconstructed from a_sc (resp. b_sc).

Theorem 1.1 provides results on the asymptotics of the scattering data at fixed energy E = ¹₂ when the external force is weaker and weaker. In other words Theorem 1.1 also provides the Born approximation of the scattering data at fixed energy E = ¹₂. More precisely, consider the Newton equation (1.1) in the external potential V^γ := γ²V where γ is a positive coupling constant and where V ∈ C²(Rⁿ,R) satisfies (1.3) and (1.4). We denote by (a^γ_sc, b^γ_sc) (resp. (a_sc, b_sc)) the scattering data related to V^γ (resp. V), and we denote byD(S) the set of definition of the scattering map related toV. Then we have

a^γ_sc(θ, x) = γa_sc(θ

γ, x), b^γ_sc(θ, x) =b_sc(θ

γ, x), (1.21)

for (θ, x)∈TSⁿ⁻¹so that (^θ_γ, x)∈ D(S). Formulas (1.21) are derived from the observation that x(γt) is a solution of the Newton equation in the potential V^γ whenever x(t) is a solution of the Newton equation in V. Then Theorem 1.1 is rewritten as follows.

Corollary 1.2. Let (θ, x)∈TSⁿ⁻¹. We have lim

γ→0⁺γ⁻²a^γ_sc(θ, x) = P F^l(θ, x) +P F^s(θ, x), (1.22) lim

γ→0⁺γ⁻²θ·

b^γ_sc(θ, x)−W(θ γ, x)

= −P V^s(θ, x). (1.23) In addition,

γ⁻²a^γ_sc(θ, x)− Z +∞

−∞

F(τ θ+x)dτ ≤γ

8n²(|x|+ 2)β² 1 + ^γ

2⁻³²−rγ

2

α²(1−r)^2α+3(2⁻³² −rγ)² , (1.24) γ⁻²θ· b^γ_sc(θ, x)−W(θ

γ, x)

+P V^s(θ, x) ≤γ

11n²(|x|+ 2)β² 1 + ^γ

2⁻³²−rγ

2

α²(α+ 1)(1−r)^2α+2(2⁻³² −rγ)³, (1.25) for (r,(θ, x)) ∈ (0,1)× TSⁿ⁻¹ and for γ < ρ0(|x|, r, β, α)⁻¹, where W is defined by (1.18).

From formulas (1.22) and (1.23) and from inversion formulas of the x-ray transform for n ≥ 2 it follows that F^s can be reconstructed from the Born approximation at fixed energy E = ¹₂ of the scattering data a^γ_sc and b^γ_sc.

Theorem 1.1 is a generalization of [13, 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 [13] to obtain our results. Note that results [13, formu- las (4.8b) and (4.9b)] also provide the approximation of the scattering data (a_sc(v−, x−), b_sc(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 in order to study these modified scattering data in the following three asymptotic regimes: at high energies, Born ap- proximation 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 [9] and references therein.

For the inverse scattering problem in quantum mechanics for the Schr¨odinger equation, see for example [5], [4], [14] and references given in [14].

Our paper is organized as follows. In Section 2 we transform the differ- ential equation (1.1) with initial conditions (1.9) into an integral equation which takes the form y− = A(y−). Then we study the nonlinear integral operator A on a suitable space (Lemma 2.1) and we give estimates for the

deflection y−(t) in (1.9) and for the scattering data a_sc(v−, x−), b_sc(v−, x−) (Theorem 2.3). Then we prove Theorem 1.1. Note that we work with small angle scattering compared to the dynamics generated by F^l with respect to the “free” solutions z−(v−, t): In particular, 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 re- sults when one replaces the “free” solutions z−(v, .) by some other functions

“z_−,N(v, .)” that may be easier to compute numerically (Formulas (2.38) and (2.39)). In Section 3 we change the definition of the scattering map so that one can obtain for the modified scattering data (˜a_sc(v−, x−),˜b_sc(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−| → +∞ (Theorem 3.3, Corollary 3.4 and formulas (3.38) and (3.39)).

Sections 4, 5 and 6 are devoted to proofs of our Theorem 2.3 and Lemmas 2.1 and 2.2.

2 Scattering solutions

2.1 Integral equation

For the rest of this Section we set β₂ := max(β₂^l, β₂^s).

Let (v−, x−)∈Rⁿ×Rⁿ, v−·x− = 0 and |v−| ≥µ. Then the function y−

in (1.9) 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 (2.1) 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 M_r defined by

M_r ={f ∈C(R,Rⁿ) | sup

(−∞,0]

|f|+ sup

t∈[0,+∞)

|f(t)|

1 +|t|

≤r}, (2.2) 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 M_r.

Lemma 2.1. Let (v−, x−) ∈ (Rⁿ\B(0, µ))×Rⁿ, v− ·x− = 0, and let r ∈

(0,min(^|v−|

2³² ,1)). Then the following estimates are valid:

kA(f)k ≤ λ₀(n, α, β₂,|x−|,|v−|, r) (2.3) := β₂(n(2|x−|+r) + 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, (2.4) 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

× 1

1−r+ ^|x√−₂^| + 1

|v−| 2³² −r

,

for (f, f1, f2)∈M_r³.

A proof of Lemma 2.1 is given in Section 5.

We also need the following result.

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

2³² ,1)). Wheny− ∈M_ris a fixed point for the mapAthenz−(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), (2.5) for t ≥0, where

a(v₋, x₋) := v₋+ Z +∞

−∞

F(z₋(v₋, τ) +x₋+y₋(τ))dτ, (2.6) b(v−, x−) := x−+l(y−) +l₁+l₂(y−), (2.7) y₊(t) :=

Z +∞

t

Z +∞

σ

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

(2.8) for t ≥0, and where

l(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, (2.9)

l1 :=− Z +∞

0

Z +∞

σ

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

(2.10) l2(y−) :=−

Z +∞

0

Z +∞

σ

F^l(z−(v−, τ)+x−+y−(τ))−F^l(z+(a(v−, x−), τ)+x−) dτ dσ, (2.11) for t ≥0.

Lemma 2.2 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 (a_sc, b_sc) and scattering solutions for the equation (1.1).

Theorem 2.3. Under the assumptions of Lemma 2.2 the following estimates are valid

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

2√

2 −r

1−r+|t| ^|v−|

2√

2 −rα+1, (2.12)

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

2√

2 −r2

1−r+|t| ^|v−|

2√

2 −rα, (2.13) for t ≤0. In addition

|a_sc(v−, x−)| ≤ 2n¹²

|v−| 2√

2 −r

(1 + ^|x√−₂^|−r)^α β₁^l

α + β₂

(α+ 1)(1 + ^|x√−₂^| −r)

.

(2.14)

|l(y−)| ≤ β₂n¹²(n¹²(|x−|+r) + 2) α(α+ 1)(1−r)^{α |v−|}

2√

2 −r2, (2.15) and

a_sc(v−, x−)− Z +∞

−∞

F(z−(v−, τ) +x−)dτ

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

2³² −r)²(1−r)^2α+3

1

|v−|

2³² −r + 12

, (2.16)

|l(y₋)−l(0)| ≤

4 max(β₂, β₃^s)²n³²(n¹²(2|x−|+r) + 2) |v−|¹ 2

3 2

−r+ 12

α²(α+ 1)(^|v−|

2³² −r)³(1−r)^2α+2 . (2.17)

In addition when

6nmax(β₁^l, β₂) α ^|v−|

2³² −r2

(1−r)^α+1 ≤1, (2.18)

then

|l₁| ≤ 8β₂n|x−|

α(α+ 1)|v−|², (2.19)

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

2³² −r)⁴, (2.20)

|y₊(t)| ≤ 4nβ₂(1 +|x−|) α(α+ 1)(^|v−|

2³² −r)²(1−r+ ^|x√−₂^|+t(^|v−|

2³² −r))^α, (2.21) for t ≥0.

A proof of Theorem 2.3 is given in Section 6. We now prove Theorem 1.1 combining Theorem 2.3, Lemma 2.1 and estimate (1.8).

2.3 Proof of Theorem 1.1

Let (v₋, x₋)∈ Rⁿ×Rⁿ, v₋·x₋ = 0 and |v₋| ≥µ. We first prove estimates (2.25) and (2.27) given below. We use the following estimate (2.22)

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

≥ |x₋|

√2 + (|v₋|

√2 − 2³²n¹²β₁^l

α|v−| )|τ| ≥ |x₋|

√2 +|τ||v₋|

2³² , (2.22)

for η ∈ (0,1) and τ ∈ R (we used (1.8) and the inequality |v−| ≥ µ). Then from (1.3), (1.4), (2.22) and (1.8) 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

≤ 2³²n³²β₃^sβ₁^l|τ| α|v₋|(1 + ^|x√−|

2 +|τ|^|v−|

2³² )^α+3, (2.23) for τ ∈R. Similarly

|F^l(z−(v−, τ) +x−)−F^l(τ v−+x−)| ≤ 2³²n³²|τ|β₂β₁^l α|v−|(1 + ^|x√−₂^| +|τ|^|v−|

2³² )^α+2, (2.24)

for τ ∈R. Then using (2.23) and (2.24) we have

Z +∞

−∞

F(z₋(v₋, τ) +x₋)dτ− Z +∞

−∞

F(τ v₋+x₋)dτ

≤ 2⁵²n³² max(β2, β^s₃)β₁^l α|v−|

Z +∞

−∞

|τ|dτ (1 + ^|x√−|

2 + ^|v−|

2³² |τ|)^α+2 ≤ 2¹³² n³² max(β2, β₃^s)β₁^l α²|v−|³(1 + ^|x√−|

2)^α . (2.25) Set

∆₁ = Z 0

−∞

Z σ

−∞

F^s z−(v−, τ)+x−

dτ dσ−

Z +∞

0

Z +∞

σ

F^s z−(v−, τ)+x−

dτ dσ.

(2.26) Then using (2.23) we have

∆₁− Z 0

−∞

Z σ

−∞

F^s τ v₋+x₋

dτ dσ+ Z +∞

0

Z +∞

σ

F^s τ v₋+x₋ dτ dσ

≤ 2⁵²n³²β₁^lβ₃^s α|v−|

Z 0

−∞

Z σ

−∞

|τ|dτ dσ (1 + ^|x√−|

2 + ^|v−|

2√

2|τ|)^α+3 ≤ 2⁷n³²β₁^lβ₃^s α²(α+ 1)|v−|⁴(1 + ^|x√−|

2)^α. (2.27) Let r∈(0,min(^|v−|

2³² ,1)). Note that max λ₀

r , λ, 6nmax(β₁^l, β₂) α ^|v−|

2³² −r2

(1−r)^α+1

≤ 6βn(|x₋|+ 1) αr(^|v−|

2³² −r)(1−r)^α+2 1 + 1

|v−| 2³² −r

,

(2.28) where λ₀ and λ are defined by (2.3) and (2.4) respectively. Assume that

|v₋| > ρ₀(|x₋|, r, β, α) where ρ₀ is the root of the equation (1.15). Then from (1.15) and Lemma 2.1 it follows that Ahas a unique fixed point in M_r denoted by y−. Then adding (2.16) and (2.25) we obtain (1.19). Note also that

l(0) = Z 0

−∞

Z σ

−∞

F^l z−(v−, τ) +x−

−F^l z−(v−, τ) dτ dσ

+∆₁.

Hence adding (2.27), (2.20) and (2.17) we obtain (1.20). 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 exists ε > 0 so that 1 + |x(t)| ≥ ε(1 +

|t|) for t ∈ R (see [3]). In the Introduction and in the previous subsections we choose to parametrize the scattering solutions of equation (1.1) by the solutions z_±(v, .) of the equation (1.6) (see the asymptotic behaviors (1.9) and (1.10)), and then to formulate the inverse scattering problem (1.14) 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 [11, Section 15] whenF^l is a Coulombian force). In general one may choose to approximate the solutionsz±(v, .) by the functionsz±,N+1(v, .) defined below. In general the approximating functions z±,N+1(v, .) are easier to compute numerically and/or analytically, 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 =bα⁻¹c the integer part of α⁻¹. Then letv ∈Rⁿ\B(0, µ). Proceeding from [8] we define by induction

z±,0(v, t) =tv, z±,m+1(v, t) = tv+ Z t

0

Z σ

±∞

F^l(z±,m(v, τ))dτ dσ, (2.29) for t ∈ R and for m = 0. . . N. Then one can prove the following estimates by induction (see Section 4 and see also [8])

|z±,m(v, t)−tv| ≤ 2³²n¹²β₁^l

α|v| |t|, t∈R, m= 1. . . N + 1, (2.30)

|z_±,m+1(v, t)−z_±,m(v, t)| ≤

2^m+1n^m+12(β₂^l)^mβ₁^l

1 +|t|^√|v|

2

1−(m+1)α

−1 α^m+1|v|^2m+2Π^m+1_j=1 (1−jα)j ,

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

|z±,N+1(v, t)−z±,N(v, t)| ≤ 2^N+1n^N+12(β₂^l)^Nβ₁^l

α^N+1|v|^2N+2Π^N_j=1⁺¹j|1−jα|, (2.32) for ±t≥0, and

|z±,m+1(v, t)−z±,m(v, t)| ≤ 2^2m+32n^m+12(β₂^l)^mβ₁^l

α^m+1|v|^2m+1 |t|, (2.33)

for t∈Rand for m= 1. . . N. 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.

(2.34) In addition when the solution x in (2.34) 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.

(2.35) In that case we define the scattering data (a_N(v₋, x₋), b_N(v₋, x₋)) := (v₊, x₊), and we consider the inverse scattering problem

Given (a_N, b_N) and given the long range tail F^l of the forceF, find F^s. (2.36) The function y− in (2.34) satisfies the following integral equation y− = A_N(y−) where

A_N(f)(t) = Z t

−∞

Z σ

−∞

F(z−,N+1(v−, τ) +x−+f(τ))−F^l(z−,N(v−, τ)) dτ dσ (2.37) for t ∈ R and for f ∈ C(R,Rⁿ), sup_(−∞,0]|f| < ∞. Then with appropriate changes in the proof of Lemma 2.1 we can study the operator AN restricted to M_r and we can obtain estimate and contraction estimate similar to (2.3) and (2.4). We also obtain the analog of Lemma 2.2 by appropriate change in its proof, and the decomposition (2.5) remains valid by replacing a, b and z−(v−, τ) +x−+y−(τ) by a_N, b_N and z−,N+1(v−, τ) +x−+y−(τ) in (2.5)–

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

byz_+,N+1 andF^l z−,N(v−, τ) in (2.5) and (2.9), and by replacingz+ byz+,N in (2.8), (2.10) and (2.11). An analog of Theorem 2.3 can be proved for the scattering solutions and scatter- ing data (a_N, b_N). Set a_sc,N(v−, x−) :=a_N(v−, x−)−v− and b_sc,N(v−, x−) :=

bN(v−, x−) − x−. Finally the following high energies limits are valid. Let (θ, x)∈TSⁿ⁻¹, then

ρ→+∞lim ρasc,N(ρθ, x) = P F^l(θ, x) +P F^s(θ, x), (2.38)

ρ→+∞lim ρ²θ·

b_sc,N −W_N

(ρθ, x) = −P V^s(θ, x), (2.39) where

W_N(v, x) :=

Z 0

−∞

Z σ

−∞

F^l(z−,N(v, τ) +x)−F^l(z−,N(v, τ))

dτ dσ (2.40)

− Z +∞

0

Z +∞

σ

F^l(z_+,N(a_N(v, x), τ) +x)−F^l(z_+,N(a_N(v, x), τ)) dτ dσ,

for (v, x)∈Rⁿ×Rⁿ, v·x= 0 and |v|> C for some constantC. The vector W_N defined by (2.40) is known from the scattering data and fromF^l.

For the Problem (2.36), from (2.38) (resp. (2.39)) and inversion formulas for the X-ray transform for n ≥2 (see [15, 6, 12, 13]) it follows thatF^s can be reconstructed from a_sc,N (resp. b_sc,N).

The Born approximation of a_sc,N, b_sc,N at fixed energy can also be de- rived from (2.38) and (2.39) by using the same observations that leaded to Corollary 1.2.

The limits (2.38) and (2.39) follow from estimates similar to (1.19) and (1.20). However these estimates do not provide the asymptotics of the scat- tering data (a_sc,N, b_sc,N) when the parameters α, n, v₋ and β are fixed and

|x−| → +∞. Motivated by this disadvantage, we introduce a new family of

“free” solutions z±(v, x, .) of equation (1.6) that will be used for parametriz- ing some unbounded solutions of the Newton equation (1.1) at nonzero energy and for measuring their deviation. Those free solutions z±(v, x, .) have the properties that limt→±∞z˙±(v, x, t) = v and z±(v, x,0) = x. In addition, for (v₋, x₋)∈Rⁿ×Rⁿ, v₋·x₋ = 0, so that the energyE = ¹₂|v₋|² is sufficiently high, then there exists a unique solution x ∈ C²(R,Rⁿ) of equation (1.1) such that

x(t) = z₋(v₋, x₋, t) +y₋(t), (2.41) where |y˙−(t)|+|y−(t)| →0, as t→ −∞. Such a solution also satisfies:

x(t) =z₊(˜a,˜b, t) +y₊(t), (2.42) where |y˙₊(t)| +|y₊(t)| → 0, as t → +∞ for a unique (˜a,˜b) ∈ Rⁿ× Rⁿ. The map ˜S defined by ˜S(v−, x−) = (˜a,˜b) at sufficiently high energies is our modified scattering map.

In the next section, we provide more details on its definition, and we study its asymptotics in the three regimes: at high energies, its Born approximation at fixed energy, and at fixed energy when the “impact parameter” |x−| → +∞.

3 A modified scattering map

First let us introduce the new family mentioned above of “free” solutions z±(v, x, .) of equation (1.6). We set

˜ µ(σ) :=

v u u t

2⁵nmax(β₁^l, β₂^l) α(¹₂ + ^σ

2³²)^α , for σ≥0. (3.1)

Let (v, x)∈(Rⁿ)² and let (w, h)∈ B(v, ^|v|

4√

2)× B(0,¹₂ + ^|x|

2³²) so that v·x= 0 and |v| ≥ µ(|x|). 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, (3.2) and

|z±(w, x+h, t)−x−h−tw| ≤ 2⁵²n¹²β₁^l

α|v|(¹₂ +₂^|x|√₂)^α|t|, (3.3) for t ∈ R. Under conditions (1.3) and (1.4), the following is valid: for any (v−, x−) ∈ Rⁿ\{0} ×Rⁿ so that |v−| ≥ µ(|x˜ −|) and v−·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), (3.4) where |y˙−(t)|+|y−(t)| →0, as t→ −∞.

In addition the function y₋ in (3.4) 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

(3.5) for t ∈ R and for f ∈ C(R,Rⁿ), sup_(−∞,0]|f| < ∞. We study the map A defined by (3.5) on the metric space M_r defined by (2.2). Set

˜k(v−, x−, f) = v−+ Z +∞

−∞

F z−(v−, x−, τ) +f(τ)

dτ, (3.6)

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

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

2³² ,1 + ^|x√−|

2 )). Then the following estimates are valid:

kA(f)k ≤ λ˜₀(n, α, β₂,|x₋|,|v₋|, r) (3.7)

= 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, (3.8) 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)

,

(3.9) for (f, f₁, f₂)∈M_r³, where λ is defined in (2.4).

For (σ,β, r,˜ α)˜ ∈ (0,+∞)³ ×(0,1], r < ¹₂ + ^σ

2³², let ˜ρ₀ = ˜ρ₀(σ, r,β,˜ α) be˜ defined as the root of the equation

1 = 20 ˜βn(1 +r)

˜ αr(^ρ˜0

2³² −r)(¹₂ −r+ ^σ

2³²)^α˜ 1 + 1

˜ ρ0

2³² −r

, ρ˜0 >2³²r. (3.10)

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

2³² ) so that

|v−|>ρ˜0(|x−|, r, β, α), whereβ = max(β₁^l, β2, β₃^s). Then from Lemma 3.1 the map A is a contraction in M_r and we denote by y− its unique fixed point, and from (3.9) one can consider the free solution z₊(˜a(v−, x−), x−, .) where

˜

a(v−, x−) = ˜k(v−, x−, y−). (3.11) Then z−(v−, x−, .) +y− is a scattering solution of (1.1) (in the sense given in Section 2.4) and 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), (3.12) for t≥0, where

G_v−,x−(h) = A(y−)(0)−H(v−, x−, y−, h,0), (3.13) H(v−, x−, y−, h, t) =

Z +∞

t

Z +∞

σ

F z−(v−, x−, τ) +y−(τ)

−F^l z+(˜a(v−, x−), x−+h, τ)

dτ dσ, (3.14) for t ≥ 0 and for |h| ≤ ¹₂ + ^|x−|

2√

2. Lemma 3.2 provides an estimate and a contraction estimate on the map G_v−,x−.

Lemma 3.2. The following estimates are valid:

|Gv−,x−(h)| ≤ β2(6(nr+√

n) +n(1 + ^|x√−₂^|)) 2α(α+ 1) ^|v−|

2√

2 −r2 1 2 +^|x−|

2√

2 −rα (3.15)

≤ 1

4+ |x₋| 10√

2, (3.16)

|Gv−,x−(h)− Gv−,x−(h⁰)| ≤ 16nβ₂^l|h−h⁰| α(α+ 1)|v−|^{2 1}₂ + ^|x−|

2√ 2

α ≤ |h−h⁰|

10 , (3.17) for (h, h⁰)∈Rⁿ×Rⁿ, |h⁰| ≤ |h| ≤ ¹₄ +^|x−|

2⁵² .

We denote by ˜b_sc(v−, x−) the unique fixed point ofG_v−,x− inB(0,¹₄+^|x−|

2⁵² ), and we set ˜b(v−, x−) := x−+ ˜b_sc(v−, x−) and ˜a_sc(v−, x−) := ˜a(v−, x−)−v−. The decomposition (3.12) becomes

z₋(v₋, x₋, t) +y₋(t) = z₊(˜a(v₋, x₋),˜b(v₋, x₋), t) +y₊(t), (3.18) y+(t) =H(v−, x−, y−,˜bsc(v−, x−), t), (3.19) for t≥0.

The map ˜S defined by ˜S(v−, x−) = (˜a(v−, x−),˜b(v−, x−)) on the set M˜ = {(v₋, x₋)∈Rⁿ×Rⁿ | v₋·x₋= 0, |v₋|>ρ˜₀(|x₋|, r, β, α)

for some r∈(0,1

2 +|x₋| 2³² )}

is our modified scattering map.

Links between (˜a,˜b) and the scattering data (a, b) defined in Section 2 are given by the 2 following formulas provided that |v−| ≥ µwhere µ is the constant defined in the Introduction:

˜

a=a v−, lim

t→−∞(z−(v−, x−, t)−z−(v−, t))

, (3.20)

b v−, lim

t→−∞(z−(v−, x−, t)−z−(v−, t))

= lim

t→+∞

z₊(˜a,˜b, t)−z₊(˜a, t)

. (3.21) We shortened ˜a(v−, x−) and ˜b(v−, x−) to ˜a and ˜b in (3.20) and (3.21).

The inverse scattering problem for equation (1.1) can now be formulated as follows

Given (˜a_sc,˜b_sc) and F^l, find F^s. (3.22) Theorem 3.3 and Corollary 3.4 provide results similar to Theorems 2.3 and 1.1.

Theorem 3.3. Let(v₋, x₋)∈Rⁿ×Rⁿ, v₋·x₋ = 0, and let r∈(0,¹₂+^|x−|

2³² ),

|v−|>ρ˜₀(|x−|, r, β, α). Then the following estimates are valid:

|y˙−(t)| ≤ β₂ nr+√

n (α+ 1) ^|v−|

2√

2 −r

1 + ^|x√−|

2 −r+|t| ^|v−|

2√

2 −rα+1, (3.23)

|y−(t)| ≤ β₂ nr+√

n α(α+ 1) ^|v−|

2√

2 −r2

1 + ^|x√−|

2 −r+|t| ^|v−|

2√

2 −rα, (3.24)