On inverse scattering at high energies for the multidimensional nonrelativistic Newton equation in electromagnetic field

On inverse scattering at high energies for the multidimensional nonrelativistic Newton

equation in electromagnetic field

Alexandre Jollivet

Abstract.We consider the multidimensional nonrelativistic Newton equation in a static electromagnetic field

¨

x=F(x,x), F˙ (x,x) :=˙ −∇V(x) +B(x) ˙x, x˙ = dx

dt, x∈C²(R,Rⁿ), (∗) where V ∈ C²(Rⁿ,R), B(x) is the n×n real antisymmetric matrix with el- ements B_i,k(x), B_i,k ∈ C¹(Rⁿ,R) (and B satisfies the closure condition), and

|∂_x^j1V(x)| + |∂_x^j2B_i,k(x)| ≤ β|j₁|(1 + |x|)^−(α+|j1|) for x ∈ Rⁿ, 1 ≤ |j₁| ≤ 2, 0 ≤ |j2| ≤1, |j2|= |j1| −1, i, k = 1. . . n and some α >1. We give estimates and asymptotics for scattering solutions and scattering data for the equation (∗) for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms P∇V and P B_i,k (on sufficiently rich sets of straight lines).

Applying results on inversion of the X-ray transform P we obtain that for n ≥ 2 the velocity valued component of the scattering operator at high ener- gies uniquely determines (∇V, B). We also consider the problem of recovering (∇V, B) from our high energies asymptotics found for the configuration val- ued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of [R. Novikov, 1999]

for (∗) with B ≡0 and of [Jollivet, 2005] for the relativistic version of (∗). We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for (∗) on the one hand and for its relativistic version on the other.

1 Introduction

1.1 The nonrelativistic Newton equation

Consider the multidimensional nonrelativistic Newton equation in an ex- ternal static electromagnetic field:

¨

x(t) =F(x(t),x(t)) :=˙ −∇V(x(t)) +B(x(t)) ˙x(t), (1.1) wherex(t)∈Rⁿ,x(t) =˙ ^dx_dt(t),andV ∈C²(Rⁿ,R) and for anyx∈Rⁿ, B(x) is a n×n antisymmetric matrix with elements Bi,k(x), Bi,k ∈ C¹(Rⁿ,R), which satisfy

∂B_i,k

∂x_l (x) + ∂B_l,i

∂x_k (x) + ∂B_k,l

∂x_i (x) = 0, (1.2)

for x= (x₁, . . . , x_n)∈Rⁿ and for l, i, k = 1. . . n.

For n = 3, the equation (1.1) is the equation in Rⁿ of motion of a non- relativistic particle of mass m = 1 and charge e = 1 in an external and static electromagnetic field described by (V, B) (see, for example, Section 17 of [LL2]). For the electromagnetic field the function V is an electric potential and B is the magnetic field. In this equation (1.1), x denotes the position of the particle, ˙xdenotes its velocity, ¨xdenotes its acceleration andtdenotes the time.

For equation (1.1) the energy E = 1

2|x(t)|˙ ²+V(x(t)) (1.3) is an integral of motion. Note that the energyE does not depend onBbecause the magnetic force B(x) ˙x is orthogonal to the velocity ˙x of the particle.

We assume that the electromagnetic coefficientsV andBare of short range onRⁿ. More precisely we assume that (V, B) satisfies the following conditions

|∂_x^j1V(x)| ≤ β|j1|(1 +|x|)^−α−|j1|, x∈Rⁿ, (1.4)

|∂_x^j2B_i,k(x)| ≤ β|j₂|+1(1 +|x|)^{−α−1−|j2|}, x∈Rⁿ, (1.5) for |j₁| ≤ 2,|j₂| ≤ 1, i, k = 1. . . n and some α > 1 (here j_l is the multiindex j_l = (j_l,1, . . . , j_l,n) ∈ (N ∪ {0})ⁿ, |j_l| = Pn

k=1j_l,k and β_|jl| are positive real constants). For example, conditions (1.4) and (1.5) are fulfilled for some real constants β_i, i = 1,2,3 when V and B are smooth and compactly supported on Rⁿ. Note also that the assumptions (1.4), (1.5) are actually similar to the assumptions of the works [HN], [ER], [Ni], [Ar] and [WY], where inverse scattering is considered for quantum non relativistic particle in electromagnetic field (or by other words for the Schr¨odinger equation in electromagnetic field).

1.2 Scattering data

Under conditions (1.4)–(1.5), the following is valid (see, for example, [S]

and see [LT] where classical scattering of particles in a long-range magnetic field is studied): for any (v₋, x₋)∈Rⁿ×Rⁿ, v₋ 6= 0, the equation (1.1) has a unique solution x∈C²(R,Rⁿ) such that

x(t) =v−t+x−+y−(t), (1.6) where ˙y−(t)→0, y−(t)→0, ast→ −∞; in addition for almost any (v−, x−)∈ Rⁿ×Rⁿ, v− 6= 0,

x(t) =v+t+x++y+(t), (1.7) wherev₊6= 0, v₊ =a(v−, x−), x₊ =b(v−, x−), y˙₊(t)→0, y₊(t)→0, ast → +∞.

The map S : (Rⁿ\{0})×Rⁿ→(Rⁿ\{0})×Rⁿ given by the formulas v₊ =a(v−, x−), x₊ =b(v−, x−), (1.8) is called 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).

By D(S) we denote the set of definition of S ; by R(S) we denote the range of S (by definition, if (v₋, x₋)∈ D(S), then v₋ 6= 0 anda(v₋, x₋)6= 0).

Under the conditions (1.4)–(1.5), the mapShas the following simple prop- erties: D(S) is an open set of Rⁿ×Rⁿ and Mes((Rⁿ×Rⁿ)\D(S)) = 0 for the Lebesgue measure on Rⁿ×Rⁿ ; the map S :D(S)→ R(S) is continuous and preserves the element of volume ; a(v−, x−)² = v²₋. (This latter property is seen to be a consequence of the conservation of energy (1.3) by combining the short range of V (1.4) for j₁ = (0, . . . ,0), and the asymptotics (1.6), (1.7) for (v−, x−)∈ D(S).)

IfV(x)≡0 andB(x)≡0, thena(v−, x−) =v−, b(v−, x−) =x−, (v−, x−)∈ Rⁿ×Rⁿ, v₋6= 0. Therefore for a(v₋, x₋), b(v₋, x₋) we will use the following representation

a(v−, x−) = v−+a_sc(v−, x−),

b(v−, x−) =x−+b_sc(v−, x−), (v−, x−)∈ D(S). (1.9) We will use the fact that, under the conditions (1.4)–(1.5), 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}.

This observation is based on the fact that if x(t) satisfies equation (1.1), then x(t+t₀) also satisfies (1.1) for anyt₀ ∈R.

1.3 X-ray transform

Consider

TSⁿ⁻¹ ={(θ, x)|θ ∈Sⁿ⁻¹, x∈Rⁿ, θx= 0}, where Sⁿ⁻¹ is the unit sphere in Rⁿ.

Consider the X-ray transform P which maps each function f with the properties

f ∈C(Rⁿ,R^m), |f(x)|=O(|x|^−β), as|x| → ∞, for someβ >1, into a function P f ∈C(TSⁿ⁻¹,R^m) where P f is defined by

P f(θ, x) = Z +∞

−∞

f(tθ+x)dt, (θ, x)∈TSⁿ⁻¹.

Concerning the theory of the X-ray transform, the reader is referred to [R], [GGG], [Na] and [No1].

1.4 Main results of the work

The main results of the present work consist in the small angle scattering estimates and asymptotics for the scattering data a_sc and b_sc (and scattering solutions) for the equation (1.1) and in application of these asymptotics and estimates to inverse scattering for the equation (1.1) at high energies. Our main results include, in particular, Theorem 1.1, Proposition 1.1 formulated below and Theorem 3.1 given in Section 3.1.

Theorem 1.1. Under conditions (1.4)–(1.5), we have

s→+∞lim a_sc(sθ, x) = W_1,1(B, θ, x) (1.10)

=

Z +∞

−∞

B(τ θ+x)θdτ, and

s→+∞lim s(a_sc(sθ, x)−W_1,1(B, θ, x)) =W_1,2(V, B, θ, x) (1.11)

=−P(∇V)(θ, x) +

+∞

Z

−∞

B(τ θ+x)(

τ

Z

−∞

B(σθ+x)θdσ)dτ

+

n

X

k=1

θk(Ω1,1,k(θ, x), . . . ,Ω1,n,k(θ, x))

for (θ, x)∈TSⁿ⁻¹, θ = (θ₁, . . . , θ_n), where Ω_1,i,k =

+∞

Z

−∞

∇B_i,k(x+τ θ)◦





τ

Z

−∞

σ

Z

−∞

B(ηθ+x)θdηdσ



dτ

for i, k = 1. . . n (◦ denotes the usual scalar product on Rⁿ); in addition, we have

s→+∞lim sb_sc(sθ, x) =W_2,1(B, θ, x) (1.12)

= Z 0

−∞

Z τ

−∞

B(σθ+x)θdσdτ − Z +∞

0

Z +∞

τ

B(σθ+x)θdσdτ,

s→+∞lim s(sb_sc(sθ, x)−W_2,1(B, θ, x)) =W_2,2(V, B, θ, x) (1.13)

=

0

Z

−∞

τ

Z

−∞

(−∇V(σθ+x))dσdτ −

+∞

Z

0 +∞

Z

τ

(−∇V(σθ+x))dσdτ

+

0

Z

−∞

τ

Z

−∞

B(σθ+x)





σ

Z

−∞

B(ηθ+x)θdη



dσdτ

−

+∞

Z

0 +∞

Z

τ

B(σθ+x)





σ

Z

−∞

B(ηθ+x)θdη



dσdτ

+

n

X

k=1

θ_k(Ω_2,1,k(θ, x), . . . ,Ω_2,n,k(θ, x)) for (θ, x)∈TSⁿ⁻¹, θ = (θ1, . . . , θn), where

Ω_2,i,k(θ, x) =

0

Z

−∞

τ

Z

−∞

∇B_i,k(σθ+x)◦





σ

Z

−∞

η1

Z

−∞

B(η₂θ+x)θdη₂dη₁



dσdτ

−

+∞

Z

0 +∞

Z

τ

∇B_i,k(σθ+x)◦





σ

Z

−∞

η1

Z

−∞

B(η₂θ+x)θdη₂dη₁



dσdτ

for i, k = 1. . . n (◦ denotes the usual scalar product on Rⁿ).

Theorem 1.1 gives the first two leading terms of the high energies asymp- totics of the scattering data. Theorem 1.1 follows from Theorem 3.1. Theorem 1.1 is proved in Section 3.2.

Note that Theorem 3.1 (see (3.16) and (3.18)) also gives the asymptotics of a_sc, b_sc, when the parameters α, n, s > 0, θ, x are fixed and the norm β_m decreases to 0 (where β_m = max(β₀, β₁, β₂)), that is Theorem 3.1 also gives the “Born approximation” for the scattering data at fixed energy when the electromagnetic field is sufficiently weak (see Section 3.3).

Proposition 1.1. Under conditions (1.4)–(1.5), the following statements are valid:

(i) W_1,1(B, θ, x) given for all (θ, x)∈TSⁿ⁻¹, uniquely determines B;

(ii) W_1,1(B, θ, x), W_1,2(V, B, θ, x) given for all (θ, x) ∈ TSⁿ⁻¹, uniquely de- termine (V, B);

(iii) if n ≥ 3 W_2,1(B, θ, x), W_2,2(V, B, θ, x) given for all (θ, x) ∈ TSⁿ⁻¹, uniquely determine (V, B);

(iv) if n = 2, then V and B are not uniquely determined by W_2,1(B, θ, x), W2,2(V, B, θ, x) given for all (θ, x)∈TSⁿ⁻¹.

Proposition 1.1 is proved in Section 3.5.

Using (1.10), (1.11) and Proposition 1.1 (i), (ii) we obtain that a_sc deter- mines uniquely ∇V and B at high energies. Actually for n ≥ 2 methods of reconstruction of f fromP f permit to reconstruct ∇V and B from the veloc- ity valued component a of the scattering map at high energies (see Sections 3.4, 3.5). The formulas (1.12), (1.13) and Proposition 1.1 (iii), (iv) show that the first two leading terms of the high energies asymptotics of b_sc do not de- termine uniquely (V, B) when n = 2 but that they uniquely determine (V, B) when n ≥ 3. Actually, (V, B) can be reconstructed from the first two leading terms of the high energies asymptotics of b_sc when n ≥ 3 (see Sections 3.4, 3.5).

1.5 Historical remarks

Note that inverse scattering for the classical multidimensional nonrela- tivistic Newton equation at high energies was first studied by Novikov [No1]

for B ≡ 0. Novikov proved, in particular, two formulas which link scattering data at high energies to the X-ray transform of −∇V and V. These formu- las are generalized by formulas (1.11), (1.13) of the present work for the case B 6≡0.

Developing Novikov’s approach [No1], the author also studied the inverse scattering for the relativistic multidimensional Newton equation at high ener- gies for B ≡ 0 [Jo1] and for B 6≡ 0 [Jo2]. In the relativistic case in contrast

with (1.10)–(1.11), the following formula was obtained in [Jo2, Theorem 1.1]:

lims→c s<c

s q

1− ^s_c²2

a^r_sc(sθ, x) = W₁^r(V, B, θ, x) (1.14)

:= −P(∇V)(θ, x) + Z +∞

−∞

B(τ θ+x)θdτ, for (θ, x)∈TSⁿ⁻¹ wherec >0 denotes the speed of light. In particular, one can see that in the relativistic case the leading high energies term ofa^r_scdepends on the both magnetic and electric fieldsB and ∇V, whereas in the nonrelativistic case the leading high energies term ofa_sc depends on the magnetic fieldB only (and is independent of the electric one). This difference is quite interesting in our opinion.

A similar difference occurs in quantim mechanics between inverse scat- tering at high energies for the nonrelativistic case (see [Ni], [Ar]) and inverse scattering at high energies for the relativistic case (see [I], [Ju], [Ha]).

Note also that for the classical multidimensional nonrelativistic Newton equation in a bounded open strictly convex domain an inverse boundary value problem at high energies was first studied in [GN].

Concerning the inverse problems for the classical multidimensional nonrel- ativistic Newton equation at fixed energy, we refer the reader to [GN], [No1], [Jo3], [DPSU] and references given in [No1], [Jo3].

Concerning the inverse problem for (1.1) in the one-dimensional case, we can mention the works [Ab], [K], [AFC].

Concerning the inverse scattering problem for a particle in electromagnetic field (with B 6≡ 0 or B ≡ 0) in quantim mechanics, see [HN], [I], [Ju], [ER], [Ni], [Ar], [Ha], [WY] and [No2] and references given in [Jo2].

1.6 Structure of the paper

Further, our paper is organized as follows. In Section 2 we transform the differential equation (1.1) with initial conditions (1.6) into a system of inte- gral equations which takes the form (y₋,y˙₋) =A_v−,x−(y₋,y˙₋). Then we study A_v−,x− on a suitable space and we give estimates for A_v−,x− and for (A_v−,x−)², and, in particular, contraction estimates for (A_v−,x−)² (Lemmas 2.1, 2.2, 2.3, 2.4). In Section 3.1 we give estimates and asymptotics for the deflection y₋(t) from (1.6) and for scattering data a_sc(v−, x−), b_sc(v−, x−) from (1.9) (Theorem 3.1). From these estimates and asymptotics the four formulas (1.10)–(1.13) will follow when the parametersβ_m, α, n,ˆv₋, x₋are fixed and|v₋|increases (where β|j|, α, n are constants from (1.4)-(1.5), β_m = max(β₀, β₁, β₂); ˆv− = v−/|v−|) (see the proof of Theorem 1.1 given in Section 3.2). In this case sup|ϑ(t)| de- creases, where ϑ(t) denotes the angle between the vectors ˙x(t) = v₋+ ˙y₋(t)

and v−, and we deal with small angle scattering. Note that, under the con- ditions of Theorem 3.1, without additional assumptions, there is the estimate sup|ϑ(t)|< ¹₄πand we deal with rather small angle scattering (concerning the term “small angle scattering” see [No1] and Section 20 of [LL1]). In Section 3.3 we also consider the “Born approximation” of the scattering data at fixed energy. In Section 3.4 we remind some results on inversion of the X-ray trans- formP and on reconstruction of a function f from its X-ray transformP f. In Section 3.5 we prove Proposition 1.1. In Section 4 we prove Lemmas 2.1, 2.2, 2.3. In Section 5, we prove Lemma 2.4.

2 Contraction maps

Ifxsatisfies the differential equation (1.1) and the initial conditions (1.6), then x satisfies the system of integral equations

x(t) = tv−+x−+ Z t

−∞

Z τ

−∞

F(x(s),x(s))dsdτ,˙ (2.1)

˙

x(t) = v−+ Z t

−∞

F(x(s),x(s))ds,˙ (2.2)

for t∈R, where F(x,x) =˙ −∇V(x) +B(x) ˙x, v₋ ∈Rⁿ\{0}.

For y−(t) of (1.6) this system takes the form

(y₋, u₋) = A_v−,x−(y₋, u₋)(t), (2.3) where u−(t) = ˙y−(t) and

A_v−,x−(f, h)(t) = (A¹_v−,x−(f, h)(t), A²_v−,x−(f, h)(t)), (2.4) A¹_v−,x−(f, h)(t) =

Z t

−∞

A²_v−,x−(f, h)(τ)dτ, (2.5) A²_v−,x−(f, h)(t) =

Z t

−∞

F(x−+τ v−+f(τ), v−+h(τ))dτ, (2.6) for v−∈Rⁿ\{0}.

From (2.3), (1.4)–(1.5) and y−(t) ∈ C¹(R,Rⁿ), |y−(t)|+|y˙−(t)| → 0, as t → −∞, it follows, in particular, that

(y−(t),y˙−(t))∈C(R,Rⁿ)×C(R,Rⁿ)

and |y˙−(t)|=O(|t|^−α), |y−(t)|=O(|t|^−α+1), as t→ −∞, (2.7)

For nonnegative real numbersRandr, consider the complete metric space M_{T ,R,r} ={(f, h)∈C(]− ∞, T],Rⁿ)×C(]− ∞, T],Rⁿ) |

sup

t∈]−∞,T]

|f(t)−th(t)| ≤r, sup

t∈]−∞,T]

|h(t)| ≤R}, (2.8) with the norm k.k∞,T defined by

k(f, h)k∞,T = max sup

t∈]−∞,T]

|f(t)−th(t)|, sup

t∈]−∞,T]

|h(t)|

!

, (2.9) whereT ∈]−∞,+∞] (ifT = +∞, ]−∞, T] must be replaced by ]−∞,+∞[)).

From (2.7) it follows that

(y₋(t),y˙₋(t))∈M_{T ,R,r} for some R and r depending on y₋(t) and T. (2.10) Lemma 2.1. Let R > 0, 0 < r ≤ 1 and let (v−, x−) ∈ Rⁿ×Rⁿ be such that |v−|>√

2R, v−x− = 0. Then under conditions (1.4)–(1.5), the following estimates are valid :

sup

t∈]−∞,T]

|A²_v−,x−(f, h)(t)| ≤ ρT ,2(n, β1, α,|v−|,|x−|, R) (2.11)

= 2^α+1β₁√

n(1 +√

n|v−|+√ nR) α(^|v√−|

2 −R)(1 +^|x√−|

2 + (^|v√−|

2 −R)|T|)^α, sup

t∈]−∞,T]

|A¹_v−,x−(f, h)(t)−tA²_v−,x−(f, h)(t)| ≤ρT ,1(n, β1, α,|v−|,|x−|, R) (2.12)

= 2^α+1β1

√n(1 +√

n|v−|+√ nR) (α−1)(^|v√−|

2 −R)²(1 + ^|x√−|

2 + (^|v√−|

2 −R)|T|)^α−1, for T ≤0 and (f, h)∈M_T,R,r;

sup

t∈]−∞,T]

|A²_v−,x−(f, h)(t)| ≤ ρ₂(n, β₁, α,|v₋|,|x₋|, R) (2.13)

= 2^α+2β₁√

n(1 +√

n|v−|+√ nR) α(^|v√−₂^|−R)(1 + ^|x√−₂^|)^α , sup

t∈]−∞,T]

|A¹_v−,x−(f, h)(t)−tA²_v−,x−(f, h)(t)| ≤ρ₁(n, β₁, α,|v−|,|x−|, R) (2.14)

= 2^α+2β₁√

n(1 +√

n|v₋|+√ nR) α(α−1)(^|v√−|

2 −R)²(1 + ^|x√−|

2 )^α−1. for T ≥0 and (f, h)∈M_T,R,r.

Remark 2.1. Note that for fixed n, β₁, α, |x−|, R, we have

ρ₁(n, β₁, α,|v−|,|x−|, R)→0, as|v−| →+∞; (2.15) ρ₂(n, β₁, α,|v−|,|x−|, R)→ 2^α+2√

2β₁n

α(1 + ^|x√−₂^|)^α, as|v−| →+∞ (2.16) (we used (2.13)–(2.14)).

Lemma 2.2. Let R > 0, 0 < r ≤ 1 and let (v−, x−) ∈ Rⁿ×Rⁿ be such that |v₋| > √

2R, v₋x₋ = 0. Then under conditions (1.4)–(1.5), for (f₁, h₁), (f₂, h₂)∈M_{T ,R,r}, the following contraction estimates are valid :

sup

t∈]−∞,T]

|A²_v−,x−(f₁, h₁)(t)−A²_v−,x−(f₂, h₂)(t)| ≤λ_4,T sup

t∈]−∞,T]

|h₁(t)−h₂(t)|

+λ_3,T sup

t∈]−∞,T]

|f₁(t)−f₂(t)−t(h₁(t)−h₂(t))| (2.17)

sup

t∈]−∞,T]

| A¹_v−,x−(f₁, h₁)−A¹_v−,x−(f₂, h₂) (t)

−t A²_v−,x−(f₁, h₁)−A²_v−,x−(f₂, h₂)

(t)| ≤λ_2,T sup

t∈]−∞,T]

|h₁(t)−h₂(t)|

+λ_1,T sup

t∈]−∞,T]

|f₁(t)−f₂(t)−t(h₁(t)−h₂(t))|, (2.18) for T ≤0, where λ_1,T, λ_2,T, λ_3,T and λ_4,T are given below by formulas (2.21)–

(2.24);

sup

t∈]−∞,T]

|A²_v−,x−(f₁, h₁)(t)−A²_v−,x−(f₂, h₂)(t)| ≤λ₄ sup

t∈]−∞,T]

|h₁(t)−h₂(t)|

+λ₃ sup

t∈]−∞,T]

|f₁(t)−f₂(t)−t(h₁(t)−h₂(t))|, (2.19)

sup

t∈]−∞,T]

| A¹_v−,x−(f₁, h₁)−A¹_v−,x−(f₂, h₂) (t)

−t A²_v−,x−(f1, h1)−A²_v−,x−(f2, h2)

(t)| ≤λ2 sup

t∈]−∞,T]

|h1(t)−h2(t)|

+λ₁ sup

t∈]−∞,T]

|f₁(t)−f₂(t)−t(h₁(t)−h₂(t))|, (2.20) for T ≥0, where λ₁, λ₂, λ₃ and λ₄ are given below by formulas (2.25).

Lemmas 2.1, 2.2 are proved in Section 4.

LetR >0, 0< r≤1 and let (v−, x−)∈Rⁿ×Rⁿbe such that|v−|>√ 2R.

ForT ≤0,real constants λ_i,T fori= 1. . .4, which appear in estimates (2.17)–

(2.18) given in Lemma 2.2, are defined by the following formulas:

λ_1,T(n, β₂, α,|v−|,|x−|, R) = 2^α+2nβ2(1 +√

n|v−|+√ nR) α(^|v√−|

2 −R)²(1 + ^|x√−|

2 + (^|v√−|

2 −R)|T|)^α, (2.21) λ_2,T(n, β₁, β₂, α,|v−|,|x−|, R) = 2^α+1n β₁(^|v√−₂^| −R) + 2β₂(1 +√

n|v−|+√ nR) (α−1)(^|v√−|

2 −R)³(1 + ^|x√−|

2 + (^|v√−|

2 −R)|T|)^α−1, (2.22)

λ_3,T(n, β₂, α,|v−|,|x−|, R) = 2^α+2nβ2(1 +√

n|v−|+√ nR) (α+ 1)(^|v√−|

2 −R)(1 + ^|x√−|

2 + (^|v√−|

2 −R)|T|)^α+1, (2.23) λ_4,T(n, β₁, β₂, α,|v−|,|x−|, R) = 2^α+1nβ₁(^|v√−|

2 −R) + 2β₂(1 +√

n|v−|+√ nR) α(^|v√−|

2 −R)²(1 + ^|x√−|

2 + (^|v√−|

2 −R)|T|)^α , (2.24) Real constants λ_i fori= 1. . .4, which appear in estimates (2.19)-(2.20) given in Lemma 2.2, are defined by the following formulas:

λ₁ = 2λ1,0

α+ 1, λ₂ = 2λ2,0

α , λ₃ = 2λ_3,0, λ₄ = 2λ_4,0. (2.25) We define real number λT(n, β1, β2, α,|v−|,|x−|, R) by

λT = max(λ1,Tλ3,T +λ2,Tλ3,T +λ3,Tλ4,T +λ²_4,T, (2.26) λ²_1,T +λ_1,Tλ_2,T +λ_2,Tλ_4,T +λ_2,Tλ_3,T),

for T ≤0 ; we define positive real number λ(n, β₁, β₂, α,|v₋|,|x₋|, R) by λ = max(λ₁λ₃+λ₂λ₃ +λ₃λ₄+λ²₄, (2.27)

λ²₁+λ₁λ₂+λ₂λ₄+λ₂λ₃).

Remark 2.2. Note that for fixed n, β₁, β₂,α, |x−|, R, T, we have λ_T(n, β₁, β₂, α,|v₋|,|x₋|, R) =O(|v₋|⁻¹), as |v₋| →+∞;

λ(n, β₁, β₂, α,|v−|,|x−|, R) =O(|v−|⁻¹), as|v−| →+∞. (2.28) Taking into account Lemma 2.1, Lemma 2.2, we obtain the following Corollary 2.1.

Corollary 2.1. Let R >0, 0< r≤1 and let (v−, x−)∈Rⁿ×Rⁿ be such that |v−|>√

2R, v−x− = 0. Then under conditions (1.4)–(1.5), the following statements are valid :

(i) for T ≤ 0, if max(^{ρT ,1}_r ,^{ρT ,2}_R ) ≤ 1 then (A_v−,x−)² is a map from M_{T ,R,r} into M_{T ,R,r} and (A_v−,x−)² satisfies the following inequality

k(A_v−,x−)²(f₁, h₁)−(A_v−,x−)²(f₂, h₂)k∞,T ≤λ_Tk(f₁−f₂, h₁−h₂)k∞,T, (2.29) for (f₁, h₁), (f₂, h₂)∈M_{T ,R,r} ;

(ii) if max(^ρ_r¹,^ρ_R²) ≤ 1 then for T = +∞, (A_v−,x−)² is a map from M_{T ,R,r} into MT ,R,r and (Av−,x−)² satisfies the following inequality

k(A_v−,x−)²(f₁, h₁)−(A_v−,x−)²(f₂, h₂)k∞,T ≤λk(f₁−f₂, h₁−h₂)k∞,T, (2.30) for (f₁, h₁), (f₂, h₂)∈M_{T ,R,r}.

(Constants ρ_{T ,1}, ρ_{T ,2}, ρ₁, ρ₂, λ_T andλare respectively defined by (2.12),(2.11), (2.14), (2.13), (2.26) and (2.27).)

Taking into account (2.10) and using Lemmas 2.1, 2.2, Corollary 2.1 (see also (3.2)–(3.4)) and the lemma about the contraction maps we will study the solution (y−(t), u−(t)) of the equation (2.3) in M_{T ,R,r}.

We will use also the following results (Lemmas 2.3, 2.4).

Lemma 2.3. Let conditions (1.4)–(1.5) be valid. Let R > 0, 0 < r ≤ 1 and let (v−, x−) ∈ Rⁿ× Rⁿ be such that |v−| > √

2R, v−x− = 0. Assume max(^ρ_R²,^ρ_r¹) ≤ 1 where ρ₁, ρ₂ are respectively defined by (2.14), (2.13). Then the following statements are valid :

[ A_v−,x−2

]₁(f, h)(t) = k_v−,x−(f, h)t+l_v−,x−(f, h) +H_v−,x−(f, h)(t), t≥0, (2.31) where A_v−,x−2

= ([ A_v−,x−2

]₁,[ A_v−,x−2

]₂) and

k_v−,x−(f, h) =

+∞

Z

−∞

F x−+sv−+A¹_v−,x−(f, h)(s), v−+A²_v−,x−(f, h)(s) ds,

(2.32)

lv−,x−(f, h) =

0

Z

−∞

s

Z

−∞

F x−+τ v−+A¹_v−,x−(f, h)(τ), v−+A²_v−,x−(f, h)(τ) dτ ds

−

+∞

Z

0 +∞

Z

s

F x−+τ v−+A¹_v−,x−(f, h)(τ), v−+A²_v−,x−(f, h)(τ)

dτ ds (2.33)

H_v−,x−(f, h)(t) =

+∞

Z

t +∞

Z

τ

F x−+τ v−+A¹_v−,x−(f, h)(τ), v−+A²_v−,x−(f, h)(τ) dτ ds,

(2.34) for t ≥ 0, (f, h) ∈ M_{T ,R,r}, T = +∞. In addition, the following estimates are valid :

|kv−,x−(f, h)| ≤ ρ2(n, β1, α,|v−|,|x−|, R), (2.35)

|l_v−,x−(f, h)| ≤ ρ₁(n, β₁, α,|v−|,|x−|, R), (2.36)

|H˙_v−,x−(f, h)(t)| ≤ ζ(n, β₁, α,|v−|,|x−|, R, t) (2.37)

= 2^α+1β1

√n(1 +√

n|v−|+√ nR) α(^|v√−|

2 −R)(1 + ^|x√−|

2 + (^|v√−|

2 −R)t)^α,

|H_v−,x−(f, h)(t)| ≤ ξ(n, β₁, α,|v−|,|x−|, R, t) (2.38)

= 2^α+1β₁√

n(1 +√

n|v−|+√ nR)

α(α−1)(^|v√−₂^|−R)²(1 + ^|x√−₂^| + (^|v√−₂^|−R)t)^α−1, fort ≥0,(f, h)∈M_T,R,r, T = +∞; in addition, for (f, h)∈M_{T ,R,r}, T = +∞, such that (f, h) = A_v−,x−(f, h), we have

|k_v−,x−(f, h)−k_v−,x−(0,0)| ≤ δ_1,1(n, β₁, β₂, α,|v−|,|x−|, R) (2.39)

= (λ₂λ₃+λ²₄)ρ₂+ (λ₁λ₃+λ₃λ₄)ρ₁,

|l_v−,x−(f, h)−l_v−,x−(0,0)| ≤ δ_2,1(n, β₁, β₂, α,|v−|,|x−|, R) (2.40)

= (λ₁λ₂+λ₂λ₄)ρ₂+ (λ²₁+λ₂λ₃)ρ₁, where λ₁, λ₂, λ₃, λ₄ are defined by (2.25).

Lemma 2.3 is proved in Section 4.

Remark 2.3. Note that for fixed n, β₁, β₂,α, |x₋|, R, we have δ_1,1(n, β₁, β₂, α,|v−|,|x−|, R) = O(|v−|⁻²), as|v−| →+∞, δ_2,1(n, β₁, β₂, α,|v−|,|x−|, R) = O(|v−|⁻³), as|v−| →+∞,

where δ_i,1, i = 1,2, are defined by (2.39)–(2.40) (we used (2.25) and (2.13)–

(2.14)).

Lemma 2.4. Let conditions (1.4)–(1.5) be valid. Let R > 0, 0 < r ≤ 1 and let (v−, x−) ∈ Rⁿ× Rⁿ be such that |v−| > √

2R, v−x− = 0. Assume max(^ρ_R²,^ρ_r¹) ≤ 1 where ρ₁, ρ₂ are respectively defined by (2.14), (2.13). Then

the following statements are valid:

|k_v−,x−(0,0)−w_1,v−,x−| ≤δ_1,2(n, β₁, β₂, α,|v−|,|x−|, R) (2.41)

= 2^α+4√

2n³(1 +√

n|v−|)(2α² +α−2)β₁(β₁+ 2β₂+β₁β₂) (α−1)α(α+ 1)^|v√−₂^|(^|v√−₂^|−R)²(1 + ^|x√−₂^|)^2α ,

|l_v−,x−(0,0)−w_2,v−,x−| ≤δ_2,2(n, β₁, β₂, α,|v−|,|x−|, R) (2.42)

= 2^α+5n³(2α+ 4)β₁(2β₂+β₁+β₁β₂)(1 +√ n|v−|) (α−1)α²(α+ 1)^|v√−|

2(^|v√−|

2 −R)³(1 + ^|x√−|

2)^2α−1 , where w_1,v−,x− and w_2,v−,x− are defined below by (2.43) and (2.45).

Lemma 2.4 is proved in Section 5.

Let (v−, x−) ∈ Rⁿ×Rⁿ, v− 6= 0. Let ˆv− = _|v^v−

−|, ˆv− = (ˆv₋¹, . . . ,vˆⁿ₋). Then vectorsw_1,v−,x− andw_2,v−,x−, which appear in (2.41) and (2.42), are defined by

w_1,v−,x− =

+∞

Z

−∞

B(τˆv−+x−)ˆv−dτ − 1

|v−|P(∇V)(ˆv−, x−)dτ (2.43) + 1

|v−|

+∞

Z

−∞

B(τvˆ₋+x₋)





τ

Z

−∞

B(σˆv₋+x₋)ˆv₋dσ



dτ

+ 1

|v−|

n

X

k=1

ˆ

v₋^k(Ω_3,1,k(v−, x−), . . . ,Ω_3,n,k(v−, x−)), where

Ω_3,i,k(v−, x−) =

+∞

Z

−∞

1

Z

0

∇B_i,k



τvˆ−+x−+ ε

|v−|

τ

Z

−∞

σ

Z

−∞

B(ηˆv−+x−)ˆv−dηdσ





◦





τ

Z

−∞

σ

Z

−∞

B(ηˆv−+x−)ˆv−dηdσ



dεdτ, (2.44)

for i, k = 1. . . n (◦ denotes the usual scalar product onRⁿ), and

w_2,v−,x− = 1

|v₋|





0

Z

−∞

τ

Z

−∞

B(σvˆ−+x−)ˆv−dσdτ −

+∞

Z

0 +∞

Z

τ

B(σˆv−+x−)ˆv−dσdτ





+ 1

|v−|²





0

Z

−∞

τ

Z

−∞

B(σˆv−+x−)





σ

Z

−∞

B(ηˆv−+x−)ˆv−dη



dσdτ (2.45)

−

+∞

Z

0 +∞

Z

τ

B(σˆv−+x−)





σ

Z

−∞

B(ηˆv−+x−)ˆv−dη



dσdτ





+ 1

|v−|²

n

X

k=1

ˆ

v₋^k(Ω4,1,k(v−, x−), . . . ,Ω4,n,k(v−, x−))

+ 1

|v−|²





0

Z

−∞

τ

Z

−∞

(−∇V(σvˆ−+x−))dσdτ −

+∞

Z

0 +∞

Z

τ

(−∇V)(σvˆ−+x−)dσdτ



,

where

Ω_4,i,k(v−, x−) = (2.46)

0

Z

−∞

τ

Z

−∞

1

Z

0

∇B_i,k



σˆv−+x−+ ε

|v−|

σ

Z

−∞

η1

Z

−∞

B(η₂vˆ−+x−)ˆv−dη₂dη₁





◦





σ

Z

−∞

η1

Z

−∞

B(η₂ˆv−+x−)ˆv−dη₂dη₁



dεdσdτ

−

+∞

Z

0 +∞

Z

τ 1

Z

0

∇B_i,k



σˆv−+x−+ ε

|v−|

σ

Z

−∞

η1

Z

−∞

B(η₂vˆ−+x−)ˆv−dη₂dη₁





◦





σ

Z

−∞

η1

Z

−∞

B(η₂ˆv−+x−)ˆv−dη₂dη₁



dεdσdτ

for i, k = 1. . . n (◦ denotes the usual scalar product onRⁿ).

3 Small angle scattering and inverse scatter- ing

3.1 Small angle scattering. Let the constants from (1.4)–(1.5) (β|j|, α and n) and r ∈]0,1] be fixed, and letrx be a nonnegative real number and letR be a positive number such that

R > 2^α+2√ 2β₁n α(1 + ^√rx

2)^α (3.1)

(see (2.16)). Consider the real numbersz₁ =z₁(n, β₁, α, R, r_x),z₂ =z₂(n, β₁, α, R, r, rx) and z3 = z3(n, β1, β2, α, R, rx) defined as the roots of the following equations

ρ1(n, β1, α, z1, rx, R)

r = 1, z₁ >√

2R, (3.2)

ρ₂(n, β₁, α, z₂, r_x, R)

R = 1, z2 >√

2R, (3.3)

λ(n, β₁, β₂, α, z₃, r_x, R) = 1, z₃ >√

2R, (3.4)

where ρ1, ρ2 and λ are respectively defined by (2.14), (2.13) and (2.27).

Note that from (2.14), (2.13) and (2.27) it follows that ρ₁(n, β₁, α, s₁, r_x, R)> ρ₁(n, β₁, α, s₂, r_x, R), for √

2R < s₁ < s₂, (3.5) ρ₂(n, β₁, α, s₁, r_x, R)> ρ₂(n, β₁, α, s₂, r_x, R), for √

2R < s₁ < s₂, (3.6) λ(n, β₁, β₂, α, s₁, r_x, R)> λ(n, β₁, β₂, α, s₂, r_x, R), for √

2R < s₁ < s₂; (3.7) in addition

ρ2(n, β1, α, s, rx, R1)

R₁ > ρ2(n, β1, α, s, rx, R2)

R₂ , for 0< R₁ < R₂ < s

√2. (3.8) As it was already mentioned in Introduction, under the conditions (1.4)–

(1.5), for any (v−, x−) ∈ Rⁿ×Rⁿ, v− 6= 0, the equation (1.1) has a unique solution x ∈ C²(R,Rⁿ) with the initial conditions (1.6). Consider the func- tion y₋(t) from (1.6). This function describes deflection from free motion. Us- ing Corollary 2.1, the lemma about contraction maps and estimate (2.11) of Lemma 2.1, and using Lemmas 2.3, 2.4 and the definition of z₁, z₂, z₃ given by (3.2)–(3.4), we obtain the following result.

Theorem 3.1.Let conditions (1.4)–(1.5)be valid. Letx−∈Rⁿand let0<

r ≤1. Let R >0and v ∈ ⁿ be such that R satisfies (3.1) (with “r ”=|x |)