Chung Shue Chen

Shing Wing

Wong
email: wswong@ie.cuhk.edu.hk

Chung Shue Chen
email: cschen@ieee.org

A ROBUST ACCESS PROTOCOL FOR WIRELESS SENSOR NETWORKS

This paper introduces a family of periodic binary sequences with interesting cross-correlation properties for potential applications in distributed wireless accessing. Following Masseys model of collision channel without feedback, a simple and robust multiple access protocol that does not require complicated processing such as back-off algorithms or random number generation is proposed for wireless sensor and ad hoc networks. In addition, energy-ef cient data reception is achievable by the protocol while continuous channel listening can be eliminated. Besides, a simple predictive collision avoidance scheme is presented for the enhancement of system throughput. Simulation studies are carried out to show its effectiveness.

I. INTRODUCTION

Technological advances in recent years enable the fabrication of miniaturized devices or sensors with built-in information processing for data communications. Building these devices into a coordinated network of sensing or controlling units could offer potentials for many innovative applications. However, these individual devices usually have quite limited processing and communication capabilities. Meanwhile, they have strict energy constraints. Existing protocols can be generally categorized into contention-based and TDMA-based solutions [START_REF] Akyildiz | A survey on sensor networks[END_REF][START_REF] Jurdak | A survey, classi cation and comparative analysis of medium access control protocols for ad hoc networks[END_REF]. When considering systems with dynamic network topology, sharing a radio channel among a large number of devices with the requirement of well-coordinated transmissions and time offsets could be very complicated particularly to thin devices.

For these reasons, it is desirable to have a simple multiple access protocol that does not require frequent monitoring of the channel for feedback information and can avoid complicated processing such as back-off algorithm or random number generation. This will provide an alternative for applications with strict system limitations. Designing such a network protocol can be traced back to Masseys work of collision channel without feedback [START_REF] Massey | The collision channel without feedback[END_REF] and the literature [47]. Some related works of binary sequence design and analysis for different communication applications can be found in [812].

Here, we focus on the recent development of a new family of periodic binary sequences, namely wobbling sequences [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF], and its applications. These sequences have nice cross-correlation properties that make them suitable for de ning multiple access protocols that do not require a feedback link. They can be easily generated and implemented. At the same time, each individual user can operate in a relatively distributed manner after an allocation of sequence key with respect to its requested transmission rate. However, the current design does not have a single peak auto-correlation property for sequence numbering acquisition. For the simplicity of operation, here we assume each data packet will contain a header of user identi cation and sequence number in a common practice. The rest of the paper is organized as follows. The system model is presented in Section II. The access protocol and resultant properties are described accordingly. A numerical analysis of the system performance is presented in Section III. For channel utilization ef ciency, a collision avoidance scheme has been investigated in Section IV. A concluding remark is drawn in Section V.

II. SYSTEM MODEL

Following Masseys model of collision channel without feedback [START_REF] Massey | The collision channel without feedback[END_REF], a communication channel is divided into time slots with equal duration. It is assumed that the channel is shared by M active users and each user follows a binary protocol sequence, W = {W (0), W (1), • • • }. An active user transmits a packet at time slot i if and only if W (i) = 1. Otherwise, it will be in an idle state. We assume the users know and align to the slot boundaries. However, users are not required to synchronize each other.

At any time slot, if more than one user transmit, the receiver is assumed to detect a collision and all transmitted packets in that time slot will be lost. On the other hand, if only one user transmits, it is assumed that the receiver can receive the packet correctly. Discussions on advanced coding and decoding schemes can be found in [START_REF] Massey | The collision channel without feedback[END_REF]. One simple approach to identify the sender is to require each packet to include a header that contains user identity and data sequence number similar to that in ALOHA. If the payload of a packet is large enough, the cost of overhead could be quite small.

The effective rate of a user is de ned as the fraction of packets it can send without suffering any collision. In the literature, many studies have focused on designing sequences that can provide high sum rate as well as support a large number of users [37]. In a random access system, one is usually most interested in the symmetric case in which all users are signaling at the same rate [START_REF] Massey | The collision channel without feedback[END_REF]. In this case, the capacity or effective sum rate of a slotsynchronized system will approach to e -1 as the number of active users, M , goes to in nity.

In addition to the system throughput, it is also important to look at the channel reliability for each individual user. It is favorable to have a design of protocol sequences which can provide reliability and robustness such that one can always ensure the availability of a communication channel for any of the active users as long as the sum of requested rates does not exceed a speci c bound. In the following, we will describe the details of wobbling sequences and their properties for wireless multiple access.

A. Linear Congruence Sequences

Let W = {W (i), i = 0, 1, 2, • • • } be a binary sequence.
The sequence W can also be represented by indexing the positions at which it has value 1, i.e., by

{I W (i), i = 1, 2, • • • } (1)
where I W (i) denotes the position at which the i-th entry of 1s in W appears. For a periodic binary sequence with a minimum period L, following [START_REF] Massey | The collision channel without feedback[END_REF], its duty factor is de ned by

r = 1 L L-1 i=0 W (i). (2)
Let b and l be two relatively prime integers with 0 ≤ b < l. A linear congruence sequence [START_REF] Titlebaum | Time-frequency hop signals, part I: coding based upon the theory of linear congruences[END_REF] generated by (b, l) can be de ned by

I W (i) = il + ib - ib l l (3)
where the integer b is known as the key generator of the sequence [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF]. It has a duty factor of 1/l. It is cyclic and has a minimum period of l 2 for b > 0. Note that, when b = 0, (3) can be written as

I W (i) = il. (4)
The sequence has a period of l.

When l is prime, the above sequence is also known as a prime sequence [START_REF] Shar | Prime sequences: quasi-optimal sequences for OR channel code division multiplexing[END_REF]. One may de ne the sub-sequence

W = {W (i), i = 0, 1, • • • , l 2 -1} (5)
as the core pattern of a linear congruence sequence. For example, the sequence W generated by (b = 2, l = 3) has a period of 9. The core pattern is given by {0, 0, 0, 0, 1, 0, 1, 0, 1}, while I W = {5, 7, 9, • • • }. Its duty factor is equal to 1/3.

B. Wobbling Sequences

Denote the linear congruence sequence generated by (b, l) by S b,l and let L represent the operator that shifts a sequence by one element to the left. Therefore,

LS(i) = S(i + 1). (6)
In the following, the discussion will focus on the case in which l = p i (7) for prime p and integer i, such that l is the power of a prime number. Consequently, we have the following de nitions.

Denition 1 [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF] Let l = p i , with i ≥ 2, be a prime power and b be an integer satisfying 0 < b < l. For 1 ≤ d < l, the wobbling sequence generated by (b, l) with duty factor d/l, W b,l,d , is de ned by

W b,l,d (i) = S b,l (i) ∨ L l S b,l (i) ∨ L 2l S b,l (i) ∨ • • • ∨ L (d-1)l S b,l (i) ( 8
)
where ∨ is the binary operator de ned by

0 ∨ 0 = 0 0 ∨ 1 = 1 ∨ 0 = 1 ∨ 1 = 1. ( 9
)
For b = 0, W b,l,d is de ned by W b,l,d (i) = S b,l (i) ∨ LS b,l (i) ∨ L 2 S b,l (i) ∨ • • • ∨ L d-1 S b,l (i). (10)
The duty factor of the above sequence W b,l,d is equal to d/l, while the minimum period is equal to l 2 . For example, given p = 3 and l = p 2 , the I W representation of S (b=1,l=9) is given by [START_REF] Prucnal | Spread spectrum beroptic local area network using optical processing[END_REF]20,30,40,50,60,70,80,81

• • • ) (13)
following [START_REF] Titlebaum | Time-frequency hop signals, part I: coding based upon the theory of linear congruences[END_REF]. Its duty factor is equal to 1/3 and the minimum period is 81.

C. Construction of the Sequence Family

Based on De nition 1, a family of wobbling sequences which can accommodate a large number of active users can be formulated in the following way [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF]. It also supports sequences with different duty factors for the purpose of multirate transmissions.

Denition 2 For any prime p and 0 < d ≤ p, construct a family of binary codes, F p (d), as follows:

1. The wobbling sequence generated by

(b = 0, l = p 2 ) is in F p (d).

For

i = 1, 2, • • • , let j = 2 i . For 0 < b < p, the wobbling sequence W b,p j ,d is in F p (d).
Consider an application of a nite subset of the code set F p (d) as protocol sequences, as long as the sum of duty factors does not exceed one, i.e.,

M j=1 r j ≤ 1, (14)
it can be shown [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF] that for any user i, a minimum number of Kr 2 i /p (15) transmission packets will not be blocked for this user within the common period encountered, says K slots. This unsuppressibility property ensures that all sensors can successfully transmit information to a guaranteed amount in every time period. For example, in the case that requested rates of users are symmetric such that

r i = 1/p (16)
for each user i, one can let d = p and employ wobbling sequences generated by (b, p 2 ),

where b = {0, 1, 2, • • • , p- 1}. Here, K = p 4 . (17)
As a result, we have a group of p different sequences with respect to different b. For each user, a number of at least p un-blocked slots are guaranteed for every cycle of p 4 slots. Based on the cross-correlation properties of wobbling sequences, a lower bound on the sum of effective rates of allowable users in the system can be derived [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF]. A numerical study on the system and individual throughputs of wobbling sequence based protocols in the model of collision channel without feedback is presented in the next section.

III. PERFORMANCE EVALUATION

An investigation on the effective rates of users is carried out. Both individual and system throughputs have been measured. We assume that there are M active users and each has a requested rate 1/p of the channel bandwidth. Their protocol sequences are used to specify their channel access time. Accordingly, a user will have a permission to transmit at a corresponding time slot if its protocol sequence refers to 1. Otherwise, it will be in an idle state.

To construct the scheme, we follow De nition 1. Let l = p 2 and d = p. The duty factor is equal to d/l, i.e., 1/p. By ( 8) and [START_REF] Prucnal | Spread spectrum beroptic local area network using optical processing[END_REF], a number of p wobbling sequences with

b = {0, 1, 2, • • • , p -1} (18)
are generated respectively. Since users are unsynchronized to each other, there will be a relative time shift between any two protocol sequences. For example, users may have different starting time. In simulations, we assume that the relative time shift between two protocol sequences is uniformly distributed in the encountered minimum period. In the above case, it is in the range of [0, p 4 -1]. One can nd that the number of combinations of relative time shifts among all the protocol sequences can be very large.

First, we observe the user throughputs (or effective rates) when the system is fully saturated, i.e.,

M i=1 r i = 1 (19)
where M = p and r i = 1/p. Fig. 1 shows the results of individual throughputs of users with respect to different p by a simulation of 10 6 runs. The minimum, mean and maximum obtained from wobbling sequence are plotted and compared with those from prime sequence and a random access respectively. A prime sequence is de ned by (3), while a random access here refers to the result of channel permissions which are determined by drawing with a probability of the desired duty factor r i from a uniform distribution.

Comparing the performance of wobbling and prime sequences, the means of their individual throughputs are almost equal. However, note that a user with prime sequence could be completely blocked by other active users in the system as shown in Fig. 1. Meanwhile, wobbling sequence has demonstrated its robustness that there cannot have enough collisions to completely block any particular sequence no matter how they are shifted with respect to one another as long as ( 14) is satis ed. It eliminates the risk of being completely blocked. The range of individual throughputs from simulation is indicated by [Min, Max], while the standard deviation is given in Fig. 2 in order to re!ect the degree of !uctuation. Both Fig. 1 and Fig. 2 show that the wobbling sequence has a much more stable performance than the prime sequence. However, it should be noted that the prime sequences have a shorter sequence period by definition, i.e., p 2 .

The effective rate of a user in the aforementioned random access is measured in a length of p 4 with reference to the common period encountered here by wobbling and prime sequences. As shown in Fig. 1 and Fig. 2, the performance of random access has quite similar behavior to that of wobbling sequences, especially when p is relatively large. It is reasonable that the means of the three schemes are almost the same after taking averages over the runs with respective to the uniformly distributed relative time shifts. As shown in Fig. 1, the wobbling sequence can guarantee a higher individual minimum throughput in general. However, the difference is insigni cant for p ≥ 7 when comparing with the random access. Also, it should be noted that a user in the random access has always a non-zero probability of being completely blocked. The probability would depend on the period considered.

Fig. 3 shows the system throughputs. It indicates similar phenomena as those occurred in Fig. 1. Besides, the system throughput will tend to e -1 as p goes to in nity. In addition to the mean value, Fig. 4 shows the standard deviation of system throughputs. By Fig. 1234, one can nd that the wobbling sequences can provide a very robust and stable service. Meanwhile, it guarantees the non-totally-blockage for every user. In addition to the above studies in symmetric rates, two case studies on multi-rate transmissions are conducted. The result is included in the Appendix for reference.

Furthermore, we have carried out a study on the system performance in different traf c loads. In the simulation, we assume that in any time instant there are at most p active users and each of them transmits following the wobbling sequences with duty factor 1/p over a xed duration of the sequence period p 4 . It is modeled as a number of p arrival processes, which correspond to the number of deployed wobbling sequences in the system. The idle time between two active users in a process is assumed to have an exponential distribution. Fig. 5 shows the system throughputs for p = 3, 5 and 7. The offered traf c is de ned as the average number of packets generated (or arrived) to the system per transmission time. The result shows similar characteristics with respect to slotted ALOHA.

To draw a summary, the wobbling sequence has the following major advantages. In contrast to the other, these sequences hold the feature of non-totally-blockage and can guarantee the availability of a communication channel for every user. Secondly, this robust access protocol does not require complicated processing of random number generation. Thirdly, as wobbling sequences are deterministic, a receiver does not need to continuously listen the channel in data reception. Once a packet is successfully received, the receiver can explicitly address to its coming packets since the transmitters wobbling sequence can be identi ed or reconstructed simply based on the information of (b, l, d) and sequence index in a header. Such an operation could be more energy-ef cient. Besides, by the periodicity of wobbling sequence, there is a room to avoid some potential channel collisions. A simple predictive scheme is presented in the next section.

IV. THROUGHPUT ENHANCEMENT

One can nd that a substantial amount of transmissions will be corrupted by packet collisions due to the nature of such an unsynchronized multiple access model. A simple listenand-avoid scheme is proposed in the following so as to enhance the effective transmission rates, particularly when the system throughput is a major concern. The idea can be integrated into the aforementioned protocol of wobbling sequence easily. From another point of view, the result also contributes to an improvement in the transmission power ef ciency for energy-constraint sensor devices.

To avoid channel collisions with existing users, a new arrival is set to listen for a duration of the minimum common period before transmitting. After channel sensing, this user will exclude its predictively unnecessary transmission slots by comparing its protocol sequence with the existing transmission activities in order to avoid coming potential collisions. In other words, the new arrival (says, user k) will follow a modi ed wobbling sequence, W (k) , which is expressible as

W (k) = W (k) -W (k)   j =k W (j)   (20)
where W (j) denotes the wobbling sequence employed by existing user j. This listen-and-avoid strategy can be manipulated individually in a distributed manner.

More importantly, such an avoidance could also help to eliminate its interference to existing active users based on the periodicity of wobbling sequences. It should be noted that the above strategy will be effective if users generally transmit in a relatively long time period as compared to their sequence period and the channel sensing duration. Otherwise, one should consider other techniques such as protocol sequence re-arrangement and precise contention resolution, if allowable. Besides, when a user nds its effective rate unsatis ed, it could be helpful to have an adaptive sequence re-scheduling for an improvement.

In simulation, we follow the same queuing model as that in Fig. 5, in which it is assumed that there are p wobbling sequences available and each has a duty factor 1/p. However, the channel holding time of an active user here is assumed to have an exponential distribution with expected value of ten times of the sequence period p 4 . We have investigated the system throughput in this model and compared the results achieved with and without the described collision avoidance mechanism respectively. As shown in Fig. 6, the system throughput enhancement is signi cant, especially in heavily loaded scenarios. The comparison indicates a substantial improvement in general due to the cooperative behavior among users. This consequently leads to an overall capacity gain. However, it should be noted that the effectiveness would generally depend on the system arrival and departure characteristics. It is worth of a further investigation.

V. CONCLUSION

In this paper, a simple and robust multiple access protocol which does not require complicated processing such as back-off algorithms or random number generation is introduced for wireless sensor and ad hoc networks. It offers an effective approach of distributed wireless access that can ensure the availability of a communication channel for any of the active users as long as the sum of duty factors does not exceed one. There cannot have enough collisions to completely block any particular user no matter how they are shifted with respect to one another. Simulation shows that both the individual and system throughputs have stable and satisfactory performance. Meanwhile, continuous channel listening during data reception can be avoided. In addition, a simple predictive collision avoidance scheme is proposed for the enhancement of system throughput. Note that it requires a channel listening in advance. The study here may also lead to other interesting schemes.

APPENDIX

Figure

  Figure The minimum, mean and maximum individual throughputs of users (r i = 1/p) from simulation are shown.

Figure 2 :

 2 Figure 2: The standard deviation of individual throughputs of users is shown. The result is associated with Fig. 1.

Figure 3 :

 3 Figure 3: The system throughput is plotted with respect to different prime p.

Figure 4 :

 4 Figure 4: The standard deviation of system throughputs is shown. The result is associated with Fig. 3.

Figure 5 :

 5 Figure 5: System throughput with respect to different offered channel traf c.

Figure 6 :

 6 Figure 6: The system throughputs achieved with and without the collision avoidance mechanism are compared.

Table 1 :

 1 Multi-rate transmissions with sequences: two 1/3, one 2/9, two 1/27. The requested rate sum = 26/27. The result is obtained from a number of 10 5 runs.

		Wobbling Random	cf
	mean sys. throughput	0.437057 0.436812 26/27
	mean indiv. throughput 0.087411 0.087362	-
	max. indiv. throughput 0.182899 0.172077	1/3
	min. indiv. throughput 0.011584 0.007926 1/27

Table 2 :

 2 Multi-rate transmissions with sequences: four 1/5, one 4/25, four 1/125. The requested rate sum = 124/125. The result is obtained from a number of 10 5 runs.

		Wobbling Random	cf
	mean sys. throughput	0.429626 0.407403 124/125
	mean indiv. throughput 0.047736 0.045267	-
	max. indiv. throughput 0.100687 0.084723	1/5
	min. indiv. throughput 0.002486 0.002383	1/125

The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project no. CUHK416906).


Wing Chung Shue Chen

Shing Wong

Resource Sharing in the Most Regular Scheduling: Deterministic Performance and Guarantee

published or not. The documents may come   L'archive ouverte pluridisciplinaire


Alexandre Jollivet

Inverse scattering at fixed energy for the multidimensional Newton equation in short range radial potentials

 we obtain a uniqueness result when B is assumed to be zero in a neighborhood of infinity and V is assumed to be spherically symmetric in a neighborhood of infinity.

Introduction

Consider the following second order differential equation that is the multidimensional nonrelativistic Newton equation with electromagnetic field ẍ(t) = F (x(t), ẋ(t)) := -∇V (x(t)) + B(x(t)) ẋ(t), (1.1) where x(t) ∈ R n , ẋ(t) = dx dt (t). In this equation we assume that V ∈ C 2 (R n , R) and for any x ∈ R n , B(x) is a n × n antisymmetric matrix with elements B i,k (x), B i,k ∈ C 1 (R n , 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 = (x 1 , . . . , x n ) ∈ R n and for l, i, k = 1 . . . n.

For n = 3, the equation (1.1) is the equation of motion in R n of a nonrelativistic particle of mass m = 1 and charge e = 1 in an external and static electromagnetic field described by (V, B) (see, for example, [START_REF] Landau | The Classical Theory of Fields Pergamon[END_REF]Section 17]).

For the electromagnetic field the function V is an electric potential and B is the magnetic field. Then x denotes the position of the particle, ẋ denotes its velocity, ẍ denotes its acceleration and t denotes the time.

For the equation (1.1) the energy

E = 1 2 | ẋ(t)| 2 + V (x(t)) (1.3)
is an integral of motion.

We assume that the electromagnetic coefficients V and B are short range. More precisely we assume that (V, B) satisfies the following conditions

|∂ j 1 x V (x)| ≤ β |j 1 | (1 + |x|) -α-|j 1 | , x ∈ R n , (1.4
)

|∂ j 2 x B i,k (x)| ≤ β |j 2 |+1 (1 + |x|) -α-1-|j 2 | , x ∈ R n , (1.5)
for |j 1 | ≤ 2, |j 2 | ≤ 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 | = n k=1 j l,k and β |j l | are positive real constants). We denote by . the norm on the short range electromagnetic fields defined by

(V, B) = sup x∈R n , j 1 ∈N n |j 1 |≤2 (1 + |x|) α+|j 1 | |∂ j 1 x V (x)| (1.6)
+ sup

x∈R n , j 2 ∈N n |j 2 |≤1, i,k=1...n

(1 + |x|) α+1+|j 2 | |∂ j 2
x B i,k (x)| .

Under conditions (1.4)-(1.5), we have the following properties (see, for example, [START_REF] Simon | Wave operators for classical particle scattering[END_REF] and [START_REF] Loss | Scattering of particles by long-range magnetic fields[END_REF] where classical scattering of particles in a short-range electric field and in a long-range magnetic field are studied respectively): for any (v -, x -) ∈ R n × R n , v -= 0, the equation (1.1) has a unique solution x ∈ C 2 (R, R n ) such that

x(t) = tv -+ x -+ y -(t), (1.7)
where | ẏ-(t)|+|y -(t)| → 0, as t → -∞; in addition for almost any (v -, x -) ∈ R n ×R n , v -= 0, the unique solution x(t) of equation (1.1) that satisfies (1.7) also satisfies the following asymptotics

x(t) = tv + + x + + y + (t), (1.8)
where

v + = 0, | ẏ+ (t)| + |y + (t)| → 0, as t → +∞. At fixed energy E > 0, we denote by S 1 E the set {v -∈ R n | |v -| 2 = 2E}
and we denote by D(S E ) the set of (v -, x -) ∈ S 1 E × R n for which the unique solution x(t) of equation (1.1) that satisfies (1.7) also satisfies (1.8). We have that

D(S E ) is an open set of S 1 E × R n and Mes((S 1 E × R n )\D(S E )) = 0 for the Lebesgue measure on S 1 E × R n . The map S E : D(S E ) → S 1 E × R n given by the formula S E (v -, x -) = (v + , x + ), (1.9)
is called the scattering map at fixed energy E > 0 for the equation (1.1). Note that if V (x) ≡ 0 and B(x

) ≡ 0, then v + = v -, x + = x -, (v -, x -) ∈ R n × R n , v -= 0.
In this paper we consider the following inverse scattering problem at fixed energy

Given S E at fixed energy E > 0, find (V, B).

(1.10)

Note that using the conservation of energy we obtain that if E < sup R n V then S E does not determine uniquely V .

We mention results on Problem (1.10). When B ≡ 0 and V is assumed to be spherically symmetric and monotonuous decreasing in |x| (V is not assumed to be short range), uniqueness results for Problem (1.10) were obtained in [START_REF] Firsov | Determination of the force acting between atoms via differential effective elastic cross section[END_REF][START_REF] Keller | Determination of the potential front scattering data[END_REF]. The scattering map S E also uniquely determines (V, B) at fixed and sufficiently large energy when (V, B) is assumed to be compactly supported inside a fixed domain of R n (see [START_REF] Novikov | Small angle scattering and X-ray transform in classical mechanics[END_REF] for B ≡ 0 and see [START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF]). This latter result relies on a uniqueness result for an inverse boundary kinematic problem for equation (1.1) (see [START_REF] Gerver | Inverse problem of mechanics at high energies[END_REF][START_REF] Novikov | Small angle scattering and X-ray transform in classical mechanics[END_REF] when B ≡ 0, and see [START_REF] Dairbekov | The boundary rigidity problem in the presence of a magnetic field[END_REF][START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF]) and connection between this boundary value problem and the inverse scattering problem on R n (see [START_REF] Novikov | Small angle scattering and X-ray transform in classical mechanics[END_REF] for B ≡ 0, and see [START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF]).

To our knowledge it is still unknown whether the scattering map at fixed and sufficiently large energy uniquely determine the electromagnetic field under the regularity and short range conditions (1.4) and (1.5) (see [16, Conjecture B] for B ≡ 0).

In this paper we propose a generalization of results in [START_REF] Firsov | Determination of the force acting between atoms via differential effective elastic cross section[END_REF][START_REF] Keller | Determination of the potential front scattering data[END_REF] for the short range case where no decreasing monotonicity is assumed. More precisely we have the following uniqueness result.

Theorem 1.1. Let (λ, R) ∈ (0, +∞) 2 and let (V, B) be an electromagnetic field that satisfies the assumptions (1.2), (1.4) and (1.5) and (V, B) ≤ λ. Assume that B ≡ 0 outside B(0, R) and that V is spherical symmetric outside B(0, R). Then there exists a positive constant E(λ, R) (which does not depend on (V, B)) so that the scattering map at fixed energy E > E(λ, R) uniquely determines (V, B) on R n .

The proof of Theorem 1.1 is obtained by recovering first the electric potential in a neighborhood of infinity using Firsov or Keller-Kay-Shmoys' result [START_REF] Firsov | Determination of the force acting between atoms via differential effective elastic cross section[END_REF][START_REF] Keller | Determination of the potential front scattering data[END_REF] and then by recovering the electromagnetic field on R n using the following proposition which generalizes [START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF]Theorem 7.2]. Proposition 1.2. Let (λ, R) ∈ (0, +∞) 2 and let (V, B) be an electromagnetic field that satisfies the assumptions (1.2), (1.4) and (1.5) and (V, B) ≤ λ. Assume that (V, B) is known outside B(0, R). Then there exists a positive constant E(λ, R) so that the scattering map at fixed energy E > E(λ, R) uniquely determines (V, B) on R n .

Concerning the inverse scattering problem for the classical multidimensional nonrelativistic Newton equation at high energies and the inverse scattering problem for a particle in electromagnetic field (with B ≡ 0 or B ≡ 0) in quantum mechanics, we refer the reader to [START_REF] Gerver | Inverse problem of mechanics at high energies[END_REF][START_REF] Novikov | Small angle scattering and X-ray transform in classical mechanics[END_REF][START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF][START_REF] Jollivet | On inverse scattering at high energies for the multidimensional nonrelativistic Newton equation in electromagnetic field[END_REF] and references therein.

Concerning the inverse problem for (1.1) in the one-dimensional case, we can mention the works [START_REF] Abel | Auflösung einer mechanischen Aufgabe[END_REF][START_REF] Keller | Inverse problems[END_REF][START_REF] Astaburuaga | The direct and inverse problem in Newtonian Scattering[END_REF].

The structure of the paper is as follows. In section 2 we prove Proposition 1.2. In section 3 we prove Theorem 1.1. In section 4 we provide similar results for the relativistic multidimensional Newton equation with electromagnetic field.

2 Proof of Proposition 1. We will use the standard Lemma 2.1 on nontrapped solutions of equation (1.1). For sake of consistency its proof is given in Appendix.

Lemma 2.1. Let E > 0 and let R E and C E be defined by

C E := 2E (nβ 1 + 2β 0 )(1 + 2(E + β 0 )) , (2.1)
sup

|x|≥R E (1 + |x|) -α ≤ C E 2 . (2.2) If x(t) is a solution of equation (1.1) of energy E such that |x(0)| < R E and if there exists a time T > 0 such that x(T ) = R E then |x(t)| 2 ≥ R 2 E + E|t -T | 2 for t ∈ (T, +∞), (2.3)
and there exists a unique

(x + , v + ) ∈ R n × S 1 E so that x(t) = x + + tv + + y + (t), t ∈ R, where |y + (t)| + | ẏ+ (t)| → 0 as t → +∞.
Note that C E → +∞ as E → +∞ while sup |x|≥R (1+|x|) -α is a decreasing function of R that goes to 0 as R → +∞. Note that Lemma 2.1 is stated for positive times t but a similar result hold for negative times t.

The inverse kinematic problem for equation (1.1)

We first formulate the inverse kinematic problem for equation (1.1) inside a ball of center 0 and radius R > 0 denoted by B(0, R). For (m, l) ∈ (N\{0}) 2 and for a function

f from B(0, R) to R m of class C l we define the C l norm of f by f C l ,R = sup x∈B(0,R), α∈N n |α|≤l |∂ α x f (x)|.
We denote by ∂B(0, R) the boundary of the ball B(0, R).

Then we recall that there exists a constant

E(R, V C 2 ,R , B C 1 ,R ) so that at fixed energy E > E(R, V C 2 ,R , B C 1 ,R
) the solutions x of equation (1.1) in B(0, R) at energy E have the following properties (see for example [START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF]):

for each solution x(t) there are t 1 , t 2 ∈ R, t 1 < t 2 , such that x ∈ C 3 ([t 1 , t 2 ], R n ), (x(t 1 ), x(t 2 )) ∈ ∂B(0, R) 2 , x(t) ∈ B(0, R) for t ∈]t 1 , t 2 [, x(s 1 ) = x(s 2 ) for s 1 , s 2 ∈ [t 1 , t 2 ], s 1 = s 2 ;
(2.4) and for any two distinct points q 0 , q ∈ ∂B(0, R), there is one and only one solution x(t) = x(t, E, q 0 , q) such that x(0) = q 0 , x(s) = q for some s > 0.

(2.5) This is closely related to the property that at fixed and sufficiently large energy E, the compact set B(0, R) endowed with the riemannian metric E -V (x)|dx| and the magnetic field defined by B is simple (see [START_REF] Dairbekov | The boundary rigidity problem in the presence of a magnetic field[END_REF]). For (q 0 , q) two distinct points of ∂B(0, R) we denote by s(E, q 0 , q) the time at which x(t, E, q 0 , q) reaches q from q 0 and we denote by k 0 (E, q 0 , q) the velocity vector ẋ(0, E, q 0 , q) and by k(E, q 0 , q) the velocity vector ẋ(s(E, q 0 , q), E, q 0 , q). The inverse kinematic problem is then Given k(E, q 0 , q), k 0 (E, q 0 , q) for all q 0 , q ∈ ∂B(0, R), q 0 = q, at fixed sufficiently large energy E, find (V, B) in B(0, R).

The data k 0 (E, q 0 , q), k(E, q 0 , q), q 0 , q ∈ ∂B(0, R), q 0 = q, are the boundary value data of the inverse kinematic problem, and we recall the following result.

Lemma 2.2 (see, for example, Theorem 7.1 in [START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF]).

At fixed E > E( V C 2 ,R , B C 1 ,R , R
), the boundary data k 0 (E, q 0 , q), (q 0 , q) ∈ ∂B(0, R) × ∂B(0, R), q 0 = q, uniquely determine (V, B) in B(0, R).

Relation between boundary data of the inverse

kinematic problem and the scattering map S E

We will prove that at fixed and sufficiently large energy E the scattering map S E determines the boundary data k 0 (E, q 0 , q), k(E, q 0 , q), q 0 , q ∈ ∂B(0, R), q 0 = q. This will prove that S E uniquely determines (V, B) in B(0, R), which will prove Proposition 1.2. Let R > 0 and λ > 0 be such that (V, B) are known outside B(0, R) and

(V, B) < λ. Note that max( V C 2 ,R , B C 1 ,R ) ≤ (V, B) < λ.
Thus there exists a constant E 0 (λ, R) such that at fixed energy E > E 0 (λ, R) solutions x(t) of equation (1.1) in B(0, R) at energy E have properties (2.4) and (2.5) and such that at fixed E > E 0 (λ, R) the boundary data k 0 (E, q 0 , q), (q 0 , q) ∈ ∂B(0, R) × ∂B(0, R), q 0 = q, uniquely determine (V, B) in B(0, R). Then using that the constant C E → +∞ as E → +∞ in Lemma 2.1 we obtain that there exists

E 1 (λ, R) such that for E > E 1 (λ, R) we have sup |x|≥R (1+|x|) -α ≤ C E 2 so that R E can be replaced by R in Lemma 2.1. Set E(λ, R) = max(E 0 (λ, R), E 1 (λ, R)) and fix E > E(λ, R). Let (x -, v -) ∈ D(S E ) and (v + , x + ) = S E (v -, x -). We denote by x(., v -, x -) the solution of equation (1.1) that satisfies (1.7) (and (1.8)). Set t -(x -, v -) = sup{t ∈ R | |x(s, x -, v -)| ≥ R, s ∈ (-∞, t)}, t + (x -, v -) = inf{t ∈ R | |x(s, x -, v -)| ≥ R, s ∈ (t, +∞)}.
Since (V, B) is known outside B(0, R) we can solve equation (1.1) with initial conditions (1.7) and (1.8) and we obtain that x(., x -, v -) is known on

(-∞, t -(x -, v -)] ∪ [t + (x -, v -), ∞). If x(s, x -, v -) ∈ B(0, R) for any s ∈ R, then t ± (x -, v -) = ∓∞. If there exists s ∈ R such that x(s, x -, v -) ∈ B(0, R) then set q 0 = x(t -(x -, v -)), q = x(t + (x -, v -)).
(2.6)

Using Lemma 2.1 and E > E 1 (λ, R) we obtain that |x(s, x -, v -)| < R for s ∈ (t -(x -, v -), t + (x -, v -)
) and q 0 = q. (Note that if x(t) satisfies equation (1.1) then x(t + t 0 ) also satisfies (1.1) for any t 0 ∈ R.) Therefore we have

x(s, x -, v -) = x(s-t -(x -, v -), E, q 0 , q) for s ∈ (t -(x -, v -), t + (x -, v -))
where x(t, E, q 0 , q) is the solution of (1.1) given by (2.5), and we have

k 0 (E, q 0 , q) = ẋ(t -(x -, v -)), k(E, q 0 , q) = ẋ(t + (x -, v -)). (2.7)
We proved that the scattering map S E uniquely determines the data k 0 (E, q 0 , q), k(E, q 0 , q), (q 0 , q) ∈ ∂B(0, R) 2 , q 0 = q, when (q 0 , q) = (x(t

-(x -, v -)), x(t + (x -, v -)))
for (x -, v -) ∈ D(S E ). And using again Lemma 2.1 we know that for any (q 0 , q) ∈ ∂B(0, R) 2 , q 0 = q, the solution x(t, E, q 0 , q) given by (2.5) satisfies (1.7) and (1.8) for some (x ± , v ± ) ∈ R n × S 1 E and that |x(t, E, q 0 , q)| > R for t < 0 and t > s(E, q 0 , q). Thus S E uniquely determines the data k 0 (E, q 0 , q), k(E, q 0 , q), (q 0 , q) ∈ ∂B(0, R) 2 , q 0 = q.

3 Proof of Theorem 1.1

In this section we assume that the electromagnetic field (V, B) in equation (1.1) satisfies (1.4) and (1.5) and is so that B ≡ 0 and V is spherically symmetric outside B(0, R) for some

R > 0. Let W ∈ C 2 ([R, +∞), R) be defined by V (x) = W (|x|) for x ∈ B(0, R). From (1.4) it follows that sup r>R (1 + r) α |W (r)| ≤ β 0 and sup r>R (1 + r) α+1 |W ′ (r)| ≤ β 1 , (3.1)
where W ′ denotes the derivative of W .

3.1 The function r min,.

Set

β := (2E + 2β 0 ) 1 2 max R, β 1 + 2β 0 2E 1 α . (3.2)
Then for q ≥ β consider the real number r min,q defined by r min,q = sup{r ∈ (R, +∞) | W (r) + q 2 2r 2 = E}.

(3.

3)

The function r min,. has the following properties.

Lemma 3.1. The function r min,. is a C 2 strictly increasing function from [β, +∞) to (R, +∞) and we have r 3 min,q W ′ (r min,q ) < q 2 for q ≥ β and W (r min,q ) + q 2 2r 2 min,q = E, dr min,q dq = qr min,q q 2 -r 3 min,q W ′ (r min,q ) > 0.

(3.4)

In addition the following estimates and asymptotics at +∞ hold

q √ 2E + 2β 0 ≤ r min,q ≤ q 2E -2β 0 q -α (2β 0 + 2E) α 2 1 2
, for q ∈ [β, +∞), (3.5) r min,q = q √ 2E + O(q 1-α ), as q → +∞.

(3.6) Lemma 3.1 and works [START_REF] Firsov | Determination of the force acting between atoms via differential effective elastic cross section[END_REF][START_REF] Keller | Determination of the potential front scattering data[END_REF] allow the reconstruction of the force F in a neighborhood of infinity. In sections 3.2 and 3.3 we develop the reconstruction procedure given in [START_REF] Firsov | Determination of the force acting between atoms via differential effective elastic cross section[END_REF][START_REF] Keller | Determination of the potential front scattering data[END_REF].

The scattering angle

Let P be a plane of R n containing 0 and let (e 1 , e 2 ) be an orthonormal basis of P.

For (v 1 , v 2 ) ∈ R 2 and for v = v 1 e 1 + v 2 e 2 we define v ⊥ ∈ P by v ⊥ = -v 2 e 1 + v 1 e 2 .
Let q ≥ β. Then set x -:= (2E) -1 2 qe 1 and v -= √ 2Ee 2 . We have x -•v -= 0 and x -• v ⊥ -= -q, and for such couple (x -, v -) we consider x q (t) the solution of (1.1) with energy E and with initial conditions (1.4)

at t → -∞. Let t -:= sup{t ∈ R | |x q (s)| ≥ R for s ∈ (-∞, t)}.
We will prove that t -= +∞. Since the force F in (1.1) is radial outside B(0, R) we obtain that x q (t) ∈ P for t ∈ (-∞, t -).

We introduce polar coordinates in P. We write x q (t) = r q (t)(cos(θ q (t))e 1 (t)+ sin(θ q (t))e 2 ) for t ∈ (-∞, t -) where the functions r q , θ q , satisfy the following ordinary differential equations

rq (t) = -W ′ (r q (t)) + q ′2 r q (t) 3 , (3.7)
r q (t) 2 θq (t) = q ′ , for some q ′ ∈ R.

Asymptotic analysis of x q (t) at t = -∞ using the initial conditions (1.4) and ẏ-(t) = o(t -1 ) as t → -∞ (see for example [START_REF] Novikov | Small angle scattering and X-ray transform in classical mechanics[END_REF]Theorem 3.1] for this latter property) shows that q ′ = q. We refer the reader to the Appendix for details.

The energy E defined by (1.3) is then written as follows 2E = ṙq (t) 2 + q 2 r q (t) 2 + 2W (r q (t)).

(3.9)

Let t q = inf{t ∈ (-∞, t -) | r q (t) = r min,q }. Then using (3.3) and (3.9) we have ṙq (t q ) = 0. Since r q satisfies the second order differential equation (3.7) we obtain that t -= +∞, r q (t q + t) = r q (t q -t) for t ∈ R, and ± ṙq (t) > 0 for ±t > t q . We thus define

g(q) = +∞ -∞ dt r q (t) 2 = 2 +∞ tq dt r q (t) 2 .
(3.10)

The integral (3.10) is absolutely convergent and from (3.8) it follows that qg(q) = R θq (s)ds is the scattering angle of x q (t), t ∈ R. Note also that from (3.8) we have

S 1,E ( √ 2Ee 2 , (2E) -1 2 qe 1 ) = √ 2E cos qg(q) + π 2 , sin qg(q) + π 2 , (3.11)
for q ∈ [β, +∞) and where S 1,E is the first component of the scattering map S E . Note that qg(q) → 0 as q → +∞ and that g is continuous on [β, +∞) (these properties can be proven by using [16, Theorem 3.1] and continuity of the flow of (1.1)). Hence using (3.11) we obtain that S 1,E uniquely determines the function g.

Reconstruction formulas

Let χ be the strictly increasing function from [0, β -2 ) to [0, r -1 min,β ), continuous on [0, β -2 ) and C 2 on (0, β -2 ), defined by χ(0) = 0, and

χ(σ) = r -1 min,σ -1 2 , for σ ∈ (0, β -2 ).
(3.12)

Let φ : (0, χ(β -2 )) → (0, β -2 ) denote the inverse function of χ. From (3.4) and (3.12) it follows that

2(E -V (χ(u) -1 )) = χ(u) 2 u , for u ∈ (0, β -2 ), (3.13) 2(E -V (s -1 ))φ(s) = s 2 , for s ∈ (0, χ(β -2 )). (3.14)
Define the function H from (0, β -2 ) to R by

H(σ) := σ 0 g(u -1 2 )du 2 √ u √ σ -u for σ ∈ (0, β -2 ), (3.15)
Hence H is known from the first component of the scattering map S E .

The following formulas are valid (see Appendix for more details)

H(σ) = π χ(σ) 0 ds 2(E -V (s -1 )) , 1 π √ σ dH dσ (σ) = d dσ ln(χ(σ)), (3.16)
for σ ∈ (0, β -2 ).

Then note that from (3.6) it follows that χ(σ) = (2E)

1 2 σ -1 2 +O(σ -1+α 2
)

-1 = (2E) -1 2 σ 1 2 +O(σ α+1
2 ) as σ → 0 + , and ln (2E)

1 2 χ(σ)σ -1 2 = ln(1+O(σ α 2
)) → 0 as σ → 0 + (note that we just need the assumption α > 0). Therefore we obtain the following reconstruction formulas

χ(σ) = (2E) -1 2 σ 1 2 e σ 0 1 π √ s dH ds (s)-1
2s ds for σ ∈ (0, β -2 ), (3.17)

W (s) = E - 1 2s 2 φ(s -1 )
for s ∈ (r min,β , +∞).

(3.18)

Set β ′ = β 2E -2β 0 β -α (2β 0 + 2E) α 2 1 2 . ( 3

The relativistic multidimensional Newton equation 4.1 Uniqueness results

Let c > 0. Consider the relativistic multidimensional Newton equation in an electromagnetic field

ṗ = F (x, ẋ) := -∇V (x) + 1 c B(x) ẋ, (4.1)
p = ẋ 1 -| ẋ| 2 c 2 , ṗ = dp dt , ẋ = dx dt , x ∈ C 2 (R, R n ),
where (V, B) satisfy (1.2), (1.4) and (1.5). The equation (4.1) is an equation for x = x(t) and is the equation of motion in R n of a relativistic particle of mass m = 1 and charge e = 1 in an external static electromagnetic field described by the scalar potential V and the magnetic field B (see [START_REF] Einstein | Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen[END_REF] and, for example, [START_REF] Landau | The Classical Theory of Fields Pergamon[END_REF]Section 17]). In this equation x is the position of the particle, p is its impulse, F is the force acting on the particle, t is the time and c is the speed of light. For the equation (4.1) the energy

E = c 2 1 + |p(t)| 2 c 2 + V (x(t)) = c 2 1 -| ẋ(t)| 2 c 2 + V (x(t)), (4.2)
is an integral of motion. We denote by B c the euclidean open ball whose radius is c and whose centre is 0. Under the conditions (1.3), we have the following properties (see [START_REF] Yajima | Classical scattering for relativistic particles[END_REF]): for any

(v -, x -) ∈ B c × R n , v -= 0, the equation (4.1) has a unique solution x ∈ C 2 (R, R n ) that satisfies (1.7) where y -in (1.7) satisfies | ẏ-(t)| + | y -(t)| → 0, as t → -∞; in addition for almost any (v -, x -) ∈ B c × R n , v -=
0, the unique solution x(t) of equation (1.1) that satisfies (1.7) also satisfies the asymptotics

x(t) = tv + + x + + y + (t), (4.3)
where

v + = 0, | ẏ+ (t)| + |y + (t)| → 0, as t → +∞. At fixed energy E > c, we denote by S E,c the set {v -∈ R n | |v -| = c 1 -c 4 E 2 }
and we denote by D(S rel E ) the set of (v -, x -) ∈ S E,c × R n for which the unique solution x(t) of equation (4.1) that satisfies (1.7) also satisfies (1.8). We have that

D(S rel E ) is an open set of S E × R n and Mes((S E,c × R n )\D(S rel E )) = 0 for the Lebesgue measure on S E,c × R n . The map S rel E : D(S rel E ) → S E,c × R n given by S rel E (v -, x -) = (v + , x + )
, is called the scattering map at fixed energy E > c 2 for the equation (4.1). Note that if V (x) ≡ 0 and B(x) ≡ 0, then

v + = v -, x + = x -, (v -, x -) ∈ B c × R n , v -= 0.
We consider the inverse scattering problem at fixed energy for equation (4.1) that is similar to the inverse problem (1.10)

Given S rel E at fixed energy E > c 2 , find (V, B). (4.4)
Note that using the conservation of energy we obtain that if E < c 2 +sup R n V then S E does not determine uniquely V . For problem (4.4) Theorem 1.1 and Proposition 1.2 still hold. In Sections 4.2 and 4.3 we sketch the proof of Theorem 1.1 and Proposition 1.2 for equation (4.1).

For inverse scattering at high energies for the relativistic multidimensional Newton equation and inverse scattering in relativistic quantum mechanics see [START_REF] Jollivet | On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies[END_REF] and references therein.

Concerning the inverse problem for (4.1) in the one-dimensional case, we can mention the work [START_REF] Funke | An inverse problem for classical relativistic mechanics[END_REF].

Proof of Proposition 1.2 for equation (4.1)

We first consider the analog of Lemma 2.1. Lemma 4.1. Let E > c 2 and let R E and C rel E be defined by

C rel E := min E -c 2 2β 0 , c 2 E-c 2 4c 2 + 1 2 -1 4β 1 n 3(E-c 2 ) 2c 2 + 1 2 , sup |x|≥R E (1 + |x|) -α ≤ C rel E 2 . (4.5) If x(t) is a solution of equation (1.1) of energy E such that |x(0)| < R E and if there exists a time T > 0 such that x(T ) = R E then |x(t)| 2 ≥ R 2 E + 1 2 c 2 E-c 2 4c 2 + 1 2 -1 3(E-c 2 ) 2c 2 + 1 2 |t -T | 2 for t ∈ (T, +∞), (4.6)
and there exists a unique

(x + , v + ) ∈ R n ×S E,c so that x(t) = x + +tv + +y + (t), t ∈ R, where |y + (t)| + | ẏ+ (t)| → 0 as t → +∞.
The proof of Lemma 4.1 is similar to the proof of Lemma 2.1.

The solutions x(t) of equation (4.1) in B(0, R) for some R > 0 also have properties (2.4) and (2.5) at fixed and sufficiently large energy. Therefore at fixed and sufficiently large energy we consider the inverse kinematic problem in a ball B(0, R) for equation (4.1) similar to the inverse kinematic problem given in Section 2.2. Then Lemma 2.2 still holds (see [START_REF] Jollivet | On inverse problems in electromagnetic field in classical mechanics at fixed energy[END_REF]Theorem 1.2]) and the connection between boundary data of the inverse kinematic problem and the scattering map S rel E is similar to the one given for the nonrelativistic case in Section 2.3 (note that the radius R has also to be chosen so that

sup |x|≥R (1 + |x|) -α < c 2 /(144β 1 n) = lim E→+∞ C rel E )
. This proves Proposition 1.2 for equation (4.1).

Proof of Theorem 1.1 for equation (4.1)

We assume that the electromagnetic field (V, B) in equation (4.1) satisfies (1.4) and (1.5) and is so that B ≡ 0 and V is spherically symmetric outside B(0, R) for some R > 0. Let W ∈ C 2 ([R, +∞), R) be defined by V (x) = W (|x|) for x ∈ B(0, R). We give the analog of Lemma 3.1. Let β = (2β 2 0 )

1 α E(2β 0 +β 1 )+β 1 β 0 -(E(2β 0 +β 1 )+β 1 β 0 ) 2 -4β 2 0 (E 2 -c 4 ) 1 2 -1 α
and set

β := max β, β 0 E -c 2 1 α E -1 c (E + β 0 ) 2 -c 4 , cR (E + β 0 ) 2 -c 4 E .
(4.7) Then for q ≥ β consider the real number r min,q defined by r min,q = sup{r ∈ (R, +∞)

| (E -W (r)) 2 -c 4 - q 2 E 2 c 2 r 2 = 0}. ( 4.8)
The function r min,. has the following properties.

Lemma 4.2. The function r min,. is a C 2 strictly increasing function from [β, +∞) to (R, +∞) and we have r 3 min,q W ′ (r min,q ) c 2 (E-W (r min,q )) E 2 < q 2 for q ≥ β and (E -W (r min,q )) 2 -c 4 -q 2 E 2 c 2 r 2 min,q = 0, (4.9)

dr min,q dq = E 2 qr min,q -c 2 (E -W (r min,q ))r 3 min,q W ′ (r min,q ) + q 2 E 2 > 0. (4.10)

In addition the following estimates and asymptotics at +∞ hold

qE c (E + β 0 ) 2 -c 4 ≤ r min,q ≤ Eq c E -β 0 qE c √ (E+β 0 ) 2 -c 4 -α 2 -c 4 , (4.11)
for q ≥ β, and

r min,q = qE c √ E 2 -c 4
+ O(q 1-α ), as q → +∞. (4.12)

Then take any plane P containing 0 and keep notations of Section 3.2. Let q ≥ β. Then set x -:= (c 1 -c 4 /E 2 ) -1 qe 1 and v -= c 1 -c 4 /E 2 e 2 , and consider x q (t) the solution of (4.1) with energy E and with initial conditions (1.4) at t → -∞. We write x q in polar coordinates: x q (t) = r q (t)(cos(θ q (t))e 1 (t) + sin(θ q (t))e 2 ) for t ∈ (-∞, t -) where t -:= sup{t ∈ R | |x q (s)| ≥ R for s ∈ (-∞, t)} and the functions r q , θ q , satisfy rq

(t) = -W ′ (r q (t)) E-W (rq(t)) c 2 3 + q 2 E 2 E -W (r q (t)) -rW ′ (r q (t)) r 3 (E -W (r q (t))) 3 , (4.13)
r q (t) 2 θq (t) 1 -ṙq(t) 2 +rq(t) 2 θq(t) 2 c 2 = qE c 2 . ( 4

.14)

The energy E defined by (4.2) is then written as follows

1 - ṙq (t) 2 c 2 - c 4 (E -W (r q (t))) 2 - q 2 E 2 c 2 r q (t) 2 (E -W (r q (t))) 2 = 0. ( 4.15)
We also have θq (t) = qE r q (t) 2 (E -W (r q (t)))

.

Similarly to Section 3.2 we have t -= +∞, r q (t q + t) = r q (t q -t) for t ∈ R, and ± ṙq (t) > 0 for ±t > t q where t q = inf{t ∈ (-∞, t -) | r q (t) = r min,q }. We thus define

g(q) = +∞ -∞ dt r q (t) 2 (E -W (r q (t))) = 2 +∞ tq dt r q (t) 2 (E -W (r q (t)))
. (4.17)

From (4.16) Eqg(q) = R θq (s)ds is the scattering angle of x q (t), t ∈ R, and

S rel 1,E (v -, x -) = c 1 -c 4 /E 2 cos Eqg(q) + π 2 , sin Eqg(q) + π 2 , (4.18)
for q ∈ [β, +∞) and where S rel 1,E is the first component of the scattering map S rel E . Since qg(q) → 0 as q → +∞ and that g is continuous on [β, +∞) S rel 1,E uniquely determines the function g. We now provide the reconstruction formulas for W from g in a neighborhood of infinity . Let χ be the strictly increasing function from [0, β -2 ) to [0, r -1 min,β ), continuous on [0, β -2 ) and C 2 on (0, β -2 ), defined by χ(0) = 0 and

χ(σ) = r -1 min,σ -1 2 for σ ∈ (0, β -2 ). Let φ : (0, χ(β -2 )) → (0, β -2 ) denote the in- verse function of χ. From (4.9) it follows that (E-V (χ(u) -1 )) 2 -c 4 -E 2 χ(u) 2 c 2 u = 0 for u ∈ (0, β -2 ), and (E -V (s -1 )) 2 -c 4 φ(s)-E 2 s 2 c 2 = 0 for s ∈ (0, χ(β -2 )
). Define the function H from (0, β -2 ) to R by (3.15). The following formulas are valid

H(σ) = π χ(σ) 0 ds (E -V (s -1 ) 2 -c 4 , E cπ √ σ dH dσ (σ) = d dσ ln(χ(σ)), (4.19)
for σ ∈ (0, β -2 ). The proof of formulas (4.19) is similar to the proof of formulas (3.16).

Then note that from (4.12) it follows that χ

(σ) = E c √ E 2 -c 4 σ -1 2 +O(σ - 1+α 2
) A Proof of Lemmas 2.1 and 3.1

-1 = c √ E 2 -c 4 E σ 1 2 + O(σ α+1
In this Section we give a proof of Lemmas 2.1 and 3.1, and we give details on the derivation of formulas (3.16) and the equality "q = q ′ " in Section 3.2. Proof Lemma 3.1. Note that for q ≥ β we have q 2 /(2R 2 ) > E+β 0 (1+R) -α ≥ E -W (R) and lim r→+∞ q 2 /(2r 2 ) = 0 < E = lim r→+∞ (E -W (r)). Hence using (3.3) we obtain r min,q ∈ (R, +∞) and W (r min,q ) + q 2 2r 2 min,q = E, (A. [START_REF] Landau | The Classical Theory of Fields Pergamon[END_REF] for q ∈ [β, +∞). Let q ∈ [β, +∞). From (A.14) it follows that q 2 /(2r 2 min,q ) ≤ E + β 0 and then r min,q ≥ q/ 2(E + β 0 ). Combining this latter estimate and estimate (3.1) we obtain that 2 E -sup

2

 2

2. 1

 1 Nontrapped solutions of equation (1.1)

2 -c 4 σ 1 2 = ln( 1 + O(σ α 2 ) 1 2s 2 , 2 1 2 .

 22121222 ) → 0, as σ → 0 + . ds , for σ ∈ (0, β -2 ), (4.21)W (s) = E -c 4 + E 2 c 2 s 2 φ(s -1 ) 1 for s ∈ (r min,β , +∞). (4.22) Set β ′ = β 2E-2β 0 β -α (2β 0 +2E)α Then note that from (4.7) and (4.11) it follows that r min,β ≤ β ′ . Therefore using (4.21) and (4.22) we obtain that W is determined by the first component of the scattering map S rel E on (β ′ , +∞). The proof of Theorem 1.1 for equation (4.1) then relies on this latter statement and on Proposition 1.2 for equation (4.1).

FF

  for t ∈ (0, +∞), where v + = ẋ(0) + +∞ 0 (x(τ ), ẋ(τ ))dτ, (A.11)x + = x(0) -(x(τ ), ẋ(τ )))dτ dσ, (A.13)for t ∈ (0, +∞), where by (A.9) the integrals in (A.11), (A.12) and (A.13) are absolutely convergent (α > 1) and |y + (t)| + | ẏ+ (t)| → 0 as t → +∞.

5 ) 2 ,

 52 and the asymptotics (3.6) follows from (A.17). Note that using (3.1) we haverW ′ (r) < β 1 r -α ≤ β 1 q -α (2(E + β 0 )) α (A.18) 2E -2W (r) > 2E -2β 0 r -α ≥ 2E -2β 0 q -α (2(E + β 0 )) α 2 , (A.19)

Proof of Lemma 2.1. We will use the following estimate. Under conditions (1.4) and (1.5) we have

Then using the conservation of energy and equation (1.1) we have

for t ∈ R. Using (A.1) and estimate on V and

for t ∈ R. Hence we have

for r ≥ q(2(E + β 0 )) -1 2 and q ≥ β. From (A.18) and (A.19) we obtain

). We have f (q, r min,q ) = 0 for q ≥ β and from the implicit function theorem and (A.17) it follows that r min,. is a C 2 strictly increasing function from [β, +∞) to (R, +∞) so that r 3 min,q W ′ (r min,q ) < q 2 for q ≥ β and the derivative of r min,. is given by (3.4).

Derivation of formulas (3.16). We first make the change of variables "r" = r q (t) in (3.10) (dr = ṙq (t)dt and ṙq (t) = 2E -q 2 rq(t) 2 -2V (r q (t)), see (3.9)) and we obtain

, for q > β.

(A.21)

Performing the change of variables "r -1 = s" in (A.21) we obtain

, for u ∈ (0, β -2 ). (A.22)

Let σ ∈ (0, β -2 ). From (A.22) and (3.15) it follows that

And performing the change of variables u = φ(s) + ε(σ -φ(s)) in (A.23) (du = (σ -φ(s))dε) and using the equality (3.14) we obtain

for s ∈ (0, χ(β -2 )) (we used the integral value π =

. Using (A.23), (A.24) and (3.14) we obtain the first equality in (3.16) We end this appendix by giving details on the equality q ′ = q in section 3.2. We keep the notations of section 3.2. We set u θ = x r ⊥ and we have where ŵ = w |w| for w = 0. Using (A.30) and (A.31) we obtain

which proves that q ′ = r 2 θ = -x -• v ⊥ -.