Thierry Daudé

François Nicoleau

Recovering the mass and the charge of a Reissner-Nordström black hole by an inverse scattering experiment

In this paper, we study inverse scattering of massless Dirac fields that propagate in the exterior region of a Reissner-Nordström black hole. Using a stationary approach we determine precisely the leading terms of the high-energy asymptotic expansion of the scattering matrix that, in turn, permit us to recover uniquely the mass of the black hole and its charge up to a sign.

Introduction

Black hole spacetimes are probably among the most fascinating objects whose existence is predicted by Einstein's general relativity theory. In the last twenty years, numerous mathematical studies have been achieved in order to better understand their properties. Propagation of fields (wave, Klein-Gordon, Dirac, Maxwell) in these peculiar geometries and direct scattering results have been extensively studied in [START_REF] Bachelot | Gravitational scattering of electromagnetic field by a Schwarzschild black hole[END_REF][START_REF] Bachelot | Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric[END_REF][START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF][START_REF] Dimock | Scattering for the wave equation on the Schwarzschild metric[END_REF][START_REF] Dimock | Scattering for massive scalar fields on Coulomb potentials and Schwarzschild metrics[END_REF][START_REF] Häfner | Sur la théorie de la diffusion pour l'équation de Klein-Gordon dans la métrique de Kerr[END_REF][START_REF] Häfner | Scattering of massless Dirac fields by a Kerr black hole[END_REF][START_REF] Melnyk | Scattering on Reissner-Nordström metric for massive charged spin 1 2 fields[END_REF] and used for instance to give rigorous interpretation of the Hawking effect [START_REF] Bachelot | The Hawking effect[END_REF][START_REF] Häfner | Creation of fermions by rotating charged black-holes[END_REF][START_REF] Melnyk | The Hawking effect for spin 1 2 fields[END_REF]. Other original phenomena related to black hole spacetimes are for instance superradiance (the possibility of extracting energy from a rotating black hole) [START_REF] Bachelot | Superradiance and scattering of a charged Klein-Gordon equation by a step-like electrostatic potential[END_REF][START_REF] Finster | An integral spectral representation of the propagator for the wave equation in the Kerr geometry[END_REF][START_REF] Finster | Decay of solutions of the wave equation in the Kerr geometry[END_REF][START_REF] Finster | A rigorous treatment of energy extraction from a rotating black hole[END_REF] or causality violation (the existence of time machine inside rotating black hole that allows to travel through time) [START_REF] Bachelot | Global properties of the wave equation on non-globally hyperbolic manifolds[END_REF]. Despite the richness of these original phenomena, a striking feature of black holes spacetimes is their simplicity. Here we refer to the fact that, whatever might be the initial configuration of a collapsing stellar body, the resulting black hole spacetime can be eventually described by "at most" three parameters -its mass, its electric charge and its angular momentum -an important uniqueness result summarized by the wellknown formula: Black holes have no hair. A natural question arises thus: assume that we are observers living in the exterior region of a black hole, at rest with respect to it and located far from it (such observers are called static at infinity and can be typically thought as a telescope on earth aimed at the black hole), can we in these conditions measure the defining parameters of the black hole by an inverse scattering experiment?

The result contained in this paper is a first step in this direction. Here we focus our attention on Reissner-Nordström black holes, that is black holes that are spherically symmetric and electrically charged and thus described by only two parameters: their mass M and their charge Q. We consider massless Dirac fields which evolve in the exterior region of a Reissner-Nordström black hole and we use the direct scattering results, already obtained in [START_REF]Scattering of linear Dirac fields by a spherically symmetric Black hole[END_REF][START_REF] Melnyk | Scattering on Reissner-Nordström metric for massive charged spin 1 2 fields[END_REF][START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF], to define the wave operators W ± and the corresponding scattering operator S. The main result of this paper is then the following. Suppose that the scattering operator S is known to observers static at infinity (accessible in theory by physical experiments), then we show that such observers can recover uniquely the mass M of the black hole as well as its charge Q, up to a sign. Note here that we cannot completely determine the charge Q of the black hole because of the simplicity of our model. The propagation of massless Dirac fields indeed is only influenced by the geometry (i.e. the metric) of the black hole which depends on Q 2 (cf. (2.1)). The consideration of massive charged Dirac fields should solve this undetermination since an explicit term corresponding to the interaction between the charge of the black hole and that of the field appear then in the equation (see [START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF]). The choice of dealing with massless Dirac fields is nevertheless meaningful since it is the simplest model we could study (in particular no long-range terms appear in the equation) that already provides significant informations. Moreover, we expect to be able to extend this result to much more complicated models such as rotating black holes (or Kerr black holes) for which there already exists a direct scattering theory [START_REF] Häfner | Scattering of massless Dirac fields by a Kerr black hole[END_REF].

The strategy we adopt to prove our main result is based on a high-energy asymptotic expansion of the scattering operator S. Such a technique was introduced by Enss and Weder in [START_REF] Enss | The geometrical approach to multidimensional inverse scattering[END_REF] and used sucessfully to recover the potential of multidimensional Schrödinger operators (note that the case of multidimensional Dirac operators in flat spacetime was treated later by Jung in [START_REF]Geometric approach to inverse scattering for Dirac equation[END_REF]). They showed that the first term of the high-energy asymptotics is exactly the Radon transform of the potential they are looking for. Since they work in dimension greater than two, this Radon transform can be inversed and the potential thus recovered. In our problem however, due to the spherical symmetry of the black hole, we are led to study a one dimensional Dirac equation and the Radon transform cannot be inversed in this case. Nevertheless the first term of the high-energy asymptotics leads to an integral that can fortunately be explicitely computed and gives exactly the radius of the event horizon, already a physically relevant information. In order to obtain the mass and the charge of the black hole, we must then calculate the second term of the asymptotics. We follow here the stationary technique introduced by one of us [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF] which is close in spirit to the Isozaki-Kitada method used in long-range scattering theory [START_REF] Isozaki | Modified wave operators with time-independent modifiers[END_REF]. The basic idea is to replace the wave operators (and thus the scattering operator) by explicit Fourier Integral Operators, called modifiers, from which we are able to compute the high-energy asymptotic expansion readily. The construction of these modifiers and the precise determination of their phase and amplitude is the crux of this paper. While this method was well-known for Schrödinger operators and applied successfully to various situations (see [START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF][START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF][START_REF] Nicoleau | An inverse scattering problem with the Aharonov-Bohm effect[END_REF][START_REF] Nicoleau | Inverse scattering for Stark Hamiltonians with short-range potentials[END_REF]), it has required some substantial modifications when applied to our model due to the specific properties of Dirac operators, essentially caused by the matrix-valued nature of the equation. A good starting point to deal with this difficulty was the paper of Gâtel and Yafaev [START_REF] Gâtel | Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential[END_REF] where direct scattering theory of massive Dirac fields in flat spacetime was studied and modifiers (at fixed energy) were constructed.

Reissner-Nordström black hole and Dirac equation

In this section, we first briefly describe the geometry of Reissner-Nordström black holes. In particular we point out the main features and properties that are meaningful from scattering theory viewpoint. We then write down the partial differential equation that governs the evolution of massless Dirac fields in the exterior region of Reissner-Nordström black holes and recall some known results in direct scattering theory.

Reissner-Nordström black holes

In Schwarzschild coordinates a Reissner-Nordström black hole is described by a four dimensional smooth manifold

M = R t × R + r × S 2 ω ,
equipped with the lorentzian metric

g = F (r) dt 2 -F (r) -1 dr 2 -r 2 dω 2 , (2.1)
where

F (r) = 1 - 2M r + Q 2 r 2 , (2.2)
and dω 2 = dθ 2 + sin 2 θ dϕ 2 is the euclidean metric on the sphere S 2 . The quantities M > 0 and Q ∈ R appearing in (2.2) are interpreted as the mass and the electric charge of the black hole. The metric (2.1) has two types of singularities. Firstly, the point {r = 0} for which the function F is singular. This is a true singularity or curvature singularity1 . Secondly, the spheres whose radii are the roots of F (note that the coefficient of the metric g involving F -1 explodes in this case). The number of these roots depends on the respective values of the constants M and Q. In this paper we only consider the case M > |Q| for which the function F has two zeros at the values r ± = M ± M 2 -Q 2 . The spheres {r = r + } and {r = r -} are called the exterior and interior event horizons of the Reissner-Nordström black hole. Both exterior and interior horizons are not true singularities in the sense given for {r = 0}, but coordinate singularities. It turns out that, using appropriate coordinate systems, these horizons can be understood as regular null hypersurfaces that can be crossed one way but would require speeds greater than that of light to be crossed the other way. Hence their name: event horizons (we refer to [START_REF]General Relativity[END_REF] for a general introduction to black hole spacetimes).

Throughout this paper we shall nevertheless use the Schwarzschild coordinates as the coordinates of our analysis. The reasons are twofold. First in this coordinate system the coefficients of the metric (2.1) don't depend on the variables t and ω, reflecting in fact the various symmetries of the black hole. We shall see in the next subsection 2.2 that the Dirac equation will take a very convenient form, namely an evolution equation (with respect to t) with time-independent Hamiltonian acting on the spacelike hypersurface Σ = R + r × S 2 ω , a nice formalism when dealing with scattering theory. Second, it corresponds implicitely to the natural notion of observers static at infinity mentioned in the introduction. Such observers are located far away from the exterior horizon of the black hole and live on the integral curves of their velocity 4-vector

U = 1 F (r) ∂ ∂t .
In particular, the time coordinate t is the proper time of such observers and thus corresponds to their true experience of time. We refer to [START_REF] O'neill | Semi-Riemanniann Geometry with Applications to Relativity[END_REF] for a more complete discussion of this notion.. This choice of coordinates appears then quite natural with the idea of scattering experiments we have in mind.

We point out now a remarkable property of the exterior horizon when described in Schwarzschild coordinates, property that has important consequences for scattering theory. From the point of view of observers static at infinity, the exterior horizon is perceived as an asymptotic region of spacetime. Precisely, it means that the exterior horizon is never reached in a finite time by incoming and outgoing null radial geodesics, i.e the trajectories followed by light-rays aimed radially at the black hole or at infinity. These geodesics are given by the integral curves of

F (r) -1 ∂ ∂t ± ∂ ∂x .
Along the outgoing geodesics we can express the time t as a function of r by the formula

t(r) = r r0 F (τ ) -1 dτ, (2.3)
where r 0 > r + is fixed. From (2.2) and (2.3), we see immediately that t(r) → -∞ when r → r + . An analogue formula holds for incoming geodesics.

In consequence we can restrict our attention to the exterior region {r > r + } of a Reissner-Nordström black hole and study the inverse problem for massless Dirac fields there. We just have to keep in mind that we won't need any boundary conditions on the exterior event horizon {r = r + } since this horizon is an asymptotic region spacetime. To make this point even clearer, let us introduce a new radial coordinate x, called the Regge-Wheeler coordinate, which has the property of straightening the null radial geodesics and will greatly simplify the later analysis. It is defined implicitely by the relation

dr dx = F (r), (2.4)
and explicitely by

x = r + r 2 + r + -r - log(r -r + ) + r 2 - r + -r - log(r -r -). (2.5)
In the coordinate system (t, x, ω), the horizon {r = r + } is pushed away to {x = -∞} and thanks to (2.4), the metric takes the form

g = F (r)(dt 2 -dx 2 ) -r 2 dω 2 . (2.6)
Observe that the incoming and outgoing null radial geodesics are now generated by the vector fields ∂ ∂t ± ∂ ∂x and take the simple form

γ ± (t) = (t, x 0 ± t, ω 0 ), t ∈ R, (2.7)
where (x 0 , ω 0 ) ∈ R × S 2 are fixed. These are simply straight lines with velocity ±1 mimicking, at least in the tx plane, the situation of a one-dimensional Minkowski spacetime.

From now on we shall work on the manifold

B = R t × R x × S 2
ω equipped with the metric (2.6).

Dirac equation and direct scattering results

Scattering theory for Dirac equations on the spacetime B has been the object of several papers [START_REF]Scattering of linear Dirac fields by a spherically symmetric Black hole[END_REF][START_REF] Melnyk | Scattering on Reissner-Nordström metric for massive charged spin 1 2 fields[END_REF][START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF] (chronological order) in the last fifteen years. We use the following convenient form for a massless Dirac equation obtained therein. Under Hamiltonian form the equation eventually reads

i∂ t ψ = Hψ, (2.8)
where ψ is a 2-components spinor belonging to the Hilbert space H = L 2 (R×S 2 ; C 2 ) and the Hamiltonian H is given by

H = Γ 1 D x + a(x)D S 2 , (2.9)
where a(x) = √ F (r) r

, D x = -i∂ x , D S 2 denotes the Dirac operator on S 2 , i.e.

D S 2 = -iΓ 2 (∂ θ + cot θ 2 ) - i sin θ Γ 3 ∂ ϕ , (2.10)
and Γ 1 , Γ 2 , Γ 3 appearing in (2.9) and (2.10) are usual 2×2 Dirac matrices that satisfy the anticommutaton relations Γ i Γ j + Γ j Γ i = 2Id. We can simplify further the expression of the Hamiltonian by using the spherical symmetry of the equation. The operator D S 2 has compact resolvent and can be diagonalized into an infinite sum of matrix-valued multiplication operators. The eigenfunctions associated to D S 2 are a generalization of usual spherical harmonics called spin-weighted harmonics. We refer to I.M. Gel'Fand and Z.Y. Sapiro [START_REF] Gel | Representations of the group of rotations of 3-dimensional space and their applications[END_REF] for a detailed presentation of these generalized spherical harmonics.

There exists thus a family of function F l n with the indexes (l, n) running in the set

I = (l, n), l -| 1 2 | ∈ N, l -|n| ∈ N which is a Hilbert basis of L 2 (S 2 ; C 2 ) and such that ∀(l, n) ∈ I, D S 2 F l n = -(l + 1 2 )Γ 2 F l n .
The Hilbert space H can then be decomposed into the infinite direct sum

H = (l,n)∈I L 2 (R x ; C 2 ) ⊗ F l n := (l,n)∈I H ln ,
where

H ln = L 2 (R x ; C 2 ) ⊗ F l n is identified with L 2 (R; C 2 )
. We obtain the orthogonal decomposition for the Hamiltonian H

H = (l,n)∈I H ln , with H ln := H |H ln = Γ 1 D x + a l (x)Γ 2 , (2.11)
and a l (x) = -a(x)(l + 1 2 ). The operator H ln is a selfadjoint operator on H ln with domain D(H ln ) = H 1 (R; C 2 ). Note also that we use the following representation for the Dirac matrices Γ 1 and Γ 2 .

Γ 1 = -1 0 0 1 , Γ 2 = 0 -1 -1 0 (2.12)
In this paper it will be enough to restrict our study to a fixed harmonics. To simplify notation we thus simply write H, H and a(x) instead of H ln , H ln and a l (x) respectively.

We summarize now the direct scattering results obtained in [START_REF]Scattering of linear Dirac fields by a spherically symmetric Black hole[END_REF][START_REF] Melnyk | Scattering on Reissner-Nordström metric for massive charged spin 1 2 fields[END_REF][START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF]. We define H 0 = Γ 1 D x as the free Hamiltonian. It is clearly a selfadjoint operator on H with domain D(H 0 ) = H 1 (R; C 2 ). The main results concerning the spectral properties of the pair (H, H 0 ) are the following. The Hamiltonian H has no eigenvalue and no singular continuous spectrum, i.e.

σ pp (H) = ∅, σ sc (H) = ∅, (2.13)
and the standard wave operators

W ± = s -lim t→±∞ e itH e -itH0 , (2.14)
exist and are complete on H. Let us make a few comments on these results. It follows directly from (2.13) and (2.14) that H has purely absolutely continuous spectrum and thus all states scatter in the asymptotic regions of spacetime where they obey a simpler equation given by the comparison dynamics e itH0 . Notice that this comparison dynamics is particularly simple for massless Dirac fields. Due to the diagonal form of the matrix Γ 1 , it becomes simply a system of transport equations with velocity +1 for the second component of the spinor and velocity -1 for the first. At last, observe that the potential a has the following asymptotics as x → ±∞:

∃κ > 0, |a(x)| = O(e κx ), x → -∞, |a(x)| = O(|x| -1 ), x → +∞.
Eventhough a is apparently a long-range potential having Coulomb decay at +∞, we don't need to modify the comparison dynamics in the definition of the wave operators. We refer to [START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF], Section 7 for a proof of this point. The scattering operator S is defined by

S = (W + ) * W -. (2.15)
We can simplify the operator S by considering incoming and outgoing initial data separately 2 . In order to define these subspaces of initial data precisely, let us introduce the asymptotic velocity operators

P ± 0 = s -C ∞ -lim t→±∞ e itH0
x t e -itH0 , and

P ± = s -C ∞ -lim t→±∞ e itH x t e -itH .
We refer to [START_REF] Dereziński | Scattering Theory of Classical and Quantum N-Particle Systems[END_REF] for a detailed presentation of these operators and their usefulness in scattering theory. By a direct calculation, we get easily P ± 0 = Γ 1 while it was shown in [START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF] that the operators P ± exist, have also spectra equal to {-1, +1} and satisfy the intertwining relations

P ± W ± = W ± Γ 1 .
(2.16)

Let us denote P in = 1 R -(Γ 1 ) and P out 1 R + (Γ 1 ) the projections onto the negative and positive spectral subspaces of the "free" asymptotic velocity Γ 1 . Then we define H 0 in = P in H and H 0 out = P out H and refer to these spaces as incoming and outgoing initial data for the free dynamics 3 . In a similar way, we define

H ± in = 1 R -(P ± )H and H ± out = 1 R + (P ±
)H the spaces of incoming and outgoing initial data for the full dynamics when t → ±∞. In our particular case it turns out that we have equalities between some of these spaces. Precisely we have Lemma 2.1 With the same notations as above, we have

H + in = H - in =: H in , H + out = H - out =: H out .
Proof : We only prove the case "out" since the proof for "in" is similar. We shall use the following characterization of the asymptotic velocity operators obtained in [START_REF] Daudé | Time-dependent scattering theory for massive charged dirac fields by a Reissner-Nordström black hole[END_REF]]

P ± = s -C ∞ -lim t→±∞ e itH Γ 1 e -itH .
(2.17)

We compute 1 R -(P -)1 R + (P + ). By (2.17) we get

1 R -(P -)1 R + (P + ) = s -lim t→±∞ e -itH 1 R -(Γ 1 )e 2itH 1 R + (Γ 1 )e -itH . Since Γ 1 H = HΓ 1 with H = Γ 1 D x -a(x)Γ 2 , we have 1 R -(P -)1 R + (P + ) = s -lim t→±∞ e -itH e 2it H 1 R -(Γ 1 )1 R + (Γ 1 )e -itH , = 0. Hence 1 R + (P + ) = 1 R -(P -) + 1 R + (P -) 1 R + (P + )1 R + (P -)1 R + (P + )
H 0 1 R ± (Γ 1 ) = ±D x 1 R ± (Γ 1
) the Hamiltonian H 0 becomes "scalar" when projected on H 0 in/out . Hence we can simplify the expression of the outgoing and incoming wave operators by

W ± out = s -lim t→±∞ e itH e -itDx P out , W ± in = s -lim t→±∞ e itH e itDx P in .
This minor remark will turn out to be important in the next section. Now from the definition (2.15) of the scattering operator, we deduce that S leaves invariant the incoming and outgoing subspaces H 0 in/out and thus can be decomposed into the direct sum

S = S in ⊕ S out , (2.18)
where

S in = (W + in ) * W - in , S out = (W + out ) * W - out . (2.19)
Clearly S in and S out are isometries on H 0 in and H 0 out .

3

The inverse problem at high energy

The main result and the strategy of the proof

We are now ready to study the inverse problem for massless Dirac fields satisfying (2.8). The question we adress is: can we determine the mass M and the charge Q of the black hole from the knowledge of the scattering operator S? The main result of this paper is summarized in the following Theorem Theorem 3.1 Assume that the scattering matrix S out (or S in ) is known. Then the corresponding potential a(x) can be entirely determined. Moreover the mass M and the square of the charge Q 2 of the black hole can be recovered by the formulas

M = lim x→+∞ x -x 3 a 2 (x) 2 , (3.1)
Q 2 = lim x→+∞ (x 4 a 2 (x) -x 2 + 2M x). (3.
2)

The strategy we adopt to prove this Theorem is based on a high-energy asymptotic expansion of S out , a well-known technique initially developed in the case of multidimensional Schrödinger operators by Enss, Weder [START_REF] Enss | The geometrical approach to multidimensional inverse scattering[END_REF]. This method can be used to study Hamiltonians with electric and magnetic potentials [START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF], the Dirac equation [START_REF]Geometric approach to inverse scattering for Dirac equation[END_REF] and Stark Hamiltonians [START_REF] Weder | Multidimensional inverse scattering in an electric field[END_REF][START_REF] Nicoleau | Inverse scattering for Stark Hamiltonians with short-range potentials[END_REF].

Precisely, we consider ψ 1 , ψ 2 ∈ H 0 out such that ψ1 , ψ2 ∈ C ∞ 0 (R; C 4 ) and we define the map

F out (λ) =< S out e iλx ψ 1 , e iλx ψ 2 >, (3.3)
for λ ∈ R. Following the ideas of [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF][START_REF] Nicoleau | An inverse scattering problem with the Aharonov-Bohm effect[END_REF], we shall obtain an asymptotic expansion of S(λ) when λ → ∞ with the exact expression of the first three leading terms. Exactly we get the following reconstruction formula:

Theorem 3.2 (Reconstruction formula) Let ψ 1 , ψ 2 ∈ H 0 out such that ψ1 , ψ2 ∈ C ∞ 0 (R; C 2 ).
Then for λ large, we obtain

F out (λ) =< ψ 1 , ψ 2 > + 1 λ < ψ 1 , L 1 (x, D x )ψ 2 > + 1 λ 2 < ψ 1 , L 2 (x, D x )ψ 2 > + O(λ -3 ), (3.4)
where L j (x, D x ) are differential operators given by

L 1 (x, D x ) = L 1 = i(l + 1 2 ) 2 2r + , L 2 (x, D x ) = a 2 l (x) 2 + (l + 1 2 ) 4 8r 2 + - i(l + 1 2 ) 2 D x 2r + ,
and

r + = M + M 2 -Q 2 .
Let us admit temporarily this result and use it to prove the main theorem. If we assume that the scattering matrix S out is known, then we can use inductively the high-energy expansion (3.4) to recover first, the radius r + of the exterior event horizon (term of order λ -1 ) and second, the potential a(x). Indeed, the term of order λ -2 (since r + is already determined) gives the quantity

< a 2 l (x) 2 ψ 1 , ψ 2 > . (3.5)
Since (3.5) can be determined for any ψ 1 , ψ 2 in a dense subset in H, we recover completely the potential a l (x) and thus a(x). Now the formulae for the mass (3.1) and for the charge (3.2) follow directly from the definition of a(x)

a 2 (x) = 1 -2M r + Q 2 r 2 r 2 ,
and the fact that r(x) ∼ x , x → +∞ an immediate consequence of (2.5). This ends the proof of our main Theorem and proves that observers static at infinity who can measure the scattering matrix are able to recover the mass of the black hole and its charge, up to a sign. Note that the first term of the expansion (3.4) permits us to determine the radius r + of the exterior event horizon, an information of physical relevance.

The main step in the proof of Theorem 3.1 is clearly the reconstruction formula 3.4 stated above. In order to prove it, we first rewrite F out (λ) as follows

F out (λ) = < S out e iλx ψ 1 , e iλx ψ 2 >, = < W - out e iλx ψ 1 , W + out e iλx ψ 2 >, = < W - out (λ)ψ 1 , W + out (λ)ψ 2 >, (3.6)
where W ± out (λ) = slim t→±∞ e itH(λ) e -it(Dx+λ) P out , and

H(λ) = Γ 1 (D x + λ) + a(x)Γ 2 .
In order to obtain an asymptotic expansion of the operators W ± out (λ), we follow the procedure exposed in [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF][START_REF] Nicoleau | An inverse scattering problem with the Aharonov-Bohm effect[END_REF], procedure inspired by the well-known Isozaki-Kitada method [START_REF] Isozaki | Modified wave operators with time-independent modifiers[END_REF] developed in the setting of long-range stationary scattering theory. It consists simply in replacing the wave operators W ± out (λ) by "well-chosen" energy modifiers J ± out (λ), defined as Fourier Integral Operators (FIO) with explicit phase and amplitude. Well-chosen here means practically that we look for J ± out (λ) satisfying for |λ| large

W ± out (λ)ψ = lim t→±∞ e itH(λ) J ± out (λ)e -it(Dx+λ) ψ, (3.7)
and

(W ± out (λ) -J ± out (λ))ψ = O(λ -3 ), (3.8)
for a fixed ψ ∈ H 0 out such that ψ ∈ C ∞ 0 (R; C 2 ). In particular if we manage to construct J ± out (λ) satisfying (3.8) then we obtain by (3.6)

F out (λ) = < J - out (λ)ψ 1 , J + out (λ)ψ 2 > + O(λ -3
), from which we can calculate the first terms of the asymptotics easily.

The rest of this paper is devoted to the construction of the modifiers J ± out (λ) and to the proof of Theorem 3.2. For simplicity of notations we omit in the next subsections the underscript "out" when naming the wave operators W ± out and modifiers J ± out .

Construction of the modifier

Let us give a hint on how to construct the modifiers J ± (λ) a priori defined as an FIO with "scalar" phase ϕ ± (x, ξ, λ) and "matrix-valued" amplitude p ± (x, ξ, λ). If we assume that (3.7) is true then we can write

(W ± (λ) -J ± (λ))ψ = i ±∞ 0 e itH(λ) C ± (λ)e -it(Dx +λ) ψdt, where C ± (λ) := H(λ)J ± (λ) -J ± (λ)(D x + λ), (3.9)
is an FIO with phase ϕ ± (x, ξ, λ) and amplitude c ± (x, ξ, λ). Hence we get the simple estimate

(W ± (λ) -J ± (λ))ψ ≤ ±∞ 0 C ± (λ) e -itDx ψ dt. (3.10)
In order that (3.8) be true it is then clear that the FIO C(λ) has to be "small" in some sense. Precisely we shall need that the amplitude c ± (x, ξ, λ) be short-range in the variable x and of order λ -3 in the variable λ.

Construction of the modifiers at fixed energy

We first look at the problem at fixed energy (i.e. we take λ = 0 in the previous formulae). Hence we aim to construct modifiers J ± with scalar phase ϕ ± (x, ξ) and matrix-valued amplitude p ± (x, ξ) such that the amplitude c ± (x, ξ) of the operator C ± = HJ ± -J ± D x be short-range in x. We follow here the treatment given by Gâtel and Yafaev in [START_REF] Gâtel | Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential[END_REF] where a similar problem for massive Dirac Hamiltonians in Minkowski spacetime was considered.

The operator C ± is clearly an FIO with phase ϕ ± (x, ξ) and amplitude

c ± (x, ξ) = B ± (x, ξ)p ± (x, ξ) -iΓ 1 ∂ x p ± (x, ξ), (3.11)
where

B ± (x, ξ) = Γ 1 ∂ x ϕ ± (x, ξ) + a(x)Γ 2 -ξ. (3.12)
As usual we look for a phase ϕ ± close to xξ and an amplitude p ± close to 1. So the term ∂ x p ± in (3.11) should be short-range et can be neglected. Moreover if p ± was exactly 1 we would have to determine ϕ ± so that

B ± = 0. (3.13)
A direct calculation of B ± = 0 leads to a matrix-valued phase ϕ ± whereas we look for a scalar one. But if ϕ ± satisfies (3.13) then it should also satisfy the equation B 2 = 0. Using the anticommutation properties of the Dirac matrices we have

(B ± ) 2 (x, ξ) = (∂ x ϕ ± ) 2 + a 2 (x) -ξ 2 -2ξB ± (x, ξ) = 0. (3.14)
Using (3.13) again, we eventually get the scalar equation

r ± (x, ξ) := (∂ x ϕ ± ) 2 + a 2 (x) -ξ 2 = 0, (3.15)
in fact an eikonal equation of Schrödinger type. Let us set ϕ ± (x, ξ) = xξ + φ ± (x, ξ) where φ ± (x, ξ) should be a priori small in the variable x. Then we must solve 2ξ∂ x φ ± + (∂ x φ ± ) 2 + a 2 (x) = 0. Neglecting (∂ x φ ± ) 2 in this last equation, we obtain

2ξ∂ x φ ± + a 2 (x) = 0. (3.16)
For ξ = 0, we get the following two solutions of (3.16) by the formulae

φ ± (x, ξ) = 1 2ξ ±∞ 0 a 2 (x + s)ds, (3.17)
and with this choice, we obtain for ξ = 0,

r ± (x, ξ) = 1 4ξ 2 a 4 (x). (3.18)
In our derivation of the phase, it is important to keep in mind that we didn't find an approximate solution of (3.13) but instead of (3.14). Therefore we cannot expect to take p ± = 1 as a first approximation and we have to work a bit more. In order to construct the amplitude p ± we follow here particularly closely [START_REF] Gâtel | Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential[END_REF]. So we look for p ± such that B ± p ± is as small as possible. What we know is that (B ± ) 2 is already small thanks to our choice of ϕ ± . Hence let's try to find a relation between B ± and (B ± ) 2 . Observe first that B ± have the following expression on Γ ±

B ± (x, ξ) = B(x, ξ) = B 0 (x, ξ) + 2ξK(x, ξ), (3.19)
where

B 0 (x, ξ) = -2ξP in , (3.20) K(x, ξ) = 1 2ξ - a 2 (x) 2ξ Γ 1 + a(x)Γ 2 . (3.21)
In particular, B ± don't depend on ± in Γ ± . Taking the square of (3.19) we get

B 2 = B 2 0 + 2ξB 0 K + 2ξKB (3.22)
But from (3.20) we see that B 2 0 = -2ξB 0 . Whence (3.22) becomes

B 2 = -2ξB 0 (1 -K) + 2ξKB. (3.23)
If we add 2ξB to both sides of the equality in (3.23) then we obtain by (3.

:= B 2 + 2ξB = -2ξB 0 (1 -K) + 2ξ(1 + K)B, 14), r(x, ξ)
or equivalently, if we isolate B on the left hand side

2ξ(1 + K)B = r(x, ξ) -4ξ 2 P in (1 -K). (3.25)
It follows immediately from the definition of K that (1±K) are inversible for ξ ∈

X := { ξ ∈ R : | ξ |≥ R }, R ≫ 1. In consequence we can write for ξ ∈ X B(1 -K) -1 = 1 2ξ (1 + K) -1 r(x, ξ)(1 -K) -1 -2ξP in . (3.26)
The first term in the right hand side of (3.26) is small thanks to our choice of phase but the second one is not. We choose p ± in such a way that it cancels this term. According to (3.26), a natural choice for p ± is thus

p ± (x, ξ) = p(x, ξ) = (1 -K) -1 P out , (3.27)
for which we have q(x, ξ)

:= B(x, ξ)p(x, ξ) = 1 2ξ (1 + K) -1 r(x, ξ)(1 -K) -1 . (3.28)
Note that p defined by (3.27) is independent of ±.

Let us summarize the situation at this stage. For ξ = 0, we have defined the phase ϕ ± (x, ξ) = xξ+φ ± (x, ξ) by (3.17) and for ξ ∈ X, the amplitude p is given by (3.27). Directly from the definitions, the following estimates hold.

Lemma 3.1 (Estimates on the phase and the amplitude)

∀(x, ξ) ∈ R × X, ∀α, β ∈ N, |∂ α x ∂ β ξ ϕ ± (x, ξ)| ≤ C αβ (1+ | x | + | ξ |) (2-α-β)+ . (3.29) ∀(x, ξ) ∈ R × X, |∂ 2 x,ξ (ϕ ± (x, ξ) -xξ)| ≤ C R 2 (3.30) ∀(x, ξ) ∈ R ± × X, ∀α, β ∈ N, , |∂ α x ∂ β ξ φ ± (x, ξ)| ≤ C αβ x -1-α ξ -1-β . (3.31) ∀(x, ξ) ∈ R × X, ∀α, β ∈ N, |∂ α x ∂ β ξ K(x, ξ)| ≤ C αβ x -1-α ξ -1-β . (3.32) ∀(x, ξ) ∈ R × X, ∀α, β ∈ N, |∂ α x ∂ β ξ (p(x, ξ) -P out )| ≤ C αβ x -1-α ξ -1-β . (3.33) ∀(x, ξ) ∈ R × X, ∀α, β ∈ N, |∂ α x ∂ β ξ r(x, ξ)| ≤ C αβ x -4-α ξ -2-β . (3.34) ∀(x, ξ) ∈ R × X, ∀α, β ∈ N, |∂ α x ∂ β ξ q(x, ξ)| ≤ C αβ x -4-α ξ -3-β . (3.35)
Thanks to (3.29), (3.30) and (3.33), for R large enough, we can define precisely our modifier J ± as a bounded operator on H ([29], Corollary IV.22

): let θ ∈ C ∞ (R) be a cutoff function such that θ(ξ) = 0 if |ξ| ≤ 1 2 and θ(ξ) = 1 if |ξ| ≥ 1.
For R large enough, J ± is the Fourier Integral Operator with phase ϕ ± (x, ξ) and amplitude

P (x, ξ) = p(x, ξ) θ( ξ R ) (3.36)
Moreover, we have

Proposition 3.1 For any ψ ∈ H 0 out such that ψ ∈ C ∞ 0 (X; C 2 ), we have W ± ψ = lim t→±∞ e itH J ± e -itDx ψ. (3.37)
Proof : We only consider the case (+). It is clear that it suffices to show

lim t→+∞ J + -θ( D x R )P out e -itDx ψ = 0. (3.38) Let η ∈ C ∞ (R) such that η(x) = 1 if x ≥ 1 and η(x) = 0 if x ≤ 0.
We write e -itDx ψ as :

e -itDx ψ = η(x) e -itDx ψ + (1 -η(x)) e -itDx ψ (3.39)
Clearly, (1η(x)) e -itDx ψ = e -itDx (1η(x + t)) ψ → 0 on H. Now, using (3.29), (3.31), (3.33) and the standard pseudodifferential calculus, we see that

T := J + -θ( D x R )P out η(x) is a Ψdo with symbol k + (x, ξ) which satisfies ∀(x, ξ) ∈ R × R, ∀α, β ∈ N, |∂ α x ∂ β ξ k + (x, ξ)| ≤ C αβ x -1-α ξ -1-β . (3.40)
Then T is a compact operator on H and since e -itDx ψ → 0 weakly, Proposition 3.1 is proved. ♦

Construction of the modifiers at high energy

Let's come back to the construction of J ± (λ) the modifiers at high energy. Recall that we look for modifiers that satisfy (3.7) and (3.8). Proposition 3.1 suggests to construct J ± (λ) close to e -iλx J ± e iλx .

Clearly, e -iλx J ± e iλx are the FIOs with phases ϕ ± (x, ξ, λ) = xξ+ 1 2(ξ+λ) ±∞ 0 a 2 (x+s)ds and amplitude P (x, ξ + λ). Since the modifiers J ± (λ) will be applied on functions ψ having compact support in the Fourier variable, the cut-off in the ξ variable in the definition of P (x, ξ + λ) disappears for λ sufficiently large enough.

Thus, if we take exactly J ± (λ) = e -iλx J ± e iλx as the modifiers, according to (3.11), for ξ in a compact set and λ ≫ 1, the amplitude of the operator C ± (λ) becomes

c ± (x, ξ, λ) = c(x, ξ, λ) = B(x, ξ + λ)p(x, ξ + λ) -iΓ 1 ∂ x p(x, ξ + λ).
We want the amplitude c(x, ξ, λ) to be of order O( x -2 λ -3 ). By (3.35) the first term B(x, ξ+λ)p(x, ξ+λ) has order O( x -4 λ -3 ). By (3.33) the second term ∂ x p(x, ξ + λ) has order O( x -2 λ -1 ) which is not a sufficient decay in λ for our purpose. So, we have to work a bit more. We look for the modifiers J ± (λ), with λ large enough, as FIOs with phases ϕ ± (x, ξ, λ) and with a new amplitude P (x, ξ, λ) under the form

P (x, ξ, λ) = p(x, ξ + λ) + 1 λ 2 P in k 1 (x, ξ) + p(x, ξ + λ)l 1 (x, ξ) + 1 λ 3 P in k 2 (x, ξ) g(ξ),
where g ∈ C ∞ 0 (R) with g ≡ 1 on Supp ψ, the functions k 1 , k 2 , l 1 (that can be matrix-valued) will be short-range of order O( x -2 ). Observe that there are two different types of correcting terms in this new amplitude: on one hand the terms which contain p(x, ξ + λ) ≃ P out (when λ → ∞) are correcting terms that "live" in H 0 out ; on the other hand the terms which contain P in are clearly correcting terms that "live" in H 0 in . Now, noting that

B(x, ξ + λ) = -2(ξ + λ)P in + M (x, ξ + λ), M (x, ξ + λ) = - a 2 (x) 2(ξ + λ) Γ 1 + a(x)Γ 2 ,
the amplitude c(x, ξ, λ) can be written for ξ ∈ Supp ψ as

c(x, ξ, λ) = B(x, ξ + λ)P (x, ξ + λ) -iΓ 1 ∂ x P (x, ξ + λ), = B(x, ξ + λ)p(x, ξ + λ) -iΓ 1 ∂ x p(x, ξ + λ) + 1 λ 2 -2(ξ + λ)P in k 1 (x, ξ) + M (x, ξ + λ)P in k 1 (x, ξ) -iΓ 1 P in ∂ x k 1 (x, ξ) (3.41) + 1 λ 2 B(x, ξ + λ)p(x, ξ + λ)l 1 (x, ξ) -iΓ 1 (∂ x p(x, ξ + λ)l 1 + p(x, ξ + λ)∂ x l 1 (x, ξ)) + 1 λ 3 -2(ξ + λ)P in k 2 (x, ξ) + M (x, ξ + λ)P in k 2 (x, ξ) -iΓ 1 P in ∂ x k 2 (x, ξ) .
Directly from the definitions we get the following asymptotics when λ → ∞

∂ x p(x, ξ + λ) = + a ′ (x) 2λ Γ 2 P out - ξa ′ (x) 2λ 2 Γ 2 P out + O( x -2 λ -3 ), p(x, ξ + λ) = P out + O( x -1 λ -1 ), B(x, ξ + λ)p(x, ξ + λ) = O( x -4 λ -3 ). This leads to c(x, ξ, λ) = -i a ′ (x) 2λ Γ 1 Γ 2 P out + i ξa ′ (x) 2λ 2 Γ 1 Γ 2 P out - 2 λ P in k 1 - 2ξ λ 2 P in k 1 + a(x) λ 2 Γ 2 P in k 1 -i 1 λ 2 Γ 1 P in ∂ x k 1 -i 1 λ 2 Γ 1 P out ∂ x l 1 - 2 λ 2 P in k 2 (3.42) +O( x -2 λ -3 ).
But the anticommutation properties of the Dirac matrices entail the relations Γ 1 Γ 2 P out = -P in Γ 2 and Γ 2 P in = P out Γ 2 . Hence the amplitude c(, ξ, λ) takes the form

c(x, ξ, λ) = 1 λ P in i a ′ (x) 2 Γ 2 -2k 1 (3.43) + 1 λ 2 P in -i ξa ′ (x) 2 Γ 2 -2ξk 1 + i∂ x k 1 -2k 2 + 1 λ 2 P out a(x)Γ 2 k 1 -i∂ x l 1 +O( x -2 λ -3 ).
Using (3.43) we can cancel the terms of order less than O(λ -3 ) inductively. We choose

k 1 (x, ξ) = i a ′ (x) 4 Γ 2 , (3.44)
which is obviously short-range of order O( x -2 ) and cancel the first term in (3.43). It remains then

c(x, ξ, λ) = 1 λ 2 P in -i ξa ′ (x) 2 Γ 2 -i 2ξa ′ (x) 4 Γ 2 - a ′′ (x) 4 Γ 2 -2k 2 (3.45) + 1 λ 2 P out i a ′ (x)a(x) 4 -i∂ x l 1 + O( x -2 λ -3 ),
for ξ ∈ Supp ψ. We choose

l 1 (x, ξ) = a 2 (x) 8 , (3.46)
and

k 2 (x, ξ) = -i ξa ′ (x) 2 Γ 2 - a ′′ (x) 8 Γ 2 , (3.47)
The three correcting terms k 1 , l 1 , k 2 are short-range of order O( x -2 ) for ξ in a compact set and cancel the two first terms in (3.45). This leads to

∀(x, ξ) ∈ R 2 , ∀α, β ∈ N, |∂ α x ∂ β ξ c(x, ξ, λ)| ≤ C αβ x -2-α λ -3 . (3.48)
Eventually the new amplitude P (x, ξ, λ) of J ± (λ) is defined as

P (x, ξ, λ) = p(x, ξ + λ) + 1 λ 2 i a ′ (x) 4 P in Γ 2 + p(x, ξ + λ) a 2 (x) 8 + 1 λ 3 -i ξa ′ (x) 2 P in Γ 2 - a ′′ (x) 8 P in Γ 2 g(ξ),
or equivalently ) for all N ≥ 0. Thus, using the continuity of FIOs again, we obtain (a) = O(< t > -N λ -3 ). Now, we evaluate the contribution (b) by using a standard non-stationary phase argument. We have

P (x, ξ, λ) = p(x, ξ + λ) + 1 λ 2 i a ′ (x) 4 Γ 2 P out + p(x, ξ + λ) a 2 (x) 8 λ 3 -i ξa ′ (x) 2 Γ 2 P out - a ′′ (x) 8 Γ 2 P out g(ξ). ( 3
(W ± (λ) -J ± (λ))ψ = O(λ -3
(x + t, ξ, λ). Let χ ∈ C ∞ 0 (R) be a cut-off function defined by χ(x) = 1 if | x |≤ 1 2 , χ(x) = 0 if | x |≥ 1 and let ζ = 1 -χ. We have D + (t, λ)ψ ≤ χ( 2x t )D + (t, λ)ψ + ζ( 2x t )D + (t, λ)ψ := (1) + (2). ( 3
ζ( 2x t )D + (t, λ)χ( 8x t )ψ(x) = (2π) -2 ζ( 2x t ) R 2
e iΨ + (t,x,y,ξ,λ) c(x + t, ξ, λ) χ( 8y t ) ψ(y) dξ dy, (3.54)

where

Ψ + (t, x, y, ξ, λ) = (x -y)ξ + 1 2(ξ + λ) +∞ x+t a 2 (s) ds.
(3.55)

In order to investigate possible critical point, we calculate

∂ ξ Ψ + (t, x, y, ξ, λ) = x -y - 1 2(ξ + λ) 2 +∞ x+t a 2 (s) ds. ( 3
ξ Ψ + (t, x, y, ξ, λ) |≥ C(1 + t+ | x |).
We conclude by a standard argument of non-stationary phase that for all N ≥ 0, (b) = O (< t > -N λ -3 ). Integrating over (0, +∞), we obtain Lemma 3.3. ♦

High energy asymptotics of the scattering operator

In this section we use the previous construction of the modifiers J ± (λ) to prove the reconstruction formula (3.4). Recall that we want to find the asymptotics as λ → ∞ of the function (see (3.6))

F (λ) =< W -(λ)ψ 1 , W + (λ)ψ 2 >,
where ψ1 , ψ2 have compact support. Using Lemma 3.3, we get

F (λ) =< J -(λ)ψ 1 , J + (λ)ψ 2 > + O(λ -3 ).
Let us first give the asymptotic expansion of J ± (λ) at high energy. By the constructions above, we have for λ large Thus J ± (λ) can be seen as pseudodifferential operators whose symbols j ± are given by j ± (x, ξ, λ) = e Hence, using Taylor expansion of e t at t = 0 and using the expansion p(x, ξ + λ) = 1 + a(x) 2λ Γ 2 -ξa(x) 2λ 2 Γ 2 P out + O(λ -3 ), we get

J ± (λ)ψ = 1 √ 2π R e iϕ ± (
j ± (x, ξ, λ) = 1 + i 2λ I ± - iξ 2λ 2 I ± - 1 8λ 2 I ± 2 + O(λ -3 ) 1 + a(x) 2λ Γ 2 - ξa(x) 2λ 2 Γ 2 + i a ′ (x) 4λ 2 Γ 2 + a 2 (x) 8λ 2 + O(λ -3 ) ,
where we denoted I ± (x) = ±∞ 0 a 2 (x + s)ds. If we put together the terms with same order, we obtain

j ± (x, ξ, λ) = 1 + iI ± 2λ + a(x) 2λ Γ 2 + - iξI ± 2λ 2 - (I ± ) 2 8λ 2 + a 2 (x) 8λ 2 + ia(x)I ± 4λ 2 - ξa(x) 2λ 2 + i a ′ (x) 4λ 2 Γ 2 + O(λ -3 ).
We denote R ± the differential operators We now can prove Theorem 3.2. We have

R ± = - iI ± D
F (λ) = < J -(λ)ψ 1 , J + (λ)ψ 2 > +O(λ -3 ) = 1 + iI - 2λ + a(x) 2λ Γ 2 + 1 λ 2 R -ψ 1 , 1 + iI + 2λ + a(x) 2λ Γ 2 + 1 λ 2 R + ψ 2 = < ψ 1 , ψ 2 > + iI - 2λ + a(x) 2λ Γ 2 ψ 1 , ψ 2 + ψ 1 , iI + 2λ + a(x) 2λ Γ 2 ψ 2 + 1 λ 2 < R -ψ 1 , ψ 2 > + < ψ 1 , R + ψ 2 > + iI - 2λ + a(x) 2λ Γ 2 ψ 1 , iI + 2λ + a(x) 2λ Γ 2 ψ 2 , = < ψ 1 , ψ 2 > + M λ + N λ 2 .
Let's compute the first order term M . Since Γ 2 ψ 1 ∈ H 0 in , the following terms vanish < Γ 2 ψ 1 , ψ 2 >=< ψ 1 , Γ 2 ψ 2 >= 0.

(3.60)

In consequence

M = < ψ 1 , i 2 (I + -I -)ψ 2 > = i 2 R a 2 (s)ds < ψ 1 , ψ 2 > .
But a direct calculation using (2.2) and (2.4) shows that R a 2 (s)ds = (l +

3 . 2 Lemma 3 . 3

 3233 .49) Mimicking the proof of Proposition 3.1, we have Lemma For any ψ ∈ H 0 out such that ψ ∈ C ∞ 0 (R) and for λ large, we have W ± (λ)ψ = lim t→±∞ e itH(λ) J ± (λ)e -it(Dx+λ) ψ. (3.50)Now, we can state the main result of this section: When λ tends to infinity, the following estimate holds:

a 2

 2 (x+s)ds p(x, ξ + λ)

  from which we deduce immediately that H + out ⊂ H - out . Analogously we can prove H - out ⊂ H + out and thus the equality H + out = H - out holds. ♦ Thanks to the intertwining relations (2.16) and Lemma 2.1 we see that the wave operators W ± are partial isometries from H 0 in/out into H in/out . We define -itH0 P in , the corresponding outgoing and incoming wave operators. Remark here that by the relations

	W ± out = s -lim t→±∞	e itH e -itH0 P out ,
	W ± in = s -lim

t→±∞

e itH e

  (which is uniformly bounded with respect to t) and with amplitude c +

	is the FIO with phase	ϕ + (x, ξ, λ, t) = xξ +	1 2(ξ + λ)	x+t +∞	a 2 (s) ds

).

Proof : Let us consider the case (+). Thanks to (3.10), we have to estimate D + (t, λ)ψ where D + (t, λ)

  .56) On Supp ζ( 2x t ), | x |≥ t 4 and on Supp χ( 8y t ), | y |≤ t 8 . Moreover the integral which appears on the (RHS) of (3.56) is uniformly bounded with respect to λ, x and t. Thus, | ∂

  We compute now the second order term N . Notice first that by (3.58) and (3.60), we get< R ± ψ 1 , ψ 2 >= + -I -)D x ψ 2 > + 1 8 < ψ 1 , (I + -I -) 2 ψ 2 > + < ψ 1 , a 2 (x) 2 ψ 2 > .The two leading terms (3.61) and (3.62) give exactly the reconstruction formula (3.4). Hence our main result is proved. ♦

														1 2 ) 2 r +	. Whence
									M =	i(l + 1 2 ) 2 2r +	< ψ 1 , ψ 2 > .	(3.61)
											-	iI ± D x 2	-	(I ± ) 2 8	-	a 2 (x) 8	ψ 1 , ψ 2 ,
	and	iI -2	+	a(x) 2	Γ 2 ψ 1 ,	iI + 2	+	a(x) 2	Γ 2 ψ 2 =< ψ 1 ,	1 4	I -I + ψ 2 > + < ψ 1 ,	a 2 (x) 4	ψ 2 > .
	Hence we obtain
	N = -< ψ 1 , (I Using i 2 R a 2 (s)ds = (l + 1 2 ) 2 r +	again,
		N = -	i(l + 1 2 ) 2 2r +	< ψ 1 , D x ψ 2 > + < ψ 1 ,	a 2 (x) 2	ψ 2 > +	(l + 1 2 ) 4 + 8r 2	< ψ 1 , ψ 2 > .	(3.62)

It means that certain scalars obtained by contracting the Riemann tensor blow up when r → 0.

The terms incoming and outgoing data have to be understood here in reference to the black hole, i.e. as the data whose associated solutions propagate, when t increases, in direction of the black hole (incoming data) or escape from it (outgoing data)

Observe that H 0 in (resp. H 0 out ) are simply the spinors with nonvanishing first component (resp. second component) only

Acknowledgements:

The first author would like to warmly thank Dietrich Häfner (who suggested him this problem) for his precious help.


Zeyneb Tadjine

Lahcen Oubahssi

Claudine Piau-Toffolon

Sébastien Iksal

Learning Scenarios' Operationalization: a Process based on Ontology and Patterns

Keywords: Operationalization, Patterns, Ontologies, Instructional Design, LMS, Teaching Situation

This article describes our work that aims to support teachers deploying automatically their instructional design, by the mean of patterns. We seek to balance between the need of expressive instructional scenarios, and the technical constraints that occur while deploying those scenarios on learning management systems (LMS). This could be seen as a need of a formal description in order to translate the concepts of a pedagogical scenario, according to those embedded in the LMS. To address this need, we propose a process of structuring formalizing, indexing and finally adapting and operationalizing the instructional scenario. This Process is based on a model of knowledge representation. Different Data levels around the scenario representation as well as the functional essence of educational systems will be presented. We also describe the way this model is structured so that it allows preserving most of the scenarios semantic.

INTRODUCTION

The deployment phase of learning scenarios allows the translation of scenarios intended by teachers, into an implemented version on learning platforms. As part of the current research in LIUM laboratory, we are interested more specifically in the dimension of designing, operationalizing and adapting educational situations. We consider the operationalization as the development of specific research procedures that will result in empirical observations, representing the learning scenario's concepts in the real world (learning management systems). We need to offer for teachers a mean to design adaptable and importable scenarios on distance learning environments such as Moodle [START_REF] Rice | Moodle 2.0 E-Learning Course Development[END_REF] Sakai [START_REF] Sakai | [END_REF] etc.

We focus on scenario-based design approach with patterns. This approach aims to provide teachers as designers with ideas that are based on broadly accepted practices [START_REF] Hernandez-Leo | A multicase study for the evaluation of a pattern-based visual design process for collaborative learning[END_REF] and help them collaboratively expressing their own pedagogic ideas [START_REF] Laurillard | Teaching as a Design Science, Building Pedagogical Patterns for Learning and Technology[END_REF]. According to Laurillard this formalism can be expressed computationally and may help teachers-designers in providing deployable scenarios with a minimum loss of semantic information [START_REF] Clayer | Assistance for Learning Design Community -A Context-awareness&Pattern-based Approach[END_REF] [START_REF] Loiseau | Abstraction of Learning Management Systems Instructional Design Semantics: A Meta-modeling Approach Applied to the Moodle Case-Study[END_REF] [START_REF] Abedmouleh | An Analysis Process for Identifying and Formalizing LMS Instructional Language[END_REF] [START_REF] Oubahssi | Design and Operationalization of Patterns-Case of a Training Situation of Personal Assistance for Public in Professional Integration[END_REF]. Despite of the significant advance of research work in the domain of learning environments, the operationalization phase of learning scenarios still remains a challenge in the technology enhanced learning (TEL) field. Teachersdesigners are still poorly assisted in this phase. Educational languages and standards were proposed [START_REF] Berggren | Practical and Pedagogical Issues for Teacher Adoption of IMS Learning Design Standards in Moodle LMS[END_REF] [START_REF] Ferraris | LDL for Collaborative Activities. Handbook of Visual Languages in Instructional Design: Theories and Practices[END_REF] to design learning scenarios in a machine-readable version. However, on the one hand this formalism may allow reproducing the author's design and running it online, but the teacher may find it difficult to express, design and adapt their pedagogical scenarios. On the other hand, their computational integration requires an extra effort, for each implementation, because of the technical and pedagogical constraints set by platforms. This complexity is due to the fact that existing platforms use instructional knowledge coded in different levels of granularity using implicit instructional language.

We aim to help teachers and designers to operationalize their learning scenarios by automating this activity. We propose a process based on patterns to guide the teachers from the learning scenarios formalization to their deployment on a target platform. Some research work has proven the efficiency of pattern approaches for learning design [START_REF] Laurillard | Teaching as a Design Science, Building Pedagogical Patterns for Learning and Technology[END_REF]. The semi-structured representation of scenarios will enable the capitalization and the reuse of teaching practices used by teachers. Most importantly, our hypothesis is that this representation allows browsing the patterns for relevant information retrieval and the deployments of this information on different learning management systems (LMS). We strive to show that regardless of the learning environment, using a pattern structure combined with a semantic representation of pedagogical objects (ontology approaches), will facilitate the operationalization.

This paper is structured as follows; the next section will gather related research work on instructional design for the operationalization of pedagogical scenarios. We focus on pattern based approaches to formalize and express scenarios and ontology based approaches for indexing and conserving the semantics of pedagogical objects. We describe in section 2 a pilot study we have conducted using a pattern editing tool. In this study, we sought to identify different needs and constraints for an automated operationalization. As a result, we present in section 3 the process we proposed based on patterns and ontologies to help achieving this automatic operationalization of learning scenarios. We conclude the paper by discussing further works.

INSTRUCTIONAL DESIGN AND OPERATIONALIZATION OF LEARNING SCENARIOS: RELATED WORKS

Although many research works addressed the learning design issues, few are those who take into account the aspect of operationalization, in particular, to facilitate the implementation of learning scenarios on existing platforms. This would be justified by the fastidiousness of this step. As a matter of fact, many difficulties and constraints are related to learning platforms, that range from the basic instructional language and rules to the implicit and complex structures related to each particular platform. Thus, these problems will create a semantic gap when considering learning scenarios concepts and platforms features. For example, designing tools based on modeling languages (EML) [START_REF] Koper | Modelling Units of Study from a pedagogical perspective: The pedagogical metamodel behind EML[END_REF], more specifically the educational standard languages (IMS-LD, 2015) such as CADMOS [START_REF] Katsamani | Designing a Moodle course with the CADMOS learning design tool[END_REF]) consider an XML notation, which is judged complex and tends to change the teachers-designers view of theirs scenarios. Also, since platforms do not follow any educational standards, deploying a standardized scenario would not be easy for a teacher to do. It will require the expertise of a pedagogical engineer. By another way, when those standardized designing tools take the operationalization step into account, it is always about one targeted learning environment (eg: CADMOS generates scripts to only deploy scenarios on Moodle). As a solution to the lack of expressivity of Educational Modeling Languages, we chose a structured and formalized pattern approach for learning designs. Patterns are used to capture expert knowledge of the teaching practice. A pattern is pictured as a three-part structure, specifying a problem and a solution addressing this problem according to a peculiar context [START_REF] Alexander | A pattern language, town, buildings, constructions[END_REF]. Defined links between patterns (association, composition, etc.) are considered as a pattern language. As shown is (Buendia & Benlock 2011), a pattern structure and formalization have been proposed in order to improve the instructional design process, by taking advantage of what patterns offer in terms of structure and ease of expression.

Educational language representation was used to help structure the proposed patterns (Anderson & Krathwohll 2001). We also note the WebCollage [START_REF] Villasclaras-Fernández | Web Collage: an implementation of support for assessment design in CSCL macro-scripts[END_REF] designing tool, it is based on pedagogical patterns. Within this approach, the implementation step still requires a platform expert assistance. [START_REF] Clayer | Assistance for Learning Design Community -A Context-awareness&Pattern-based Approach[END_REF] elaborates a framework of an engineering design process and an editing tool based on patterns, however, the operationalization aspect of the patterns is not addressed. Finally, GLUE!-PS is a tool dealing with instantiation and deployment that allows deploying learning designs from multiple learning design language/authoring tool to multiple learning environments, yet, the design languages are based on IMS LD, which is too complex for the teachers [START_REF] Prieto | GLUE!-PS: a multi-language architecture and data model to deploy TEL designs to multiple learning environments[END_REF].

However, having as a main goal to automate learning scenario's operationalization, we have noticed that most of the proposed design languages and tools do not preserve the semantic meaning of teachers' intention while transposing it on a learning system. There will always be a lack of information, and as consequence a need for adapting and modifying the initial learning scenario. Moreover, we believe that the use of ontologies for both designing as well as operationalizing scenarios can solve this problem. Ontologies allow having one same semantic base which will retain the essence of the scenario during the transition between learning design and deployment phases. In educational fields, ontologies have played an important role as knowledge representation and sharing mechanism. We find ontologies based on IMS LD language [START_REF] Amorim | A learning design ontology based on the IMS specification[END_REF], as well as ontologies around the learning scenario [START_REF] Paquette | A Competency-Based Ontology for Learning Design Repositories[END_REF] and also ontologies to describe common modules of learning platforms [START_REF] Montenegro | Modeling and comparison study of modules in open source lms platforms with cmapstool[END_REF]. We noticed that the main advantages of these ontologies take place during the learning design phase. We note that we highly believe that it would simplify the implementation phase and help us automate the deployment of patterns based scenarios.

The main concern of this work is to study the mechanisms supporting instructional design and scenario's deployment activity by teachers-designers. We are adopting a co-participative and iterative approach with teachers-researchers. The approach is called "Design Based Research" [START_REF] Wang | Design-based research and Technology-Enhanced Learning[END_REF], a methodology suitable to both research and design of technology-enhanced learning environments (TELEs). Especially those design experiments involve both scientific and educational values (through scientific processes of discovery, exploration, confirmation, and transmission that create strong links among researching, designing, and engineering). By this approach, we try to reduce the gaps between what a technology enhanced learning environment is and how it should be used theoretically (comparing what it is and how it is used in practice). We propose in this work to link learning design and operationalization of pattern-based scenarios. The main goal is to automate operationalization of scenario without losing its semantic information.

NEEDS AND CHALLENGES FOR A PATTERN-BASED SCENARIO OPERATIONALIZATION

The present section provides an introduction to issues related to the deployment of pattern-based scenarios in TEL environment. As shown in Figure 1, we worked on an example following the process of operationalization of learning scenarios based on patterns design, in order to study its feasibility and hen, identify the problems to deal with in our research.

In this direction, we put our focus on patternsbased learning design approaches as they offer a high level of expressivity and formalization for learning design concepts [START_REF] Laurillard | Teaching as a Design Science, Building Pedagogical Patterns for Learning and Technology[END_REF].

We expect to benefit from their formalized structure to achieve an operationalization of scenarios on LMS. Ontologies are also a very important part of this work, considering the knowledge representation and the sharing mechanisms they offer, we model and browse all the learning vocabulary and language embedded in learning platforms as well as in learning scenarios. Ontologies allow making a description of learning scenario's context, taking into account the level of granularity used in it (teaching program, course, learning unit, etc.).

We performed a pilot study aiming to explain how would both patterns and ontologies be a key solution for the automation of learning scenario's deployment reducing the semantic loss of information. The two starting points were the following: in a first step we collect the textual version of a learning scenario (as intended by the teacher) and model it with a patternbased design tool [START_REF] Clayer | Assistance for Learning Design Community -A Context-awareness&Pattern-based Approach[END_REF]. Then we looked at the version of the same scenario already operationalized manually on the Moodle platform. The idea behind this example is to identify the different needs in terms of technical and theoretical constraints around the deployment of learning scenarios on TEL environments.

The learning scenario is about an algorithmic introductory course for students in computer science in first university degree. The work was carried out in three steps, as a first step we extracted a list of learning concepts identified in the textual version of our scenario. We modelled the learning scenario using a pattern-based design tool [START_REF] Clayer | Assistance for Learning Design Community -A Context-awareness&Pattern-based Approach[END_REF]. During this step, we noticed that even though we consider the same textual version as a base of our design, numerous pattern formalizations could be realized (without any loss of the learning concepts identified earlier).

After that, we studied the "manually" operationalized version of the scenario on Moodle platform, and following the same logic, we identified the learning concepts in this deployed scenario. Once again, the concepts list remained unchanged.

Figure 1: A process for formalizing and adapting learning scenarios for an operationalization purpose.

Though, we have noticed a lack of a set of information needed for the operationalization (they were missing in the initial textual description). The information would make the connection between concepts describing the scenario and their equivalents on the platform. (eg courses, course structure etc.)

Those two first steps results and the literature leads us to conclude that the use of ontologies and meta-modeling when defining patterns for scenarios would reduce the semantic gap due to the transformation steps from the teacher's pedagogical intention to the platform. A pattern-based formalization -considering its semi structured dataallows teachers-designers expressing their pedagogical needs without extensive loss of semantic information while representing their pedagogical intention with a pattern-based editing tool. On the other side, this open way of expressivity raises some difficulties for automating the learning scenario operationalization phase [START_REF] Bézivin | Ontology-based layered semantics for precise OA&D modeling[END_REF] In fact, learning platforms have their own pedagogical structure and language. So, the mapping of each element of the scenario with the relevant concepts in the platform is not obvious. We noticed that several solutions could be possible. Then we need to guide the teacher-designer toward a learning design approach that considers the operational needs and constraints, without forcing them to use any specific platform formalism.

The third and final step was to confront the XML file obtained from Moodle scenario (after transforming the backup file according to Moodle meta-model [START_REF] Abedmouleh | An Analysis Process for Identifying and Formalizing LMS Instructional Language[END_REF]) with the XML file generated from the pattern-based editing tool (we kept two versions of the learning scenario formalization). We noticed that: a pattern component corresponds, sometimes, to more than one educational concept. The identification is not" unique". Also, the same pedagogical concept is identified in different locations for each formalization (according to the teacher's point of view), this makes it difficult to automatically implement the scenario on a computer environment. Our conclusions lead us to identify the need for techniques to establish the best correspondence between pattern's components and educational concepts for the targeted learning platform. We suggest to use an indexing service [START_REF] Dietrich | An ontology based representation of software design patterns[END_REF].

According to our study we formulate more precisely our main research concerns as: which approaches models and / or techniques to consider for transforming the pattern-based scenarios into implementable models on different learning platforms? To answer this, we should deal with three major questions:

• How could we provide to the teacher-designer some predefined components or "patterns" that would be used to gradually build a learning scenario ready to be directly implemented on any learning platform?

• Which formalism can be proposed for the patterns so that the learning design process delivers a structure helping the automatic operationalization without limiting the degree of expressivity and reuse?

• How to successfully maintain the semantics of learning scenarios while transforming its pedagogical concepts into learning platforms features?

A PROCESS FOR OPERATIONALIZING PATTERN-BASED SCENARIOS USING ONTOLOGIES

To answer our research questions presented in the previous section, we propose to define a process that aims at assisting teachers-designers to manage the instructional learning design activities (Figure 03). We believe that we should offer to the teachers a certain merging of expressivity, but it should be structured enough to make the scenarios deployable on learning platforms. This process consists of five steps. The first two: structuring (1) and indexing (2) allow a mapping of the educational concepts (coming from the teaching practices and needs of designers) and the learning platforms concepts and features. Formalizing (3), which consists on developing pattern-based scenarios by teachers-designers? Then we have the step to automate the implementation of scenarios (5). Before that, an adaptation step ( 4) is conducted to reduce the gap between the pedagogical language embedded in the platforms and the one used by teacher-designers.

Structuring

This step aims at structuring the learning scenario. It consists of two phases: the identification phase and classification phase. On the basis of the work presented in section 3, one can observe that the patterns-structured learning design scenarios as used, cause some difficulties while deploying on computer environments. Some pedagogical concepts could be missing or ignored, or poorly linked to the concepts of the targeted learning platforms. This lack of information prevents the automatic implementation of the scenarios (eg. not mentioning the course structure, activities dependencies etc.). In this phase, different concepts of the learning scenario are identified in order to be formalized as patterns (Course, structuring unit, Sequence of activities etc.). The identification is based on research work about ontologies and educational standards (see in particular work presented by [START_REF] Paquette | A Competency-Based Ontology for Learning Design Repositories[END_REF]) [START_REF] Neven | Reusable learning objects: a survey of LOM-based repositories[END_REF]. Each of the scenario concepts are also linked to other concepts such as course outline, study plan, learning method, pedagogical method, strategy or tactics of teaching.

We also mention that a pedagogical activity is defined as a series of steps or sub-activities that could be considered too as a learning scenario. Once the vocabulary for the scenario is built, we proceed for the classification phase. In order to offer pedagogically correct, significantly related and well structured patterns, we relied on the research work Figure 3: A process for formalizing and adapting learning scenarios for an operationalization purpose.

about the different dimensions and classification levels of the a learning scenario [START_REF] Pernin | Dispositifs d'apprentissage instrumentés par les technologies : vers une ingénierie centrée sur les scénarios[END_REF]. We consider different levels of granularity for a learning scenario: a structuration unit, an instructional sequence and even an elementary activity. We also rely on Bloom's taxonomy to classify the educational knowledge (Anderson & Krathwohll, 2001) and the different types of learning scenarios and activities [START_REF] Paquette | A Competency-Based Ontology for Learning Design Repositories[END_REF]. This classification will help the indexing work (presented next), because the structure of the learning scenario should satisfy the requirements of its implementation on a learning platform. We are talking about how to facilitate the detection and extraction of the relevant pedagogical information in order to map it to the most suitable platform feature, having a minimal semantic gap.

In this step, we aim to classify the learning scenario's patterns on several levels.Meaningful semantic relations should be defined between the different levels (Hierarchy, Typology, Compositions, Use etc.).

Indexing

In our context, we consider indexing as matching the pedagogical features offered by learning platforms with the learning scenario content. This phase consists on a mapping between both learning platforms and learning scenarios pedagogical language. Through our confrontation work (section 3) between a pattern-based learning scenario and its operationalized version, we identified a component of the learning scenario that has been translated into several features on the learning platform. Consequently, we need this indexation as a necessary intermediate phase between the structuring/formalizing the pattern-based learning scenario and its operationalization on a learning management system. We initially use an ontological description about the learning scenario concepts [START_REF] Paquette | A Competency-Based Ontology for Learning Design Repositories[END_REF] [START_REF] Montenegro | Modeling and comparison study of modules in open source lms platforms with cmapstool[END_REF]. The ontology description should respect the structure of patterns previously explained. Defining semantic relations intra and extra packages is very important in order to facilitate the detection of any relevant information from the learning scenario. Once the ontology is built, we proceed on matching every concept, every semantic relation and every constraint with the paradigm offered by each learning platform to consider. This indexation will create an extension to our ontology. It is part of the originality of our proposal. The ontology extension allows adapting any set of patterns designed by the teacher-designer, to a directly implemented scenario on any learning platform. We note that, the pedagogical language of the learning platform is extracted from its pedagogical meta-model. To identify this metamodel, we use the process of identification and formalization of the LMS instructional design language [START_REF] Abedmouleh | An Analysis Process for Identifying and Formalizing LMS Instructional Language[END_REF].

Formalizing

Once the structure of our learning scenario is defined, comes then the step of formalizing it. The formalism to offer is different from a pattern package to another. We are mainly inspired by the design patterns that have been adopted in e-learning context [START_REF] Goodyear | Handbook of research on learning design and learning objects: Issues, applications and technologies[END_REF] [START_REF] Clayer | Assistance for Learning Design Community -A Context-awareness&Pattern-based Approach[END_REF] and more particularly pedagogical patterns (ppp 2015). While modeling our scenarios with the pattern-based editing tool [START_REF] Clayer | Assistance for Learning Design Community -A Context-awareness&Pattern-based Approach[END_REF] (Part 3), we noticed that it is more likely an open tool for learning design that allows a free expression, this leads to some difficulties for detecting a specific needed concept. For example, the course duration could be set differently from one version of a scenario to another. Therefore, and in order to get over the automated operationalization problems, the composition of a pattern, should not compromise the detection of the relevant information, it should be well formalized while offering for teachers some freedom to design their scenarios. To be able to locate any information in a pattern is the key to an automatic operationalization, also, ensured by the use of an ontology allowing combining the pedagogical language concepts of a teacher-designer and the learning platform concepts as defined in the next phase of indexing.

Adapting

This step reflects the adjustments to apply on the learning scenario aiming to allow its automatic operationalization. The starting point is the patternbased learning scenario, it should be structured and organized according to the previous step of the process (for each pattern used and each component filled by the teacher). Depending on the target LMS, an instantiation of the indexed concepts (as presented in section 4.3) should be executed as well, thus, providing an XML file in accordance with the metamodel of the learning platform. XSLT transformations are applied to cover the missing information and properties if necessary. Finally, we get a scenario file ready to be deployed on the target platform.

Operationalizing

Consequently of the steps above, the learning scenario is formalized as structured patterns, well adapted to instructional language of the target platform. The next step is to implement this resulted scenario. To do so, we use the module of learning scenario importation. This importation is automatic and does not require any intervention of the learning platform expert. All that the teacher has to do, is to express his/her intention and pedagogical need in a semi-open structured language. We will then offer tool support to guide the automatic operationalization with a minimal semantic gap, especially without requiring him/her to master the complicated instructional language of the learning platform. The progression of this process is based on data about the pedagogical scenario and learning platforms (inputs / outputs). This data is organized and represented as a model introduced in the following paragraph.

Data Representation

Considering the data level point of view, the process of operationalization of learning scenarios involved is specified on different levels of representations (From a logical level to the physical level). As shown in the following illustration (Figure 4), we define three levels of representation, depending on our operationalization needs of pattern-based scenarios 4.

1. Conceptual and Semantic Level

This first level of representation stands with the "Computationally Independent" viewpoint, and enables us to have an instructional design knowledge representation as closer as it could be to the language used by a human teacher as well as the language embedded in a learning management system. this layer is about the "Learning scenario" modeled in an ontology inspired. This ontology should reflect the different teaching strategies and the diffenrent levels of granularity in a learning scenario (a course, a learning unit etc.). Keeping in mind our main objective to automate the operationalization process of learning scenarios, this ontology should include in its definition of concepts, the features provided by the various LMS to consider. This extension is an indexation of the instructional language of a learning platform. Building this semantic level ensures a common vocabulary for all teachers-designers and facilitates the interoperability between different LMS.

6. 2. Instructional Design Level

The previously presented process's phases "structuring" and ''formalizing" provide a representation of the pattern-based scenario. This result is what makes the content of the instructionl design level. Each element of instructional design level is connected to one or more nodes from the semantic representation (level 1). The elements are linked through "Instructional Relations" that establish the function and identify the various features to use on the learning platform while deploying the scenario.

4.

3. Content and Ressources Level

This content layer consists of different learning objects (documents and material resources) used in different contexts [START_REF] Churchill | Towards a useful classification of learning objects[END_REF] Such as: course notes, exam's forms, the use of software and any mean of communication etc. A classification of these objects has to be considered [START_REF] Churchill | Towards a useful classification of learning objects[END_REF] (presentation, practice, simulation, conceptual models, contextual information and representation objects). This level is strongly related to levels 2 and 3, it allows to instantiate the objects on learning platforms depending on the choice of use of the teacher-designer (as a support resource, mediation, building knowledge or as course application ).

CONCLUSION AND FUTURE WORK

This paper presented the research about a process of designing deployable learning scenarios, based on a model of data representation. We tried to answer our main question of research, which concerns the way of transforming pattern-based scenarios into implementable models ready to be automatically operationalized on a learning platform. It is important to consider the use of the semantic web advantages. We leaned on a case study that helped us to highlight the problems facing the operationalization of learning scenarios based on patterns. We intend to provide a structure and classification of scenarios to help the LMS's features indexing purpose. We take into account the taxonomy of activities, resources and other educational concepts, according to the teacher's intentions while designing. At the present time, we prepare an experiment, which will allow us to experiment a pattern-based formalism and collect multiple versions of patterns structures for the same learning scenario. Then, we plan to apply the process presented earlier on these resulting scenarios to study their operationalization in a design-based research approach.

Figure 2 :

 2 Figure 2: Case study and methodology.

Figure 4 :

 4 Figure 4: Data layers for the automatic operationalization of pattern-based learning scenarios.