The Scattering Amplitude for the Shrödinger Operator in a Layer

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The Scattering Amplitude for the Shrödinger Operator

in a Layer

Michel Cristofol, Patricia Gaitan

To cite this version:

Michel Cristofol, Patricia Gaitan. The Scattering Amplitude for the Shrödinger Operator in a Layer.

2003. �hal-00128231�

The Scattering Amplitude for the Schr¨

odinger Operator in a Layer

Michel Cristofol

Patricia Gaitan

Abstract

For a layer with compactly supported inhomogeneity we give an appropriate definition of the scattering amplitude. Then we recover the relation between the far field pattern and the scattering amplitude.

1

Introduction

The scattering amplitude is a wellknown way to solve some inverse scattering problem. In the litterature we actually find the definition of the scattering amplitude in the whole homogeneous perturbed space [5],[3] or half space [1]. For a stratified space, this definition is more complicated [2]. The situation in the layer, not yet studied, introduces also difficulties. In this paper we are looking for a definition of the scattering amplitude in the layer. Such wave guides can modelize problems of wave propagation in geophysic, underwater acoustic and so on.

One consider the Schr¨odinger operator A = −∆ + V (X) in L2_{(Ω) which derives from A}

0= −∆ in L2(Ω), where Ω = Rn_{× (0, π), n ≥ 1 and D(A) = D(A}

0) = H01(Ω) ∩ H2(Ω). We study the boundary value problem: 

(−∆ + V (X))u(X) = λu(X) in L2_(Ω), u(X)|∂Ω= 0.

(1.1)

We define the scattering amplitude and the outgoing solutions of (1.1). A work in progress uses the scattering amplitude to establish uniqueness theorem for the potential V (X).

The paper is organized as follow. In section 2, we carry out the generalized eigenfunctions of A and we define the wave operators. We state results on the existence and completeness of these operators. This allows us to prove that the system of generalized eigenfunctions of A is dense. In section 3, we define the scattering operator, the scattering matrix and finally the scattering amplitude. In section 4, we give an asymptotic development of the generalized eigenfunctions of A and we link it with the scattering amplitude.

2

Spectral Study of A

2.1

Generalized Eigenfunctions of A

By a standard Fourier technique, we obtain the generalized eigenfunctions ψ0

j of A0associated to the eigen-values λj(|p|) in the following form

ψ0j(x0, y, p) = (2π)n r 2 πe ix0·p_{sin(jy), and λj(|p|) = j}2_{+ |p|}2_{, j ≥ 1,} with X = (x0, y), x0= (x1, ..., xn_{) ∈ R}n , y ∈ (0, π) and p ∈ Rn_. This family of generalized eigenfunctions {ψ0

j}j≥1is dense in L2(Ω). Then to obtain a spectral representation

∗_{Centre de Math´ematiques et d’Informatique, Universit´e de Provence, 39 rue Joliot Curie, 13453 Marseille cedex 13, France,}

of A0, if f ∈ L2_{(Ω), we define the following limit in L}2_{(Ω) and the spaces Hj:} e fj(p) = lim M →+∞ Z |x0_|<M ψ0 j(x0, y, p)f (x 0_{, y)dx}0_dy, Hj= {u ∈ L2(Ω); ∃ efj ∈ L2 (Rn), u(x0, y) = Z Rn ψ_j0(x0, y, p) efj(p)dp}. (2.1) The unitary map Φ : f 7→ ( ef1(p), ef2(p), ...) from L2(Ω) into H1⊕ H2⊕ ... is a spectral representation of A0 in the following sense

Φ(A0f ) = ( ]A0f1, ]A0f2, ...) = (λ1(|p|) ef1(p), λ2(|p|) ef2(p), ...).

2.2

Generalized Eigenfunctions of A

We look for the generalized eigenfunctions of A in the form:

ψ±_j(x0, y, p) = ψ0_j(x0, y, p) + w±_j (x0, y, p), (2.2) We denote

N (λ) = ]{j > 1; 2≤ λ}. (2.3) The function ψ_j± satisfies the eigenvalue equation

(−∆ + V (x0, y) − λj(|p|))ψ±j(x

0_{, y, p) = 0. (2.4)} Then, taking into account the equations (2.2), (2.4) and the limiting absorption principle, we obtain

w±_j(x0, y, p) = −R±(λj(|p|))(V ψ_j0)(x0, y, p), (2.5) where R±(z) is the resolvent of A. If we introduce the resolvent equation in (2.5), we obtain

w_j±(x0, y, p) = −R±₀(λj(|p|))(V ψ0j)(x0, y, p) + R±0(λj(|p|))V R±(λj(|p|))(V ψ0j)(x0, y, p). (2.6)

2.3

Wave Operators

We first consider the wave operators

W±= s − lim t→±∞e

itA_e−itA0_{, (2.7)}

and the scattering operator S = W₊∗W−. The wave operators actually exist. The proof is an adaptation of the theorem XIII-31 of [6]. Then the limits (2.7) exist, moreover we obtain the completeness of the wave operators W± from H0ac(Ω) = L2(Ω) (subspace of absolute continuity of A0) into Hac(Ω) (subspace of absolute continuity of A), see [4]. To obtain a spectral representation of A, if f ∈ L2_{(Ω), we define the} following limit in L2_(Ω): f f_j±(p) = lim M →+∞ Z |x0_|<M ψ_j±(x0_{, y, p)f (x}0_{, y)dx}0_dy,

and H±_j = W±Hj, where Hj is defined by (2.1). The unitary map Φ± : f 7→ ( ff1±(p), ff ±

2(p), ...) from L2(Ω) to H±₁ ⊕ H±₂ ⊕ ... is a spectral representation of A|Hac.

3

The Scattering Amplitude in a Layer

We have defined in the previous section the scattering operator S = W+∗W−. Using the relations Φ+= ΦW−∗ and Φ−= ΦW+∗, the scattering operator can be written

S = Φ∗Φ−Φ+∗Φ. (3.1) In a first step we define the scattering matrix as follows

g

Suj(p) = ( eS(λj(|p|))uj)(ω) with ω ∈ Sn−1. (3.2) Note that eS(λ) ∈ L((L2_(Sn−1₎₎N (λ)_{). We obtain an N (λ) × N (λ)-matrix, where N (λ) was defined by (2.3).} So we have to determine gSuj(p). Using (3.1) we obtain ΦSu = Φu + (Φ−− Φ+)θ+u where θ+ = Φ+∗Φ. Then we have

g

Suj(p) =uj(p) +_e lim M →+∞

Z

|x0_{|<M, y∈(0,π)}

[ψ−_j(x0_{, y, p) − ψ}+

j (x0, y, p)]θ

+_u(x0_{, y)dx}0_dy.

After several technical transformations, we obtain:

g Suj(p) =uj(p) +_e P k2_<j2_+|p|2iπαn−2  R Rn×(0,π)V (x 0_{, y)ψ}0 j(x0, y, p) R Sn−1ψ 0 k(x0, y, αω0)ufk(αω 0_)dω0_dx0_dy −R Rn×(0,π)V (x 0_{, y)ψ}0 j(x0, y, p) R Sn−1[R+(λk(αω0))V ψk0(·, ·, αω0)](x0, y, αω0)uk(αωf 0_)dω0_dx0_dy, (3.3)

with α = (λj(|p|) − k2₎1/2_{. So taking into account (3.2) and (3.3) we obtain explicitely the scattering matrix} and the scattering amplitude A(ω, ω0, λj(|p|)) = [Aj,k(ω, ω0, λj(|p|))]1≤j,k≤N (λ) is defined as the kernel of this operator ((S(λj^(|p|)) − I)uej)(ω) = Z Sn−1 X k2_<j2_+|p|2 Aj,k(ω, ω0, λj(|p|))ufk(αω 0_)dω0_{. (3.4)}

Proposition 3.1 The scattering amplitude is a N (λ) × N (λ)-matrix A(ω, ω0, λ) = [Aj,k(ω, ω0, λj(|p|))]1≤j,k≤N (λ) Aj,k(ω, ω0, λj(|p|)) = Bj,k(ω, ω0, λj(|p|)) + Cj,k(ω, ω0, λj(|p|)), (3.5) Bj,k= iπαn−2 Z Rn×(0,π) V (x0, y)ψ0_j(x0, y, p) Z Sn−1 ψ0 k(x0, y, αω0)fuk(αω 0_)dω0_dx0_dy Cj,k= −iπαn−2 Z Rn×(0,π) V (x0, y)ψ0_j(x0, y, p) Z Sn−1 [R+_(λ k(αω0))V ψ0k(·, ·, αω0)](x 0_{, y, αω}0₎ f uk(αω0)dω0dx0dy.

Remark: It should be mentioned the matrix-form of the scattering amplitude which differs radically from the classical one.

4

Asymptotic Behaviour of the Generalized

Eigenfunctions

The functions w±_j see (2.2) are defined by w±_j(x0, y, p) = −R±₀(λj(|p|))(V ψ0

j)(x0, y, p) + R ±

0(λj(|p|))V R±(λj(|p|))(V ψ0j)(x0, y, p) = I1,j(x0, y, p) + I2,j(x0, y, p).

We are looking for the asymptotic behaviour of each term of this sum. Let us consider the kth_component of the first term

[I1,j(x0, y, p)]k= Z Rn ψk0(x0, y, p) 1 |p0_|2_{+ k}2_{− λj− i0}[V ψ^ 0 j(·, p)](p 0_)dp0_. Then [I1,j(x0, y, p)]k= (2π)−3n(2 π) 3/2_sin(ky) Z Rn eix0·p0 |p0_|2_{+ k}2_{− λj− i0}f1(p 0_)dp0 with f1(p0) = Z Rn×(0,π)

ei ex0·(p−p0)sin(ky) sin(j_e y)V ( e_e x0_, e y)d ex0_d

e y.

Remark: To obtain an asymptotic development of [I1,j(x0, y, p]k, we use the lemma A1 of [2]. For this we have to replace f1∈ C∞

(Rn) by a C₀∞_(Rn) function.

Let p0₀= αω0, (p0₀verify |p0₀|2_{+ k}2_{= λj(|p|)), we define the C}∞ 0 (R

n_{) function χ equal to 1 in a neighborough} of p00and 0 outside. Then

[I1,j(x0, y, p)]k= (2π)−2n(2 π) 3/2 sin(ky)(J1(x0) + K1(x0)) J1(x0) = 1 (2π)n Z Rn eix0·p0 χ(p0) |p0_|2_{+ k}2_{− λj− i0}f1(p 0_)dp0_, K1(x0) = 1 (2π)n Z Rn eix0·p0(1 − χ(p0)) |p0_|2_{+ k}2_{− λj− i0}f1(p 0_)dp0_.

Then, we apply the lemma A1 of [2] and we obtain

J1(x0) = Cp0 0 ei|x0|·|p0 0| |x0_|n−12 (f1(p00) + O(|x0|−1)) where Cp0 0 = (4π) −1_αn−3 4 (2π) 3−n 2 e−iπ n−3 4

. Moreover, we prove that K1(x0) = O(|x0|−n+12 )_{. So the} asymp-totic behaviour of the first term of w_j+, denoted [I1,j(x0, y, p)]k, is given by

[I1,j(x0, y, p)]k = (_π2) 1+5n 2 sin(ky)α n−3 2 e−iπ n−3 4 ei|x0 |·|p00| |x0_|n−12 ×R Rn×(0,π)e i ex0_·(p−p0 0)_sin(k e y) sin(j_ey)V ( ex0_, e y)d ex0_d e y.

In the same way, we prove that the asymptotic behaviour of the second term of w+_j, denoted [I2,j(x0_{, y, p)]k,} is given by [I2,j(x0, y, p)]k = (2π)1−3n2 π sin(ky)α n−3_{V (x}0_{, y)e}−iπn−3 4 ei|x0 |·|p00| |x0_|n−12 ×R Rn×(0,π)e −i ex0_·p0 0sin(k e y)R(λj(|p|)(V (·, ·)ψj0(·, ·, p))( ex0,y, p)d ee x 0_d e y.

In conclusion, we obtain the asymptotic behaviour of the kthcomponent of w+_j:

[w+_j(x0_{, y, p)]}

k = [I1,j(x0, y, p)]k+ [I2,j(x0, y, p)]k. (4.1) So we have to link the far field pattern with the scattering amplitude. For this, we compare (3.5) with (4.1).

Proposition 4.1 If we set [G(x0, y, p)]k = r 2 π(2π) −n−1 2 sin(ky)α −n+1 2 e−iπ n−1 2 e i|x0_|·α |x0_|n−1₂ , with α = (λj(|p|) − k2)1/2, we can write

ψ+_j(x0, y, p) = ψ0j(x0, y, p) + N (λ)

X k=1

Aj,k(ω, ω0, λj(|p|))[G(x0, y, p)]k+ o(|x0|−n+12 _).

This relation allows us to give a physical sense to the scattering amplitude.

Our goal is to prove that exists an injection between the scattering amplitude and the potential V (X). For the layer (R2× (0, π)) [3] prove an uniqueness result but using the Dirichlet to Neumann map. In a strip (R × (0, π)) does not exist any result. We think that the knowledge of the scattering amplitude at fixed energy is not sufficient to determine the potential. We want to prove that a countable infinity of energies will be sufficient.

References

[1] G. Eskin and J. Ralston, Inverse coefficient problems in perturbed half space, Inverse Problems, 1999 vol 3, p.683-699.

[2] J.C. Guillot and J. Ralston, Inverse scattering at fixed energy for layered media, J. Math. Pures Appl., 78, 1999, p.27-48 .

[3] M. Ikehata , Reconstruction of a obstacle from the scattering amplitude at fixed energy, Inverse Prob-lems, vol 14, (1998) p.949-954.

[4] W.C. Lyford, Spectral Analysis of the Laplacian in Domains with Cylinders, Math. Ann., (1975) p.213-251.

[5] R.G. Novikov, On determination of the Fourier transform of a potential from the scattering amplitude, Inverse Problems, vol 17, (2001) p.1243-1251.

[6] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol IV, Analysis of Operators , Academic Press, 1978.