9 pagesWe study an inverse scattering problem for a pair of Hamiltonians $(H(h) , H_0 (h))$ on $L^2 (\r^n )$, where $H_0 (h) = -h^2 \Delta$ and $H (h)= H_0 (h) +V$, $V$ is a short-range potential with a regular behaviour at infinity and $h$ is the semiclassical parameter. We show that, in dimension $n \geq 3$, the knowledge of the scattering operators $S(h)$, $h \in ]0, 1]$, up to $O(h^\infty)$ in ${\cal{B}} (L^2(\r^n ))$, and which are localized near a fixed energy $\lambda >0$, determine the potential $V$ at infinity

Nicoleau, François

HAL-Univ-Nantes

An inverse scattering problem for the Schro¨dingerequation in a semiclassical process.Franc¸ois NicoleauTo cite this version:Franc¸ois Nicoleau. An inverse scattering problem for the Schro¨dinger equation in a semiclassicalprocess.. Journal de Mathe´matiques Pures et Applique´es, Elsevier, 2006, 86, pp. 463-470.<hal-00015615>HAL Id: hal-00015615https://hal.archives-ouvertes.fr/hal-00015615Submitted on 12 Dec 2005HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00015615, version 1 - 12 Dec 2005An inverse scattering problem for the Schro¨dingerequation in a semiclassical process.Franc¸ois NicoleauDe´partement de Mathe´matiquesU.M.R 6629 - Universite´ de Nantes2, rue de la Houssinie`re BP 92208F-44322 Nantes cedex 03e-mail : nicoleau@math.univ-nantes.frAbstractWe study an inverse scattering problem for a pair of Hamiltonians (H(h),H0(h))on L2(IRn), where H0(h) = −h2∆ and H(h) = H0(h)+V , V is a short-range potentialwith a regular behaviour at infinity and h is the semiclassical parameter. We showthat, in dimension n ≥ 3, the knowledge of the scattering operators S(h), h ∈]0, 1], upto O(h∞) in B(L2(IRn)), and which are localized near a fixed energy λ > 0, determinethe potential V at infinity.1 Introduction.This work is a continuation of [10]. In the present paper, we study an inverse scatteringproblem for the pair of Hamiltonians (H(h), H0(h)) on L2(IRn), n ≥ 2, where the freeoperator is H0(h) = −h2∆, h ∈]0, 1] is the semiclassical parameter, and the perturbedHamiltonian is given by :(1.1) H(h) = H0(h) + V.We assume that V is a real-valued smooth short-range potential satisfying :(H1) ∀α ∈ INn , ∃ Cα > 0 , | ∂αxV (x) |≤ Cα < x >−ρ−|α| , ρ > 1.1Under the hypothesis (H1), it is well-known that the wave operators :(1.2) W±(h) = s− limt→±∞ eitH(h)e−itH0(h),exist and are complete, i.e Ran W±(h) = Hac(H) = subspace of absolute continuity of H(h).Let S(h) be the scattering operator defined by :(1.3) S(h) =W+∗(h) W−(h).Since S(h) commutes with H0(h), we can define the scattering matrices S(λ, h), λ > 0, asunitary operators acting on L2(Sn−1), where Sn−1 is the unit sphere in IRn. We denote byΦ0(x, λ, ω, h), (λ, ω) ∈]0,+∞[×Sn−1, the generalized eigenfunction of H0(h) :(1.4) Φ0(x, λ, ω, h) = eih√λx·ω.The unitary mapping F0(h) defined by :(1.5) (F0(h)f)(λ, ω) = (2πh)−n2 λn−24∫IRnΦ0(x, λ, ω, h) f(x) dx ,gives the spectral representation for H0(h), i.e H0(h) is transformed into the multiplicationby λ in the space L2 (IR+ ; L2(Sn−1)).Then, the scattering matrices are defined by :(1.6) F0(h)S(h)f(λ) = S(λ, h) (F0(h)f)(λ),and we have the Kuroda’s representation formula :(1.7) S(λ, h) = 1− 2iπF0(λ, h)V F0(λ, h)∗ + 2iπF0(λ, h)V R(λ+ i0, h)V F0(λ, h)∗ ,where for f ∈ L2s(IRn), s > 12 , F0(λ, h)f(ω) = (F0(h)f)(λ, ω) and R(λ + i0, h) is given bythe well-known principle of limiting absorption.We recall the non-trapping condition. Let (z(t, x, ξ), ζ(t, x, ξ)) be the solution to the Hamil-ton equations :(1.8) z˙(t, x, ξ) = 2ζ(t, x, ξ) , ζ˙(t, x, ξ) = −∇V (z(t, x, ξ)),with the initial data z(0, x, ξ) = x, ζ(0, x, ξ) = ξ.We say that the energy λ is non-trapping, if for any R ≫ 1 large enough, there existsT = T (R) such that | z(t, x, ξ) |> R for | t |> T , when | x |< R and λ = ξ2 + V (x).When λ is a non-trapping energy, and if V satisfies (H1) with ρ > 0, we have the followingestimate when h→ 0, (see [12], [14]),(1.9) ∀s > 12, ||< x >−s R(λ + i0, h) < x >−s||= O ( 1h).2The goal of this paper is to give a partial answer to the following conjecture for a class ofpotentials which are regular at infinity. We denote by B(H) the space of bounded operatorsacting on a Hilbert space H, and by f+(x) = max (f(x), 0).Conjecture.Let V1 and V2 be potentials satisfying (H1) and let Sj(λ, h), j = 1, 2, be the correspondingscattering matrices at a fixed non-trapping energy λ > 0. Assume that, in the semiclassicalregime h→ 0, S1(λ, h) = S2(λ, h) +O(h∞), in B(L2(Sn−1)).Then, the potentials are equal in the classical allowed region, i.e(1.10) (λ− V1(x))+ = (λ− V2(x))+.According to the author, this result is not known in the general case. In [11], for potentialssatisfying (H1) with ρ > n, Novikov shows, without the non-trapping condition and using the∂-method, that if S1(λ, h) = S2(λ, h) , ∀h ∈]0, 1[, then V1 = V2. It is clear that this approachis not adapted to study our conjecture in the semiclassical setting since, physically, we cannot obtain any information on the potential outside the classical allowed region. To givean example, in [13], Robert and Tamura obtain, under suitable assumptions, the completeasymptotic expansion of the scattering amplitude f(λ, θ, ω, h), with θ, ω fixed, θ 6= ω, andwhere λ is a fixed non-trapping energy. The coefficients of this expansion depend only onthe values of V (x) in {x ∈ IRn : V (x) ≤ λ}. It follows that if we modify the potentialV outside the classical allowed region, the scattering amplitude remains unchanged up toO(h∞).It seems to be difficult to prove our conjecture for a fixed energy without assuming thenon-trapping condition. The difficulty comes from the estimate of the resolvent R(λ+ i0, h)due to resonances converging exponentially to the real axis. To avoid the problem due tothese resonances, one can average with respect to the energy λ. In [16], Yajima obtains,without the non-trapping condition and with the additional condition ρ > max (1, n−12), theasymptotic expansion of the scattering amplitude, averaging with respect to λ and θ . In[6], Michel proves for a similar result when one averages only with respect to λ, i.e. θ 6= ωare fixed. In [7], he obtains for a fixed trapping energy, but with an additional assumptionon the resonances, the asymptotic expansion of the scattering amplitude similar to the oneestablished in the non-trapping case.For potentials with compact support, an inverse scattering problem close to our conjectureis studied in [1]. Assuming that λ > supx∈IRnV+(x) and that the metric g(x) = (λ− V (x)) dx2is conformal to the Euclidian, Alexandrova shows in [1] that the scattering amplitude in thesemiclassical regime, (actually, the scattering relations), determines V uniquely. This resultis obtained by comparing the inverse scattering problem for the Schro¨dinger equation withthe problem of the wave equation with variable speed for which the result is well-known. Thescattering relations for the metric g(x) determine the boundary distance function uniquely,and then we determine g and therefore V (x). This method can not be used for generalpotentials satisfying (H1).3Now, let us explain more precisely the setting of this paper. We consider the class ofpotentials V which are asymptotic sums of homogenous terms at infinity, i.e. we assumethat V ∈ C∞(IRn) and satisfies the regular behaviour at infinity :(H2) V (x) ≃∞∑j=1Vj(x)where the Vj(x) are homogeneous functions of order −ρj with 1 < ρ1 < ρ2 < · · · .Inverse scattering at a fixed energy with regular potentials at infinity and when the Planck’sconstant h = 1 was first studied in [5] with ρj = j + 1. In [15], Weder and Yafaev studya similar problem for general degrees of homogeneity ρj . They show that the completeasymptotic expansion of the potential is determined by the singularities in the forwarddirection of the scattering amplitude. In particular, they obtain the following result :Let V1, V2 be potentials satisfy the assumption (H2) and let Sj(λ), j = 1, 2, be thecorresponding scattering matrices. Assume that the kernel of S1(λ) − S2(λ) belongs toC∞(Sn−1 × Sn−1), then V1 − V2 belongs to the Schwartz class S(IRn).If we want to study the inverse scattering problem in the semiclassical setting with a fixednon-trapping energy and for potentials having a regular behaviour at infinity, it seems to theauthor that the technics developed in [15] are not well adapted. Indeed, as it was pointedin ([12], [13]), for potentials satisfying (H1) with ρ > n, the forward amplitude f(λ, ω, ω, h)is of order O (h−µ) with µ =(n− 1)(n+ 1)2(ρ− 1) and f(λ, θ, ω, h) with θ 6= ω is of order O(1).Thus, the scattering amplitude has a strong peak in a neighborhood of ω.Moreover, let us emphasize that, generically, we do not have any information on the kernel ofS1(λ, h)−S2(λ, h) = O(h∞). These are the reasons why we prefer to work with an arbitrarysmall interval of positives energies and we use the approach given in ([8], [9], [10]) which isa time-independent version of [2].So, our goal in this paper is to recover all functions Vj from the knowledge of the scatteringoperator S(h) localized near a fixed energy λ, up to O(h∞) in the sense of the operatornorm in L2(IRn). Since we do not work with a fixed energy, but in an averaged sense, weemphasize that the non-trapping condition is not necessary.In order to localize the scattering operator near a fixed energy λ > 0, we introduce a cut-offfunction χ ∈ C∞0 (]0,+∞[), χ = 1 in a neighborhood of λ > 0.We show that we obtain the complete asymptotic expansion of the potential from the knowl-edge of S(h)χ(H0(h)) up to O (h∞) in B(L2(IRn)), when h → 0. More precisely, we de-termine, for n ≥ 3, the asymptotics expansion of the potential at infinity, by studying theasymptotics of :(1.11) F (h) =< S(h)χ(H0(h))Φh,ω,Ψh,ω > ,where <,> is the usual scalar product in L2(IRn), and Φh,ω, Ψh,ω are suitable test functions,(see section 2). We need n ≥ 3 in order to use the uniqueness for the Radon transform inhyperplanes.42 Semiclassical asymptotics for the localized scatteringoperator.2.1 Definition of the test functions.First, let us define the unitary dilation operator U(hδ), δ > 0, on L2(IRn) by :(2.1) U(hδ) Φ(x) = hnδ2 Φ(hδx).We also need an energy cut-off χ0 ∈ C∞0 (IRn) such that χ0(ξ) = 1 if | ξ |≤ 1, χ0(ξ) = 0 if| ξ |≥ 2.For ω ∈ Sn−1, we write x ∈ IRn as x = y + tω, y ∈ Πω = orthogonal hyperplane to ω andwe consider :(2.2) Xω = {x = y + tω ∈ IRn : | y |≥ 1 }.For δ >1ρ1 − 1 and ǫ < 1 + δ, we define :(2.3) Φh,ω = eih√λx.ω U(hδ) χ0(hǫD) Φ ,where Φ ∈ C∞0 (Xω) and D = −i∇, ( Ψh,ω is defined in the same way with Ψ ∈ C∞0 (Xω)).2.2 Semiclassical asymptotics for the scattering operator.In this paper, we prove the following theorem :Theorem 1Let V be a potential satisfying (H2). Then, there exists an increasing positive sequence(νk)k≥1, (depending only on δ), with ν1 = δ(ρ1 − 1)− 1 and limk→+∞νk = +∞ such that :(2.4) < (S(h)− 1) χ(H0(h)) Φh,ω , Ψh,ω > ≃∞∑k=1hνk < Φ , Ak(x, ω,D) Ψ > ,when h→ 0, where Aj(x, ω,D) are differential operators defined in a recursive way.Moreover, ∀j ≥ 1, ∃ kj ≥ 1, (with k1 = 1), such that :(2.5) Akj (x, ω,D) =i2√λ∫ +∞−∞Vj(x+ tω) dt + Bj(x, ω,D),with B1 = 0 and for j ≥ 2, Bj(x, ω,D) is a differential operator only depending on thefunctions Vk, 1 ≤ k ≤ j − 1.52.3 Proof of Theorem 1.Step 1 :Let us begin by an elementary lemma.Lemma 2∀h≪ 1 small enough, we have :(2.6) χ(H0(h))Φh,ω = Φh,ω.ProofWe easily obtain :(2.7) F [χ(H0(h))Φh,ω](ξ) = h−nδ2 χ((hξ)2)χ0(hǫ−δ−1(hξ −√λω))FΦ(h−δ−1(hξ −√λω)),where F is the usual Fourier transform. Then, on Supp χ0, we have | hξ −√λω |≤ 2h1+δ−ǫ.Since ǫ < 1 + δ, we have for h small enough, χ((hξ)2) = 1. 2Then, by Lemma 2, we obtain,(2.8) F (h) =< W−(h)Φh,ω , W+(h)Ψh,ω >,and an easy calculation gives :(2.9) F (h) =< Ω−(h, ω)χ0(hǫD)Φ , Ω+(h, ω)χ0(hǫD)Ψ >,where(2.10) Ω±(h, ω) = s− limt→±∞ eitH(h,ω)e−itH0(h,ω),with(2.11) H0(h, ω) = (D +√λh−(1+δ)ω)2,and(2.12) H(h, ω) = H0(h, ω) + h−2(1+δ)V (h−δx).So, by (2.9), we have to find the asymptotics of Ω±(h, ω)χ0(hǫD)Φ. We follow the samestrategy as in [10], and we only treate only the case (+).Step 2 :As in [10], we construct, for a suitable sequence (νk) defined below, a modifier J+(h, ω)as a pseudodifferential operator, (actually a differential operator close to Isozaki-Kitada’sconstruction [4]), having the asymptotic expansion :(2.13) J+(h, ω) = op1 +∑k≥1hνk d+k (x, ξ, ω) .6We denote :(2.14) T+(h, ω) = H(h, ω)J+(h, ω)− J+(h, ω)H0(h, ω).A direct calculation shows that the symbol of T+(h, ω) is given by :(2.15) T+(x, ξ, h, ω) = h−1−δ−∑k≥12i√λω · ∇dk hνk + (2iξ · ∇dk +∆dk) hνk+1+δ+∑k≥1Vk h−(1+δ)+δρk +∑j,k≥1Vj dk h−(1+δ)+δρj+νk .Roughly speaking, we construct νk and d+k in order to obtain T+(h, ω) = O(h∞).First, we choose ν1 = δ(ρ1 − 1)− 1 and we solve the transport equation :(2.16) ω.∇d+1 (x, ω) =12i√λV1(x).The solution of (2.16) is given by :(2.17) d+1 (x, ω) =i2√λ∫ +∞0V1(x+ tω) dt.Then, we choose ν2 = min (ν1 +1+ δ, −(1 + δ) + δρ1, 2ν1) and we solve the correspondingtransport equation in order to construct d2. The functions dk and the coefficients νk fork ≥ 3 are determined in a recursive way. 2As in [10], we have the following proposition :Proposition 3(2.18) Ω+(h, ω)χ0(hǫD) Φ = limt→+∞ eitH(h,ω) J+(h, ω) e−itH0(h,ω)χ0(hǫD) Φ.(2.19) || (Ω+(h, ω)− J+(h, ω)) χ0(hǫD) Φ ||= O (h∞).Then, Theorem 1 follows from Proposition 3 in the same way as in [10].73 The inverse scattering problem.Let Vj, j = 1, 2, be two potentials satisfying (H2) and let Sj(h), j = 1, 2 be the scatteringoperator associated with the pair (H0(h) + Vj, H0(h)).We have the following result :Corollary 4For n ≥ 3, assume that in B(L2(IRn)),(3.1) S1(h)χ(H0(h)) = S2(h)χ(H0(h)) +O (h∞) , h→ 0.Then : V1 − V2 ∈ S(IRn) .Proof :Corollary 4 follows from Theorem 1, the uniqueness for the Radon transform in hyperplanes[3], and the fact that || Φh,ω ||, (resp. || Ψh,ω ||) are uniformly bounded with respect to h.We refer to [10] for the details. 2Comments.It is not difficult to generalize the previous results to the case of long-range potentials, i.efor potentials satisfying (H2) with ρ1 > 0, using modified wave operators, close to Isozaki-Kitada’s ones [4], (see [10], Theorem 8, for details).Acknowledgment.The author is grateful to Xue-Ping Wang for suggesting this subject and fruitful remarks.References[1] I. Alexandrova, “Structure of the Semi-Classical Amplitude for General ScatteringRelations”, Comm. in P.D.E, , vol. 30, Issues 10 to 12, p. 1505-1535, (2005).[2] V. Enss - R. Weder, ‘”The geometrical approach to multidimensional inversescattering”’, J. Math. Phys. 36 (8), p. 3902-3921, (1995).[3] S. Helgason, “The Radon Transform”, Progress in Mathematics 5, Birkha¨user,(1980).[4] H. Isozaki - H. Kitada, “Modified wave operators with time-independent modi-fiers”, Papers of the College of Arts and Sciences, Tokyo Univ., Vol. 32, p. 81-107,(1985).[5] M. S. Joshi - A.S. Barreto, “Recovering asymptotics of short range potentials”,Communications in Mathematical Physics, 193, p. 197-208, (1998).8[6] L. Michel: “Semi-classical limit of the scattering amplitude for trapping pertur-bations”, Asymptotic Analysis , 32, p. 221-255, (2003).[7] L. Michel: ”Semi-classical behavior of the scattering amplitude for trapping per-turbations at fixed energy”, Can. J. Math., 56, p. 794-824, (2004).[8] F. Nicoleau, ‘”A stationary approach to inverse scattering for Schro¨dinger oper-ators with first order perturbation”, Comm. in P.D.E, Vol 22 (3-4), p. 527-553,(1997).[9] F. Nicoleau, “An inverse scattering problem with the Aharonov-Bohm effect”,Journal of Mathematical Physics, Issue 8, p. 5223-5237, (2000).[10] F. Nicoleau, “A constructive procedure to recover asymptotics of short-range orlong-range potentials”, Journal in Differential Equations 205, p. 354-364, (2004),[11] R. Novikov, “∂-method with non-zero background potential application to inversescattering for the two-dimensional acoustic equation”, Comm. in P.D.E, Vol 21,(3-4), p. 597-618, (1996).[12] D. Robert - H. Tamura, “Semi-classical estimates for resolvents and asymptoticsfor tottal scattering cross-sections”, Ann. Inst. Henri Poincare´, tome 46, (1), p.415-442, (1987).[13] D. Robert - H. Tamura, “Asymptotic behaviour of scattering amplitudes in semi-classical and low energy limits”, Annales de l’institut Fourier, tome 39, (1), p.155-192, (1989).[14] X. P. Wang, “ Time-decay of scattering solutions and classical trajectories”, Ann.Inst. Henri Poincare´ Phys. The´or. 47, no. 1, 25–37, (1987).[15] R. Weder - D. Yafaev, “On inverse scattering at a fixed energy for potentials witha regular behaviour at infinity”, Inverse Problems 21, p. 1937-1952, (2005).[16] K. Yajima, ”The quasi-classical limit of scattering amplitude : L2 approach forshort-range potentials”, Japan. J. Math., 13, (1), p. 77-126, (1987).9

An inverse scattering problem for the Schrödinger equation in a semiclassical process.

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An inverse scattering problem for the Schrödinger equation in a semiclassical process.

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