Reconstruction of the singularities of a potential from backscattering data in 2D and 3D

We prove that the singularities of a potential in the two and three dimensional Schr\"odinger equation are the same as the singularities of the Born approximation (Diffraction Tomography), obtained from backscattering inverse data, with an accuracy of $1/2^-$ derivative in the scale of $L^2$-based Sobolev spaces. The key point is the study of the smoothing properties of the quartic term in the Neumann-Born expansion of the scattering amplitude in 3D, together with a Leibniz formula for multiple scattering valid in any dimension.


Introduction
The inverse scattering problem for Schrödinger potentials deals with the uniqueness, reconstruction and stability of the potential q in the Hamiltonian H = ∆ + q from the far field pattern of the generalized eigenfunctions or scattering solutions. These are the unique solutions of the asymptotic boundary value problem    (∆ + q + k 2 )u = 0 where the function u out satisfies the outgoing Sommerfeld radiation condition, which means, for compactly supported potential q, that u has asymptotics as |x| → ∞ u(x, θ, k) = e ikx.θ + C|x| where θ ′ = x/|x|.
The inverse problem for whole data is formally overdeterminate, as one easily can see by counting variables. For this reason, to avoid redundancies, some kinds of partial data are selected for the inverse problems. The selection of these data is motivated by numerical experience and applications.
The most celebrated sets of partial data are the following: • Fixed energy data. We assume as data the values A(θ ′ , θ, k) for fixed k and free θ, θ ′ ∈ S n−1 .
Uniqueness of the inverse problem in this case was studied by [16], [17], [23], [32]. The approach to this problem is related to the Calderón-Sylvester-Uhlmann complex exponential solutions, used in the Electrical Impedance Tomography inverse problem. The stability happens to be very weak.
• Fixed angle data. The knowledge of A(θ ′ , θ, k) for fixed θ, free θ ′ ∈ S n−1 and free k > 0 is assumed. The uniqueness of the inverse problem is open and only generic and local uniqueness is proved under a priori regularity assumptions on the potential, see [30].
In practical applications the actual potential is substituted by the so called Born approximation of the scattering amplitude. The procedures to imaging the Born approximation from the scattering data are known as Diffraction Tomography.
For backscattering data the Born approximation is given in the frequency domain by the polar coordinatesq B (−2kθ) = A(−θ, θ, k). (

1.3)
This fact makes backscattering data more natural and simpler than fixed angle data for diffraction tomography.
The use of the Born approximation is not, in general, justified on a mathematical basis: one would like to know how much information on the actual potential q is contained in the Born approximation.
This problem has been treated by several authors. In full data case see [19], [21], [20] and [2]. For fixed angle data and backscattering data, both of which are formally well determinate, the justification of diffraction tomography was studied in [26] (fixed angle) and [11], [18], [28], [24] (backscattering). We would like to remark that each of the last two types of data require the analysis of special multilinear operators which are not related.
In this work we study the case of backscattering data in dimension two and three, we continue and complete the research of [18], [28] and [24], by removing some constrains in their results. We prove that the diffraction tomography is a migration scheme, see [3], within an accuracy of at least 1/2. This is to say that the most singular parts of the actual potential can be reconstructed from the Born approximation up to a certain order (the accuracy of the migration). The determination of this accuracy is very important to design numerical methods, adapted to the spaces in which one expects to obtain the information on the actual potential from real scattering data. We prove Theorem 1. Assume that n ∈ {2, 3}, q is a compactly supported function in W α , 2 (R n ) and α ≥ 0. Then q − q B ∈ W β , 2 (R n ) + C ∞ (R n ) , for any β ∈ R such that 0 ≤ β < α + 1 2 .
In Theorem 1 the regularity is measured in the scale of L 2 -based Sobolev spaces. The optimality of this accuracy in this scale of spaces is, so far, an open and interesting question.
The procedure to justify the migration scheme is to study the smoothing properties of the multilinear terms in the Neumann-Born expansion of the scattering amplitude (multiple scattering).
Physical evidence suggests that multiple scattering is strong in the case of backscattering data. The control of double and triple scattering in 3D, within an accuracy of 1/2, was obtained in [28], but their result together with the general estimates for multiple scattering do not suffice to assure that, for a potential q a priori in the Sobolev space W α,2 , the error q − q B is in W β,2 for any β < α + 1/2; the restriction 0 ≤ α < 3/4 is needed. In the range α ≥ 3/4 known estimates of quadruple scattering became worse than those of double or triple scattering. The study of the accuracy of the Born approximation requires, then, to improve the estimates of the quartic term in the series. We accomplish this in the present work. We also extend the results, which previously were only studied for α < 3/2 in 3D and for α < 1 in 2D, to any α ≥ 0 by using a Leibniz' type formula for the derivatives of multiple scattering terms (see §C.1 in [25]).
In dimension three, we only are able to prove that the errors due to double, triple and quadruple scattering are a half of a derivative better than the actual potential, as opposite to the 2D case where the regularity increases with the order.
Result from [28], [24] together with Corollary 1 allow us to state the following result concerning reconstruction of classical discontinuities from backscattering in 2D: In fact, it was proved (Theorem 2 in [28]), that, for such a q, the quadratic term is a continuous function. Hölder continuity of the cubic term is obtained since it is in W β,2 for all β < α + 1, see Theorem 1 in [24]. The remainder is controled by Corollary 1.
In the three dimensional case, it follows from Theorem 1 that the whole non continuous part of the actual potential can be reconstructed from the Born approximation, assuming a priori that q is in the Sobolev space W α,2 for some α > 1. Notice q might have some discontinuities if α is between 1 and the 3D Sobolev exponent 3/2: From the previous work [28] it follows that in 3D the discontinuities in the case of a piecewise regular potential can be reconstructed from the Born approximation (the result is not stated in [28] but it is similar to Corollary 0.1 in [18] in the 2D case). By using the evolution equation the reconstruction of conormal singularities was achived in [11]. On one hand Theorem 3, as far as we know, is the first result of reconstruction of discontinuities in 3D, without assuming special structure of the singular set but, on the other hand, one expects that q ∈ W α,2 , for any α > 1/2 suffices for the reconstruction of discontinuities. So far, this improvement has not been achieved. We know from Corollary 1, that the high frequency Neumann-Born series for j ≥ 5 converges to a Hölder continuous function for α > 1 2 .
An important feature of Theorem 1 is the fact that, regardless of the a priori regularity assumptions on the potential, the accuracy of the migration scheme is at least 1/2. This independency is important to construct any recurrence scheme, in order to obtain further information on the actual potential from scattering data. In the case of fixed angle data, one can define a modified Born approximation by inserting the error q − q B in the quadratic form, see [26]. This increases the known accuracy for rough potentials q, but an inconvenient to iterate the procedure is the dependency on α of the accuracy.
Finally, we remark that in the higher dimensional case, the order of accuracy is an open question.
We believe that 1/2 also applies, but the technical complexity of our approach makes it necessary to look for a new point of view on the problem. The treatment of the 3D problem due to Lagergren [14], [15], based upon a time dependent expansion of the backscattering operator, also requires a very technical treatment of its multilinear term. See also [4] and [5].
Notation and definitions. We will write F f orf to denote the Fourier transform of f . F −1 denotes the inverse Fourier transform. The letter M denotes the Hardy-Littlewood maximal operator.
Let F (η) given by the integral on a manifold A(η) of some function. Since our proofs are based upon a decomposition of A(η) in several subdomains D(η) ⊂ A(η), we will denote by F D (η) the same expression when we restrict the integration to the subdomain D(η).
The outgoing resolvent operator for the Laplacian is defined, in terms of the Fourier transform, We define the operator Q j in the following way where k = |ξ|/2, θ = −ξ/|ξ|. With these expressions for k and θ, we define the multi-linear form Q j (f 1 , . . . , f j ) in the FT side as We denote the high frequency version a certain constant C 0 > 0 to be chosen (see section 4). Notice that the cutoff near the origin allows us to reduce the estimates of Sobolev norms to the estimation of homogeneous Sobolev norms.
The permutation group of order k is denoted by S k . For multi-indexes β and γ in N n , we use the standard definitions of β!, |β| and β ≤ γ.
We use the letter C to denote any constant that can be explicitly computed in terms of known quantities. The exact value denoted by C may change from line to line in a given computation.
The key operator in the above integral equation is There are several a priori estimates for R k that allow to prove existence and uniqueness of Lippmann-Schwinger integral equation. Usually, Fredholm theory applies and everything follows from compactness arguments, Rellich uniqueness theorem and unique continuation principles, in the case of real valued potentials. The solution can be obtained in several situations (these cases do not require q to be real) by perturbation arguments, assuming that the energy is sufficiently large, k > k 0 ≥ 0 , where k 0 depends on some a priori bound of the potential q . As an example we may consider compactly supported q ∈ L r (R n ) for some r > n 2 . In this case, which is the one considered in this work, the resolvent operator R k is bounded from L p (R n ) to L p ′ (R n ) with norm decaying to 0 as k → ∞ when 1 p − 1 p ′ = 1 r , see [1], [13] and see also [26]. This together with Hölder inequality proves that for big k the operator T k is a contraction in L p and then existence and uniqueness of solution of (2.1) easily follow and u can be expressed as a convergent Neumann-Born series.
Once the scattering solution is obtained we may prove that the far field pattern can be expressed as A(θ ′ , θ, k) = R n e −ikθ ′ · y q(y)u(y, θ, k)dy , (2.2) see [7] where this is used as a definition for non compactly supported potentials.
By inserting the series u in (2.2) one obtains the Neumann-Born series of the scattering amplitude for k large enough (high frequency Born series): and χ * is a cutoff function near the origen (see the notations).
We deal with the backscattering inverse problem, for which one assumes the data with the direction of the receiver opposed to the source direction (echoes), i.e. A(−θ, θ, k) . The inverse problem is then formally well determined. In this case the Neumann-Born series for the scattering amplitude is where ξ = −2kθ and the j-adic term in the Neumann-Born series Q j (q) is given by the operator (1.7). We define the Born approximation for high frequency backscattering data as where ξ = −2kθ . Notice that the series (2.4) is addapted to the reconstruction of singularities, since q B,H − q B and q − F −1 (q(·)χ * (| · |/2)) are C ∞ functions.
We denote the remainder term in the high frequency series as The main part of this work, which is §3, is due to obtain the control of the term Q 4 (q) in dimension three: Theorem 4. Let us assume that q is a compactly supported function in W α , 2 (R 3 ) , for 0 ≤ α < 3/2 . Then Q 4 (q) ∈ W β , 2 (R 3 ) , for any β such that 0 ≤ β < α + 1/2 .
From Sobolev embedding theorem we obtain, Corollary 1. In the hypothesis of Theorem 5, assume also α > 0 in 2D and α > 1 2 in 3D. Then R l is a Hölder continuous function.
In §5 we give the procedure to extend the above results to the case α ≥ n/2. The key is Theorem 6 which is a Leibniz' type formula for derivatives of multiple scattering terms.
The proofs are very involved and technical. For this reason, we only include the details of the proof in the key case of Theorem 4, see Proposition 3 in §3.1. In other cases, we just sketch the proof and try to convince the reader that similar arguments work.
The quartic term in the Neumann-Born series for backscattering data is given by for any ξ ∈ R 3 , where ξ = −2kθ , that is, k = |ξ| 2 and θ = − ξ |ξ| . From Lemma 3.1 in [27], this term can written as Proposition 2. For any dimension n and η ∈ R n \ {0} , The key to understand the structure of the quartic term is the pure spherical measures part (3.8). Hence we define, for any η ∈ R 3 \ {0} Notation: We prove in section §3.1: where ε := α + 1 2 − β > 0 and the constant C > 0 just depends of α, β, ǫ and the support of q.
Now we sketch the estimates of principal value terms (3.3)-(3.7) We use a decomposition of the space into diadic shelves, as it was done for the cubic term in 2D, see [24], and for the quadratic and cubic terms in 3D in [28]. More detail can be seen in [25].
Let us state, as a model, the main features to control the principal value term Q ′ (q), given by The key to estimate this principal value operator is to control the term: . Comparing (3.11) with (3.9), we observe that we replace the sphere Γ(η), in which the variable φ runs, by its tubular neighborhood Γ δ (η) of width δ|η| in the normal direction. Notice that dσ . In this way we may expect estimates for the Sobolev norm of Q δ (q) obtained from estimates of Q(q) multiplied by δ. If one follows the lines of the proof of Proposition 3, one gets the following where C just depends on α, β, δ 1 and the support of q.
Now, to estimate the term (3.10), we use a decomposition of the Euclidean space R 3 in a similar way as was done in [24] for 2D: Remark. Technically this partition only makes sense for j 1 ≥ 3, but this is not a constraint if we This decomposition is used to split the operator (3.10). To control the operator corresponding to the annulus terms Lemma 3.1, with δ = 2 −j+1 , suffices. To deal with the central term, corresponding to Γ * ∞ (η), which is close to the singularity Γ(η), we use again Lemma 3.1 and the following Then for any β < α + 1/2, To estimate this central term, dealing with the principal value, one needs to use the cancelation. We must replace the integral on the ring Γ * , given by symmetry with respect to Γ(η) allows us to pass to the limit when ε → 0 + . To cancel the singularities we use an estimate, due to Calderón, for first differences in terms of the Hardy-Littlewood maximal operator M (as in [24], several standard reductions are also needed): After some changes of variables in the integrals involving F , we reduce to study the following terms: The proof of Lemma 3.2 follows the lines of the proof of Proposition 3. Heuristically, Lemma 3.2 is derived from Proposition 3 replacing the domain Γ(η) for the variable φ by the tubular neighborhood Γ ∞ (η) which is the result of widening the sphere Γ(η) a distance 1 in the normal direction. Nevertheless there is an additional difficulty which has to be managed: the fact that neither f 3 nor f 4 are compactly supported and their Fourier transform can not be controlled by the maximal operator using Lemma 6.2. But we must keep in mind that we can apply Lemma 6.2 to two functions, f 1 , f 2 , which are compactly supported, and the integral of | f 3 | 2 or | f 4 | 2 in φ can be bounded by the L 2 -norm using that the variable φ is solid. After these comments, we omit the long and tedious proof of Lemma 3.2. The reader can see all the details in a similar situation for the cubic term in 2D (Lemma 2.2.3 of [25]).
The key to control the principal value term (3.3) remains in the following lemma whose proof follows the lines of the proof of Proposition 3.
Let q and α as in Lemma 3.1 and β < α + 1/2, where C only depends on α, β, δ 0 and the support of q.
Analogously to the comment about Lemma 3.1 above, this result should not be surprising since To estimate the term (3.3) we have to take the partition (3.15) of R 3 with j 1 the lowest integer such that j 1 ≥ 1 − log 2 (δ 0 ), for δ 0 from Lemma 3.4. In particular, the control of the ring terms where c > 0 and we use definition (3.12).
To estimate the central term we must replace each integral on the ring Γ * ∞ (η) by Γ + ε (η) + Γ − ε (η) . We, then, use the map F : . We need again Calderón estimate for first differences and its analogous estimate for second differences: where D 2 u denotes the matrix of derivatives of order two and These tools allow us to reduce to a sum of integrals, analogous to those written after Lemma 3.3 for the case (3.7).

Proof of Proposition 3.
Let us split the set Γ(η) 3 into the following regions In this way, we can write Q(q) = Q I (q) + Q II (q) + Q III (q) + Q IV (q). We will prove that Proof of estimate (3.24). Taking the change of variable φ = η − φ ′ , we have We decompose where for any k ∈ N, we denote and then to prove (3.24) we use For each k ≥ 1 we claim In the following, we use the notation in Lemma 6.3, which is the key of the proof of the above claims.
We go on with the region II 2 k (η) . Let us split it as follows: On the region II 2 k,a (η) , we know that if |ξ| ≥ 2 −k |η| we can follow the lines of the case II k (η) . So, splitting once more as II We may write In this way, we reduce to the case II 2 k,a,2 (η) , where |φ| ≤ 3 · 2 −k |η| holds.
Remark. From the proof of claim 6 in [28] one deduces to the following estimates: By Cauchy-Schwartz inequality as in (3.26), (3.27), the fact that Lemma 6.2 implies (3.28), applying the change ζ = η − τ , since by (3.35), and changing the order of integration in ξ and η by Lemma 6.1, we have , 2 , where the last inequality follows from part (i) of Lemma 6.3 and A k (η) , F k (ξ) are defined in (3.30), (6.1).
Let us go on with the domain We conclude the estimate This ends the proof of estimate (3.24).
Proof of estimate (3.23) . Taking the change of variable φ = η − φ ′ , we have For η ∈ R 3 fixed, we take the decomposition where with (ξ, τ, φ) ∈ Γ(η) 3 . It holds and also, We claim the following: The estimate (3.23) follows from these three claims. In their proofs we use the notation introduced in the key Lemma 6.3 located in the appendix.
Let η ∈ R 3 \ {0} . The occurrences of q in the term Q I2 (q)(η) interact with each other, so that we only may bound by the maximal operator just once. We need to consider an extra splitting carried out by taking the set of points {θ j : 1 ≤ j ≤ N } in the unitary sphere S 2 , where N is large enough, to get a covering of the sphere Γ(η) with N spherical cups J j (η) centered at Ω j of radius C 1 |η| , for a certain constant C 1 > 0 to be chosen later (with N ∼ 1 C 2 1 ). We define for every j We fix j ∈ {1, . . . , N }. In this part, we take an orthonormal reference of R 3 {e 1 , e 2 , e 3 } such that e 1 = θ j , according to the notation used in (3.42). On the integral expression R Jj (q)(η), we apply Cauchy-Schwartz inequality as in (3.26)-(3.27). For each j, η fixed, by Fubini's theorem, we have
Remark. We choose N large enough in order to take the radius of the spherical cup J j (η) with Estimate for the domain A 1 .
Next for each j, ζ ′ , γ ′ fixed we change variables (s, Θ, θ) → µ = (µ 1 , µ 2 , µ 3 ) given by The Jacobean of this transformation is given by Notice that the mentioned change involves an expression for s in terms of µ which depends on the parameters j, ζ ′ , γ ′ . Hence, the function h has the same parametric dependence h(s) = h(µ, j, ζ ′ , γ ′ ).
Proof of estimate (3.25) . This case is inspired on the method used to control the piece Q ′ II (q) of the cubic term from the Neumann-Born series in the three-dimensional case in [28].

It holds
hence estimate (3.25) follows from the following claims:
In this section we are going to extend Theorem 4, Theorem 5 and estimates (2.5), (2.6), (2.7) and (3.1)-(3.2) for any α ≥ 0. Then Theorem 1 will follow from these estimates for any α ≥ 0. We start with a Leibniz' type formula for derivatives of Q j (q) which we state as follows Theorem 6. Assume that α ∈ N n , j ∈ Z , j ≥ 2 and let q ∈ W |α|,2 (R n ) be a compactly supported function. Then Remark. From the proof of Theorem 6 one also deduces the formula for the same hypotheses on q.

Appendix.
In this section we state two results, Lemma 6.1 and 6.2, which are often used in this work and state and prove an important result, Lemma 6.3, in order to demonstrate Proposition 3.
The following lemma in [28] is used several times in this work.
(3) For 0 < γ < n 2 , q Ẇ −γ, 2 ≤ C q L 2 , where C depends on the size of the support of q.
Proof of Lemma 6.3.
We obtain