SCATTERING PROBLEMS FOR PERTURBATIONS OF THE MULTIDIMENSIONAL BIHARMONIC OPERATOR

. Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by ﬁrst and zero order perturba- tions, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space H 2 − δ . The main result of this paper is the proof of Saito’s formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito’s formula is based on norm estimates for the resolvent of the direct operator in H 1 − δ .

1. Introduction. We consider the following n-dimensional (n ≥ 2) biharmonic operator where ∆ is the Laplacian and · denotes the dot-product x · y = n j=1 x j y j for x, y ∈ C n . The bi-Laplacian is perturbed by first and zero order perturbations, vector-valued function q and a scalar function V , that may be complex-valued.
The motivation to study operators of order 4 appears for example in the study of elasticity and the theory of vibrations of beams. As a concrete example, the (linear) beam equation [9] ∂ 2 t U + ∆ 2 U + mU = 0 under time-harmonic assumption U (x, t) = u(x)e −iωt results in the equation For scattering in the nonlinear case, see e.g. [20] and the references therein. Other examples of biharmonic problems include hinged plate configurations, described by equations of form ∆ 2 u = f in Ω u = ∆u = 0 on ∂Ω, where u satisfies Navier boundary conditions [9] and scattering by grating stacks, described by the equation ∆ 2 u − β 4 u = 0 [17]. In terms of inverse problems for bi-and poly-harmonic operators we mention some solutions to inverse boundary value problems, see e.g., [14] and also [3,4]. In these papers the aim is to recover the coefficients q and V in operator ∆ m + q ·∇+V , m ≥ 2 in some bounded domain Ω ⊂ R n , n ≥ 3. The main idea is to define the socalled Dirichlet-to-Neumann map and then to recover the coefficients q and V from it. In [22] it was proved that the Dirichlet-to-Neumann map uniquely corresponds to q and V and also to a second order perturbation F . The motivation for the approach in the above texts is based on the fundamental papers [24,25].
The present work is concerned with the following scattering problem for H 4 given by (2) H 4 u = k 4 u, k > 0, u = u 0 + u sc , u 0 (x, k, θ) = e ik(x,θ) , , |x| → ∞, for both f = u sc and f = ∆u sc , where the scattered wave u sc should be from some suitable function space. Here θ ∈ S n−1 is the angle of incident wave and (·, ·) denotes the usual real inner product. The second line in (2) is interpreted as an analogue of Sommerfeld's radiation condition at infinity for this biharmonic operator. To the best of our knowledge this radiation condition has not appeared in the literature before. We use it to reduce (2) to a related integral equation. The authors were originally motivated to start studying scattering for fourth order operators by the article of Aktosun and Papanicolaou [2], where the timeevolution of several scattering coefficients for the 1D biharmonic operator was studied. In terms of inverse scattering problems for fourth order operator we mention K. Iwasaki's results [12,13]. Iwasaki studied the scattering problem in one-dimension and considered the inverse problem as a Riemann-Hilbert boundary value problem with respect to the wavenumber k in the complex cone arg([0, π/4]) \ {0}. Given certain reflection and connection coefficients he then showed that it is possible to uniquely recover the potentials q and V of the scattering operator. In abstract setting, Kuroda [15] has studied scattering for zero-order perturbations for self-adjoint operators. However, we are not aware of other results regarding inverse scattering for higher order operators even with just zero-order perturbations.
Let us now turn to (2) and define H 0 := ∆ 2 − k 4 . Due to the equality H 0 = (−∆−k 2 )(−∆+k 2 ) the operator H 4 inherits some Schrödinger-like properties which allows us to conclude that a fundamental solution to H 0 is given by Here H are the Hankel function of the first kind and Macdonald's function of orders n−2 2 . The function G + k is also the kernel of the integral operator (∆ 2 − k 4 − i0) −1 . By applying the fundamental solution to (1) we obtain a Lippmann-Schwinger integral equation Since u 0 = e ik(θ,x) is just a bounded function it is more convenient to study the equivalent integral equation for the scattered wave, namely The aim of the present text is to study the classical scattering theory for (1) and in particular to prove an analogue of so-called Saito's formula. The approach is similar to scattering theory of Schrödinger operators see e.g., books by Cakoni and Colton [6] and Eskin [7] and to scattering theory of operators with constant coefficients, see the classic book of Hörmander [11]. We start by showing that a solution to the scattering problem (2) indeed also satisfies equation (3). This translates the study of the scattering problem to the somewhat simpler study of an integral equation for which we can then show the unique solvability by extending some well-known estimates for the Schrödinger operator to biharmonic case. This solution admits the asymptotic representation where θ ∈ S n−1 is the angle of measurement and the function is called the scattering amplitude. From the point of view of inverse problems we regard this scattering amplitude as the relevant scattering data. For our purposes we require the scattering amplitude to be known for all possible incident and measurement angles and all arbitrarily high frequencies (k > 0 large). Then Saito's formula is given by the following Theorem 1.1 (Saito's formula). Assume that q ∈ W 1 p,2δ (R n ) and V ∈ L p 2δ (R n ), where 2δ > n − n p and n < p ≤ ∞. Then the limit lim k→∞ k n−1 S n−1 ×S n−1 e −ik(θ−θ ,x) A(k, θ, θ )dθdθ = 2 n π n−1 R n β(y) |x − y| n−1 dy holds uniformly in x. Here β := − 1 2 ∇ · q + V . To the best of our knowledge a formula of this type first appeared in [21] for the Schrödinger operator with real short-range potential |V (x)| ≤ C(1+|x|) −µ , µ > 1 in R 3 . This formula has since then been generalized to Schrödinger operators in any dimensions with more general singular potentials and in some cases to nonlinear coefficients, see e.g., [18,10,23]. The significance of Saito's formula for inverse problems is apparent from its corollaries.
Corollary 1 (Uniqueness). Let q 1 , V 1 and q 2 , V 2 be as in Theorem 1.1. If the corresponding scattering amplitudes for these coefficients coincide for some sequence k j → ∞ then the corresponding coefficients β 1 and β 2 are equal in the sense of distributions.
Corollary 2 (Representation formula). Under the same assumptions as in Theorem 1.1 we have in the sense of tempered distributions.
This paper is organized as follows. In Section 2 we fix some notations and recall asymptotic formulas for Hankel and Macdonald functions. Then in Section 3 we show that a solution to (2) satisfies (3). In Section 4 we prove the existence and the uniqueness of the solution to (3). Several estimates for the solution and its related operators are given. Section 5 concerns the asymptotic behaviour of the solution u. In particular, we define the scattering amplitude which is regarded as the scattering data for the inverse problem. Finally, in Section 6 we give a proof for Saito's formula.
2. Preliminaries. We use the following notations throughout the text. The letter C will be used to denote a generic constant C > 0 which may have different values from step to step. The symbol p denotes the Hölder conjugate of p, i.e. 1/p +1/p = 1. The weighted Lebesgue spaces L p δ are defined by the norm The Sobolev spaces are defined by We say that q ∈ W 1 p,δ (R n ), if each scalar-valued component of q belongs to W 1 p,δ (R n ) in the above sense. As usual, in the case of L 2 δ -based Sobolev space, we write W s 2,δ (R n ) =: H s δ (R n ). Later we will need some knowledge about the behaviour of the function G + k . We start by recalling that the asymptotic behaviour of Hankel functions is as x → 0+ (see e.g., [26,16]). The Macdonald function K ν has the same behaviour at x → 0+. Moreover, for all ν ≥ 0 Correspondingly, Macdonald's function has the asymptotic representation The explicit form of the fundamental solution G + k allows us to conclude that for k > 0 the function G + k behaves like On the other hand, the relations and chain rule imply that for all n ≥ 2. This leads to the asymptotic behaviour , k|x| → +∞.

Remark 1.
In the three dimensional case the fundamental solution G + k (k > 0) has a compact explicit representation see e.g., [16].
3. From scattering problem to Lippmann-Schwinger equation. This section is devoted to showing that a solution to the scattering problem (2) must satisfy the Lippmann-Schwinger equation. The proof is classical and uses similar techniques as Cakoni and Colton [6] and Eskin [7].

Proof. By Hölder's inequality we obtain
We choose 2r = p which implies 2r = 2p p−2 . By the Sobolev embedding theorem we have the continuous embedding for some constant c 0 > 0 depending only on p and n. Since for some constant C > 0 and for fixed k > 0.
Proof. By using the radiation condition for u sc and ∆u sc and then applying Green's second identity we calculate (with slight abuse of o-notation) Using the Cauchy-Schwartz inequality in both terms gives where the first term on the right-hand side is finite according to Lemma 3.1. This inequality then implies the claim.
denotes the open ball of radius R centered at origin. Let ε > 0 be such that B ε (x) ⊂ B R (0) and denote Ω R,ε := B R (0)\B ε (x). Then by Lemma A.1 we calculate Letting ε → 0+ we obtain the equality Note that for fixed x ∈ R n and large R > 0 we have . Therefore, letting R → +∞ we see that the latter integral tends to zero. This follows from Lemma 3.2 and the fact that all u sc , ∆u sc , G + k and ∆G + k satisfy the radiation condition.

Corollary 3.
If v ∈ H 4 loc (R n ) satisfies the radiation condition at infinity and fulfils the homogeneous equation 4. Solution to Lippmann-Schwinger equation. Next we show that when k > 0 the integral equation (4) can be solved by iterations defined by u (j) sc = L j k u 0 for j = 0, 1, . . . We denote by G + k the convolution operator with kernel G + k . Our first lemma gives the required mapping properties for this integral operator. Similar estimates for Schrödinger operator are familiar in the literature and can be met for example in [7] and [19]. The first lemma is analogous to the limiting absorption principle.
Here the constant C 0 only depends on n and δ. Proof. The claim follows from Agmon's estimate [1, Appendix A, Remark 2] for p = 2, δ > 1 2 and k > 1 given by where the constant C 0 > 0 only depends on n and δ. We use this result to the opposite direction and take only the terms where |α| equals 0,1 or 2 to get the claim.
We now proceed to prove norm estimates for the operator L k .
where 2δ > n − n p and n < p ≤ ∞. Then the following properties are satisfied 1. The function u 0 belongs to H 2 −δ (R n ) with the estimates is bounded and satisfies the norm estimates Proof. In view of Lemma 4.1 we consider only the case j = 0 and the estimates for derivatives follow. We apply Lemma 4.1 to estimate By Lemmata 3.1 and 4.1 we see that . The proof is then finished. Corollary 4. Let q ∈ W 1 p,2δ (R n ) and V ∈ L p 2δ (R n ) where n < p ≤ ∞ and 2δ > n− n p . Then there exists a constant k 0 > 1 such that the function u sc (x, k, θ) defined by the series Proof. We show that . . By Theorem 4.2 the claim holds for j = 0. Suppose then that the claim is proved for some j > 0. By induction hypothesis , so the claim holds for all j = 0, 1, . . . We may now choose any k 0 > max{1, √ C 2 } to conclude that the series (6) converges in H 1 −δ (R n ). Because the operator L k is linear and maps continuously in H 1 −δ (R n ) the series solves (4). Similar calculation shows that u sc ∈ H 2 −δ (R n ).
To verify the uniqueness of the solution, suppose that u and v are solutions to (4). Then by Theorem 4.2 we have Our With these results the limiting absorption principle can be extented for the full operator H 4 .
Corollary 5. Let q ∈ W 1 p,2δ (R n ) and V ∈ L p 2δ (R n ) with n < p ≤ ∞ and 2δ > n− n p . Then the operator . By (formally) applying this equation repeatedly we obtain The convergence of this series in H 1 −δ follows from the estimates of Theorem 4.2 and the norm estimates follow from those of G + k , so there exists a unique solution. Then it is straight-forward to confirm that G p satisfies (H 4 − k 4 ) G p = I.

Remark 2. The previous proof yields a useful identity
Proposition 1. Suppose n = 2 or 3 and let q ∈ W 1 p,2δ (R n ) and V ∈ L p 2δ (R n ), with 2δ > n − n p and n < p ≤ ∞. Then the operator L k maps W 1 for k ≥ 1.
Proof. We start from the case n = 3 and choose some f ∈ W 1 ∞ (R 3 ). Then a direct estimation by using Remark 1 yields For the gradient of L k f we have These integrals converge uniformly in x when n < p ≤ ∞, which yields the claim in R 3 .
Consider next the case n = 2. Suppose that f ∈ W 1 ∞ (R 2 ). Then by using the asymptotic representation obtained in Section 2 we get Since q, V ∈ L 1 (R 2 ) then I 2 satisfies the claim. In the first integral we may use Hölder's inequality to get The above integral converges when 1 ≤ p < ∞, that is, 1 < p ≤ ∞, and therefore the claim holds also for I 1 . The gradient of L k f can be estimated straight-forwardly as Again, J 2 clearly satisfies the claimed estimate. In the term J 1 Hölder's inequality yields This integral converges when 1 ≤ p < 2, or 2 < p ≤ ∞, concluding the proof.
Proof. The proof is essentially the same as the proof of Corollary 4, but uses the estimates of Proposition 1.

Asymptotics for Lippmann-Schwinger equation.
Let us first prove an elementary fact about an integral that appears in the sequel. Proof. For all θ ∈ S n−1 and y ∈ R n it holds that (θ , y) ≤ |(θ , y)| ≤ |y|. Therefore e k(θ ,y) ≤ e k|y| . Then we estimate the integral This concludes the proof.
Next we study the asymptotic behaviour of the solution u obtained in the previous section.
Theorem 5.2. Let q ∈ W 1 p,2δ (R n ) and V ∈ L p 2δ (R n ) with 2δ > n − n p and n < p ≤ ∞. Then for fixed k > 0 the solution u(x, k, θ) to (3) admits the representation , |x| → ∞, if n = 2 or 3. The same representation holds for any n ≥ 4 if we assume in addition that q and V are compactly supported. The constant C n is given by The function A(k, θ , θ) is called the scattering amplitude and is defined by where θ = x |x| is the direction of observation.
Proof. We begin the proof along the lines of [10, Lemma 3.1]. Let 0 < a < 1 be a parameter, so that we can decide its value to suit our interests later. Since we can divide the region of integration in two parts as |y| ≤ |x| a and |y| > |x| a . In the former case we apply the Maclaurin series (1 + ω) s = 1 + sω + O(ω 2 ) to see that Then, for fixed k > 0 and |y| ≤ |x| a we take a < 1 2 . Because |y| ≤ |x| a ≤ |x| → +∞ then we may assume that k|x − y| > 1. This allows us to use the large argument asymptotic behaviour of Hankel and Macdonald functions and the Maclaurin expansion e ω = 1 + O(ω) to compute Let us apply these estimates to u sc . We have Now integral I 1 equals As |x| → ∞ the second integral above is o |x| − n−1

2
, since q · ∇u and V u are integrable. On the other hand, by Lemma 5.1 the third term in this expression is also o |x| − n−1 2 .
Next we consider I 2 . We first split the region of integration as In the case J 1 we have |x−y| ≥ |x|−|y| > |x| 2 . Thus as |x| → +∞ by the asymptotic of G + k we have Note that so far we have only used the fact that q ∈ W 1 p,2δ (R n ) and V ∈ L p 2δ (R n ), n ≥ 2. If these coefficients are compactly supported, then choosing large enough |x| immediately yields J 2 = 0, giving the claimed asymptotic representation. On the other hand, even if the coefficients are not compactly supported, in dimensions n = 2, 3 we know that the solution u to (3) is in W 1 ∞ (R n ). We study first the case n = 3. By Remark 1 we have for a fixed k > 0 that where we did the change of variables y = z|x|. Dividing the region of integration to 2 > |z| > 1 2 and |z| > 2 we get because p < n/2. In the region |z| > 2 we have |(x/|x|) − z| ≥ |z| − 1 > |z|/2 and therefore Recalling that 3 p − 2δ < 0 we conclude that J 2 is o 1 |x| . In the case n = 2 we use the asymptotic behaviour of G + k . We consider the regions k|x − y| < 1 and k|x − y| > 1 separately, with the respective integrals denoted by J 2 and J 2 . In the first case In the second case J 2 we pick some σ > 0 such that 0 < 2δp −2 < σ < p 2 +2δp −2. Then By changing the variables y/|x| = z in the above integral we get If the above integral converges, we have that J 2 = o |x| − 1 2 , since −2δ + 2 p < 0. And indeed, the integral converges because where the first integral is finite due to the choice of σ. The second integral also converges, as can be seen by splitting This concludes the proof.

Remark 3.
In similar spirit as in the proof of Theorem 5.2 one can confirm that u sc and ∆u sc satisfy the radiation condition at infinity at least if q and V are smooth and compactly supported. For such coefficients, in light of Section 3 we can then conclude that the scattering problem (2) and Lippmann-Schwinger equation (3) are equivalent.
6. Proof of Theorem 1.1. Note that (unlike the proof for the asymptotic formula of u) this theorem does not require us to assume that q and V have compact supports.
Consider I 1 first. We can further split I 1 in two terms as In I (1) 1 we can first split the integral into two parts and then integrate the first one by parts and do a change of variables θ = −γ and θ = −γ in the second term as follows R n e −ik(θ −θ,y) ik(θ − θ ) · q(y)dydθdθ Here ∇ · q denotes the divergence of q. The integration by parts is valid, since . But now the last integral is equal to −I (1) 1 , so that and An integral of this form was studied in [10] and we carry out the analogous analysis here. It is known that , and one proof for this fact can be found for example from [10, Lemma 3.6]. Here J ν (x) is the Bessel function of first kind and of order ν. From [16] we find that We split the region of integration as (k|x − y|) n−2 β(y)dy =: I 1 + I 1 .
The term I 1 → 0 as k → +∞. Indeed, where n−1− n p = n p −1 < 0 when p > n. The term I 1 converges to the claimed limit of this theorem as k → +∞. To see this we utilize the large argument behaviour of Bessel function to get Here the first integral gives the claimed limit as k → +∞, since R n |β(y)| |x − y| n−1 dy < ∞ uniformly in x when p > n. The second integral above gives The latter integral I * * 1 → 0 as k → +∞, while the former integral requires little more effort. We do the change of variables y = x − z and then swap to polar coordinates to obtain Define Now it suffices to conclude that f x ∈ L 1 (0, ∞) uniformly in x and then apply the Riemann-Lebesgue lemma to conclude that I * 1 → 0 as k → +∞. We calculate the L 1 -norm of f x to see that β(y) |x − y| n dy → 0 as k → +∞. Let ε > 0 be a parameter whose value will be chosen later. Estimate where we required ε < n p . This estimate shows that all of the terms of I 1 tend to 0 as k → ∞.
Next we confirm that the term I 2 → 0 as k → ∞. To achieve this we calculate the integrals with respect to θ and θ as in [18] and use the mapping properties of G p . First we separate the integral into two parts with respect to q and V as Recall from Remark 2 that u sc = − G p ( q · ∇u 0 + V u 0 ). This allows us to write From [26] we find that d dx (x −ν J ν (x)) = −x −ν J ν+1 (x) and therefore in I Next, we may write q = ( q/| q|)| q|   2 . Now we define operator K 1 as the integral operator with the kernel |V (y)| 1 2 G p (y − ·)q 1 2 (·). Actually, K 1 : L 2 (R n ) → L 2 (R n ) uniformly in x with the same norm estimates as G p . Indeed, let f ∈ L 2 (R n ). Then, since q ∈ W 1 p,2δ (R n ), we have that Due to the Sobolev embedding theorem we have the continuous embedding . Thus, taking a function g ∈ H 1 −δ (R n ) and applying Hölder's inequality gives These estimates and the mapping property of G p finally yield . Next we show that the function from (7) satisfies Here it appears to be simpler to conclude directly that and to not factorize the integrals into new operators. We estimate the square of the above norm by Dividing the region of integration to |x − z| < 1 and |x − z| > 1 we see that . The above integral converges uniformly in x, when n < p ≤ ∞. Thus, since we can use the previous calculations and the fact that ∇ G p maps L 2 δ to L 2 −δ with the same estimates as earlier. Now for g ∈ L 2 −δ (R n ) we have that . Replacing the first |V | 1 2 by | q| 1 2 in the previous calculations and using the above norm estimate we may conclude that I (1) 2 satisfies the same estimate in k as I 2 . Therefore, I 2 → 0 as k → +∞ and the theorem is proved.
The proofs of Corollaries 1 and 2 are well-known in literature and can be found for example in [18] or [10] (case n = 3).

Conclusion.
A direct scattering problem for a first-order perturbation of the biharmonic operator was studied. It was shown that a solution to scattering problem with certain radiation conditions satisfies the Lippmann-Schwinger equation. This integral equation has a unique solution in the weighted Sobolev space H 1 −δ in any dimension and in particular this solution is bounded if the dimension is 2 or 3. The proofs of these results are based on a Green-type formula for bi-Laplacian and Agmon's estimate [1]. The asymptotic behaviour of the solution for fixed k > 0 as |x| → +∞ was studied and a formula for the scattering amplitude was obtained.
The main result of this paper, Saito's formula, was proved under quite general assumptions on the coefficients. More precisely, the formula for scattering amplitude was proved under the assumption that the coefficients of the direct operator are compactly supported, but this assumption was not necessary for the proof of Saito's formula. The proof itself was based on explicit calculations starting from the formula for scattering amplitude and utilizing the properties of certain resolvent operator related to the direct operator. Some consequences of Saito's formula were discussed. Namely, the scattering amplitude uniquely determines a combination of the coefficients for the direct problem and in turn gives a uniqueness result for the inverse problem. A representation formula for this combination of the coefficients was given.
Let f ∈ W 1 p (R n ). If a = b then by the mean value theorem we have the equality |a p − b p | = p|ξ| p−1 |a − b| for some ξ ∈]a, b[. Thus for the function |f | we find that a.e., and therefore |f | p can be chosen to be uniformly (even (1−n/p)-Hölder) continuous. Next, assume in contrary that |f | p does not vanish at infinity. It means that there exists a constant c > 0 and points x j ∈ R n so that |f (x j )| p ≥ c. We may pick a sequence of these points with the property that for each m ∈ N we have |x m | ≥ m. Since |f | p is uniformly continuous, there exists δ > 0 (that only depends on c) such that |f (x)| p ≥ c 2 when |x m − x| < δ. This is a contradiction since |f | p is integrable, but |x m | → ∞ and |x−xm|<δ where |B δ (x m )| denotes the volume of a ball of radius δ centered at x m .
Finally, if f did not vanish at infinity then we would find constant c > 0 and points x m ∈ R n so that |f ( x m )| ≥ c. But this means that |f ( x m )| p ≥ c p , which is a contradiction, as we just saw.
Remark 4. In dimension n = 1 it is possible to prove straight-forwardly that W 1 1 (R) ⊂Ċ(R). Moreover, Theorem A.2 does not hold for p = ∞, since for example all constant functions belong to W 1 ∞ (R n ), but the only constant function vanishing at infinity is the constant C ≡ 0.