1 Introduction and Main Result

This paper concerns inverse scattering problem at fixed angles for the high order Schrödinger operator with potential, defined by

$$\begin{aligned} H=(-\Delta )^m+q(x),\ \ x\in {\mathbb {R}}^n, \end{aligned}$$

where \(m=1, 2, \ldots\), and \(n\ge 3\).

We will assume that the potential q(x) is \(C_0^\infty ({\mathbb {R}}^n)\) with small norm in certain Sobolev spaces \(W^{\alpha ,2}\), i.e.,

$$\begin{aligned} q\in C_0^\infty (B(0,R)), ||q||_{W^{\alpha ,2}}<A. \end{aligned}$$
(1)

Here B(0, R) denotes the ball centered at the origin with radius \(R>0\), A is a constant small enough.

We start by recalling the definition of the scattering amplitude associated with q. We consider the high order Schrödinger equation,

$$\begin{aligned} (H-k^{2m})u(x)=0,\,\,\,k>0,\,\,\,x\in {\mathbb {R}}^n. \end{aligned}$$
(2)

One assumes that for each \(k>0\) and each \(\omega \in S^{n-1}\), the Eq. (2) has a unique solution of the form \(u=e^{ik\omega \cdot x}+v(x,w,k)\) such that \(v=\lim _{\varepsilon \downarrow 0}v_\varepsilon\), where \(v_\varepsilon \in L^2({\mathbb {R}}^n)\) is the solution of

$$\begin{aligned} (-\Delta )^mv_\varepsilon +qv_\varepsilon -(k+i\varepsilon )^{2m}v_\varepsilon =-e^{ik\omega \cdot x}q. \end{aligned}$$
(3)

From (3) it follows that

$$\begin{aligned} v=((-\Delta )^m-(k+i0)^{2m}))^{-1}(-qu). \end{aligned}$$
(4)

Since \(q\in C_0^\infty ({\mathbb {R}}^n)\), the following expression holds (see Lemma 3 in [1]):

$$\begin{aligned} v(x)=C_nk^{(n+1)/2-2m}|x|^{-(n-1)/2}e^{ik|x|}a(\theta ,\omega ,k)+o(|x|^{-(n-1)/2}), \text {as} |x|\rightarrow \infty , \end{aligned}$$
(5)

where \(\theta =x/|x|\) is the reflecting angle and

$$\begin{aligned} a(\theta ,\omega ,k)=\int _{{\mathbb {R}}^n}e^{-ik(\theta ,y)}(-qu)(y)dy. \end{aligned}$$
(6)

The function \(a(\theta ,\omega ,k)\) on \(S^{n-1}\times S^{n-1}\times (0,\,\infty )\) is known as the scattering amplitude.

The main result states that the scattering amplitude at two fixed angles can determines the potential.

Theorem 1

Suppose q(x) satisfy (1), \(n\ge 3,\ m\) and \(\alpha\) satisfy

$$\begin{aligned} max\{0,\frac{n}{2}-\frac{n}{n-1}(4m-3)\}<\alpha \le 1 \ \ \ and \ \ \ n>2m-1, \end{aligned}$$
(7)

then the fixed angles scattering amplitude \(a(\theta ,\pm \theta _0,k)\) can uniquely determines the potential q(x), where \(\theta _0 \in S^{n-1}\) is a fixed direction.

Remark 1

For \(m=1\), this result holds for \(n=3(0<\alpha \le 1)\) and \(n=4(2/3<\alpha \le 1)\), this comes from Barceló, Castro, Luque and Vilela [2, 3]. We extend this result to the case \(m\ge 2\).

Remark 2

For the expression (5), from the condition \(q(x)\in C^\infty ({\mathbb {R}}^n)\) and Eq. (3), one can show that \(v_\varepsilon \in C^\infty ({\mathbb {R}}^n)\) by iteration, and that \(u\in C^\infty ({\mathbb {R}}^n)\). Since q(x) is compact supported, we have \(uq\in C_0^\infty ({\mathbb {R}}^n)\), which ensured ‘the condition (a)’ in the proof of Lemma 3 in [1], thus, the expression (5) holds automatically.

We now mainly describe earlier results on the inverse scattering problem.

In the case \(m=1\), there has been considerable work devoted to recovering information of the potential from the backscattering data for Schrödinger operators. The inverse backscattering problem was considered by Prosser [4] for small potentials in the so-called Friedrich class in three dimensions. Lagergren has also proved similar results for smooth compact supported potentials in three dimensions, in which different Banach spaces were used, see Lagergren [5 Chap. 8]. Eskin and Ralston [6,7,8] obtained the local uniqueness of the potential from the backscattering data. For the inverse scattering problems, the magnetic field and electric potential were uniquely determined in Eskin and Ralston [9], and this result were improved by Päivärinta, Salo and Uhlmann [10] from a different method. The uniqueness results for the fixed energy scattering problem with compact electric potential were prove in [11, 12]. As for the fixed angle scattering, we refer to Barceló, Castro, Luque and Vilela [2, 3], they recovered a real compact supported potential from the fixed angle scattering data by a new iterative method.

In the case \(m=2\), we mention that, Tyni and Harju [13,14,15] considered inverse backscattering problem for bi-harmonic operators with lower order perturbations and for Schrödinger operators with potentials without compact supports, where they were able to recover jumps and singularities of an unknown combination of potentials by using the inverse Born approximation.

In the case \(m\ge 2\), we refer to [16], Huang, Duan and Zheng proved that the potentials can be uniquely determined by the scattering matrix of the higher order Schrödinger operator. We also refer to [1], the author consider the inverse backscattering problem for the higher order Schrödinger operator. For the higher order or fractional order Schrödinger operator case, we also refer to the work [17,18,19,20,21], they considered the inverse boundary value problem.

In this paper, we consider the case \(m\ge 1\) and prove the uniqueness of the potential from the scattering amplitude at two fixed angles, which extends the second order case to the high order case on this problem.

The rest of this paper is organized as follows. Section 2 contains the construction of a iteration sequences and several known results. In Sect. 3, we are able to prove Theorem 1. In Sect. 4, we will prove the Proposition 1 and Proposition 2 used in the proof of Theorem 1.

2 Preliminaries

We first rewrite the inverse scattering problem in equivalent integral formulation, and denote the outgoing resolvent operator of the high order Laplace operator as

$$\begin{aligned} R_0(f)=((-\Delta )^m-(k+i0)^{2m}))^{-1}(f). \end{aligned}$$
(8)

Then, from (4), v is the solution of the Lippmann-Schwinger integral equation

$$\begin{aligned} v(x,w,k)=-R_0(qe^{ik\omega \cdot x})-R_0(qv(x,w,k)). \end{aligned}$$
(9)

Moreover, from (6) we have that

$$\begin{aligned} -a(\theta ,\omega ,k)=\int _{{\mathbb {R}}^n}e^{-ik((\theta -\omega ),y)} q(y)dy+\int _{{\mathbb {R}}^n}e^{-ik(\theta ,y)}(qv(x,w,k))(y)dy. \end{aligned}$$
(10)

The first term on the right hand of (10) can be interpreted as a suitable Fourier transform that can be inverted to obtain the so-called Born approximation. For fixed \(\theta _0\), we have

$$\begin{aligned} {\mathbb {R}}^n=H_{\theta _0}\cup H_{-\theta _0}=\{\xi \in {\mathbb {R}}^n: \xi \cdot \theta _0<0\}\cup \{\xi \in {\mathbb {R}}^n: \xi \cdot \theta _0>0\}. \end{aligned}$$
(11)

Then, for \(\xi \in H_{\pm \theta _0}\), there exists unique \(\theta (\xi )\in S^{n-1}\) and \(k(\xi )>0\) such that

$$\begin{aligned} \xi =k(\xi )(\theta (\xi )\mp \theta _0). \end{aligned}$$

Let

$$\begin{aligned} \theta _0(\xi )= {\left\{ \begin{array}{ll} {\theta _0} &{} {\xi \in H_{\theta _0}},\\ {-\theta _0} &{} {\xi \in H_{-\theta _0}}. \end{array}\right. } \end{aligned}$$

Then, the Born approximation for fixed angle scattering amplitude of a potential q is defined by

$$\begin{aligned} \widehat{q_{\theta _0}}(\xi )=-a(\theta (\xi ),\theta _0(\xi ),k(\xi )). \end{aligned}$$

We insert iteratively the Lippmann-Schwinger integral equation (9) into (10),

$$\begin{aligned} \begin{aligned} -a(\theta ,\theta _0,k)=&\int _{{\mathbb {R}}^n}e^{-ik(\theta -\theta _0)\cdot y}q(y)dy +\sum _{j=1}^{\tau }\int _{{\mathbb {R}}^n}e^{-ik(\theta \cdot y)}(qR_0)^j(qe^{ik\theta _0\cdot (\cdot )})(y)dy\\&+\int _{{\mathbb {R}}^n}e^{-ik(\theta ,y)}(qR_0)^\tau (qv)(y)dy. \end{aligned} \end{aligned}$$
(12)

For convenience, we rewrite (12) as follows

$$\begin{aligned} {\widehat{q}}(\xi )={\widehat{q}}_{\theta _0}(\xi ) -\sum _{j=1}^{\tau }\widehat{Q_j(q)}(\xi )-\widehat{q_\tau ^r}(\xi ), \end{aligned}$$
(13)

where

$$\begin{aligned} \widehat{Q_j(q)}(\xi )=\int _{{\mathbb {R}}^n}e^{-ik(\xi )\theta (\xi )\cdot y}(qR_0)^j(qe^{ik(\xi )\theta _0(\xi )\cdot (\cdot )})(y)dy, \end{aligned}$$
(14)

and

$$\begin{aligned} \widehat{q_\tau ^r}(\xi )=\int _{{\mathbb {R}}^n}e^{-ik(\xi )\theta (\xi )\cdot y}(qR_0)^\tau (qv)(y)dy. \end{aligned}$$
(15)

The convergence of this series suggests that the last term in (13) should be small for large \(\tau\). Based on this suggestion we use the following family of reduced equations for \(q_m\), where we have removed this last term,

$$\begin{aligned} \widehat{q_\tau }(\xi )={\widehat{q}}_{\theta _0}(\xi ) -\sum _{j=1}^{\tau }\widehat{Q_j(q_\tau )}(\xi ), \ \ \ \xi =k(\xi )(\theta (\xi )\mp \theta _0), \end{aligned}$$
(16)

with \((\theta ,k)\in S^{(n-1)}\times (0,\infty )\). These reduced equations have the advantage that they do not involve v(x, w, k), avoiding the solution of Lipmann-Schwinger equation (9). However, for each \((\theta ,k)\in S^{(n-1)}\times (0,\infty )\), (16) is still nonliner in \(q_\tau\). Moreover, it is not clear if there exists a unique function \(q_\tau\) satisfying (16).

For this reason, we propose a fixed point produce to find approximations of \(q_\tau\). and define a linear operator \(L_\tau\) as,

$$\begin{aligned} \widehat{L_\tau (q)}(\xi )={\widehat{q}}_{\theta _0}(\xi ) -\sum _{j=1}^{\tau }\widehat{Q_j(q)}(\xi ). \end{aligned}$$
(17)

Then, if \(q_\tau\) is a solution of (16), it must be a fixed point of \(L_\tau\) and we can try the usual iterative method based on powers of \(L_\tau\) to approximate \(q_\tau\). This requires that \(L_\tau (q)\) is of compact support. Therefore, instead of \(L_\tau\) we consider the modified operator

$$\begin{aligned} T_\tau (q):=\phi {L_\tau (q)}=\phi q_{\theta _0}-\phi \sum _{j=1}^{\tau }{Q_j(q)}, \end{aligned}$$
(18)

where \(\phi \in C^\infty\) is a cut-off function with compact support satisfying

$$\begin{aligned} \phi (x)=1 \ if\ |x|<R \ and \ \phi (x)=0 \ if\ |x|>2R. \end{aligned}$$
(19)

For each \(\tau \in N\), we consider the sequence \(\{q_{\tau ,l}\}_{l\in N}\) defined recursively by

$$\begin{aligned} q_{\tau ,1}=0,\ \ \ q_{\tau ,l+1}=T_\tau (q_{\tau ,l}),\ l\ge 1. \end{aligned}$$
(20)

Note that \(q_{\tau ,2}=\phi q_{\theta _0}\), which is a good approximation to a potential q with support in B(0, R).

In next subsection we will show that the sequence of approximations \(\{q_{\tau ,l}\}_{\tau ,l\in N}\) converges to the potential q in some sense.

More precisely, under the assumption in Theorem 1, we will prove the following convergence in Section 3, there exists \(q_\tau \in W^{\alpha ,2}({\mathbb {R}}^n)\) satisfying

$$\begin{aligned} q_\tau =\lim _{l\rightarrow \infty } q_{\tau ,l} \ \ \ in \ W^{\alpha ,2}({\mathbb {R}}^n), \end{aligned}$$

and the sequence \(\{q_{\tau }\}_{\tau \in N}\) satisfies

$$\begin{aligned} q=\lim _{\tau \rightarrow \infty } q_{\tau } \ \ \ in \ W^{\alpha ,2}({\mathbb {R}}^n). \end{aligned}$$

To estimate (14) for every \(j\in N\), we define the following multilinear operator defined via its Fourier transform:

$$\begin{aligned} P_j(f)(x)=\int _{{\mathbb {R}}^n}e^{ix\cdot \xi }\widehat{P_j({\textbf {f}})}(\xi )d\xi =(\int _{H_{\theta _0}} +\int _{H_{-\theta _0}})e^{ix\cdot \xi }\widehat{P_j({\textbf {f}})}(\xi )d\xi , \end{aligned}$$
(21)

where \({\textbf {f}}=(f_1,f_2,...,f_{j+1}),\theta _0\in S^{n-1}\), and \(\widehat{P_j({\textbf {f}})}(\xi )\) defined as

$$\begin{aligned} \widehat{P_j({\textbf {f}})}(\xi )=\int _{{\mathbb {R}}^n}e^{-ik\theta \cdot y}(f_{j+1}R_0...f_2R_0)(f_1e^{ik\theta _0\cdot (\cdot )})(y)dy. \end{aligned}$$
(22)

Observe that in the particular case \(f_i=q,i=1,2,...,j+1\), we have that

$$\begin{aligned} \widehat{P_j({\textbf {f}})}(\xi )=\widehat{Q_j(q)}(\xi ), \ \ \ \xi \in S^n. \end{aligned}$$
(23)

Use the methods in [2], we can prove the following two propositions, which extend their result(see Proposition 2.1, Proposition 2.2 and Corollary 2.3 in [2]) to the higher order case.

Proposition 1

Let \(n\ge 3,\ m\) and \(\alpha\) satisfying

$$\begin{aligned} 0<\alpha \le 1,\ \ \ \frac{n}{2}-\frac{n}{n-1}(4m-3)<\alpha <\frac{n}{2} \ \ \ and \ \ \ n>2m-1. \end{aligned}$$
(24)

For each \(j\in N\) fixed, \({\textbf {f}}=(f_1,f_2,...,f_{j+1})\) with \(f_l\in W^{\alpha ,2}\) and compactly supported with support in B(0, R) for \(l=1,...,j+1\). Then, there exists a constant \(C_1\) such that

$$\begin{aligned} ||P_j({\textbf {f}})||_{W^{\alpha ,2}}\le C_1^j\prod _{l=1}^{j+1}||f_l||_{W^{\alpha ,2}}. \end{aligned}$$
(25)

Use Proposition 1, we have the following estimates, which will be used in Section 3.

Proposition 2

Let \(n\ge 3,\ m\) and \(\alpha\) satisfy (24), \(q\in W^{\alpha ,2}\) be a real valued function with compact support in B(0, R) and such that

$$\begin{aligned} ||q||_{W^{\alpha ,2}}< A, \end{aligned}$$
(26)

where A is the constant small enough. Then, for every \(\tau \in N\) there exists a constant \(C_2\) such that

$$\begin{aligned} ||q_\tau ^r||_{W^{\alpha ,2}}\le C_2^\tau ||q||_{W^{\alpha ,2}}^{\tau +1}, \end{aligned}$$
(27)

and there exists a constant \(C_3\) such that

$$\begin{aligned} ||q_{\theta _0}||_{W^{\alpha ,2}}\le C_3||q||_{W^{\alpha ,2}}. \end{aligned}$$
(28)

Next, we introduce the following known result concerning the product of functions in Sobolev spaces due to Zolesio [22], which will be used in the proof of Theorem 1.

Lemma 1

Let \(0\le \alpha \le s\) and \(s>n/2\), there exists a constant \(C>0\) such that

$$\begin{aligned} ||\phi g||_{W^{\alpha ,2}}\le C||\phi ||_{W^{s,2}}||g||_{W^{\alpha ,2}}\le C||g||_{W^{\alpha ,2}}, \ \ \ \forall g \in W^{\alpha ,2}({\mathbb {R}}^n), \end{aligned}$$
(29)

where \(\phi\) is defined in (19).

3 Proof of Theorem 1.

We shall adapt the strategy in [2, 3], and state the main steps as follows.

Step 1. Boundedness of the sequence \(\{q_{\tau ,l}\}_{l\in N}\) in \(W^{\alpha ,2}({\mathbb {R}}^n)\).

From (18) and (20),we have that

$$\begin{aligned} ||q_{\tau ,2}||_{W^{\alpha ,2}}=||\phi q_{\theta _0}||_{W^{\alpha ,2}}. \end{aligned}$$

Using Lemma 1 and (29) we get

$$\begin{aligned} ||q_{\tau ,2}||_{W^{\alpha ,2}}\le C || q_{\theta _0}||_{W^{\alpha ,2}}\le C C_3|| q||_{W^{\alpha ,2}}, \end{aligned}$$

for \(\alpha\) and q under the assumptions of Proposition 2.

Form (18), (20), (29), (23) and Proposition 1, we obtain

$$\begin{aligned} \begin{aligned} ||q_{\tau ,l+1}||_{W^{\alpha ,2}}&\le ||\phi q_{\theta _0}||_{W^{\alpha ,2}} +||\phi \sum _{j=1}^{\tau }{Q_j(q_{\tau ,l})}||_{W^{\alpha ,2}}(\ use \ (18), (19))\\&\le C C_3 || q||_{W^{\alpha ,2}}+ C\sum _{j=1}^{\tau } ||{Q_j(q_{\tau ,l})}||_{W^{\alpha ,2}}(\ use \ (29))\\&\le CC_3 || q||_{W^{\alpha ,2}}+C \sum _{j=1}^{\tau }C_1^j(2CC_3)^{j+1}||{q}||_{W^{\alpha ,2}}^{j+1}( \ use\ (23), Proposition 2). \end{aligned} \end{aligned}$$

Since for q satisfying (26), we can get

$$\begin{aligned} 2C \sum _{j=1}^{\tau }C_1^j(2CC_3)^{j}||{q}||_{W^{\alpha ,2}}^{j}\le 1, \end{aligned}$$

then, for each \(\tau \in N\), we have the following estimate,

$$\begin{aligned} ||q_{\tau ,l}||_{W^{\alpha ,2}}\le 2C C_3||q||_{W^{\alpha ,2}}, \ \ \ \ \forall l\ge 2. \end{aligned}$$
(30)

Step 2. Convergence of the sequence \(\{q_{\tau ,l}\}_{l\in N}\) for l in \(W^{\alpha ,2}({\mathbb {R}}^n)\).

From (18) and (20), using (30), we have that

$$\begin{aligned} ||q_{\tau ,l+1}-q_{m,\iota +1}||_{W^{\alpha ,2}} \le C ||\sum _{j=1}^{\tau }({Q_j(q_{\tau ,l})}-Q_j(q_{\tau ,\iota }))||_{W^{\alpha ,2}}. \end{aligned}$$

Using (23), the fact that \(P_j\) is a multi-linear operator, and the triangular inequality, we get

$$\begin{aligned} ||q_{\tau ,l+1}-q_{\tau ,\iota +1}||_{W^{\alpha ,2}} \le C \sum _{j=1}^{\tau }\sum _{i=1}^{j+1}||P_j(f_{\tau ,l,\iota ,i})||_{W^{\alpha ,2}}, \end{aligned}$$
(31)

where

$$\begin{aligned} f_{\tau ,l,\iota ,i}=(q_{\tau ,l},...,q_{\tau ,l},\underbrace{q_{\tau ,l}-q_{\tau ,\iota }} _{i-position},q_{\tau ,\iota },...,q_{\tau ,\iota }). \end{aligned}$$

Using (25) and (30) in (31) we obtain

$$\begin{aligned} \begin{aligned} ||q_{\tau ,l+1}-q_{\tau ,\iota +1}||_{W^{\alpha ,2}}&\le C \sum _{j=1} ^{\tau }(j+1)(2CC_1 C_3||q||_{W^{\alpha ,2}})^j||q_{\tau ,l}-q_{\tau ,\iota }||_{W^{\alpha ,2}} \\&\le B||q_{\tau ,l}-q_{\tau ,\iota }||_{W^{\alpha ,2}}, \end{aligned} \end{aligned}$$
(32)

where \(B<1\) is a small constant.

Therefore, for such a q, we have proved that \(\{q_{\tau ,l}\}_{l\in N}\) is a Cauchy sequence. As a consequence, there exists \(q_\tau \in W^{\alpha ,2}(R^n)\) such that

$$\begin{aligned} q_\tau =\lim _{l\rightarrow \infty }q_{\tau ,l} \ \ \ in \ \ W^{\alpha ,2}(R^n). \end{aligned}$$

From (30), we have that \(q_\tau\) satisfies

$$\begin{aligned} ||q_\tau ||_{W^{\alpha ,2}}\le 2C C_3||q||_{W^{\alpha ,2}}. \end{aligned}$$
(33)

Moreover,

$$\begin{aligned} q_\tau =\phi q_{\theta _0}-\phi \sum _{j=1}^{\tau }{Q_j(q_{\tau })}, \end{aligned}$$
(34)

since arguing as we did to get (32), we obtain

$$\begin{aligned} ||q_{\tau ,l+1}-(\phi q_{\theta _0}-\phi \sum _{j=1}^{\tau }{Q_j(q_{\tau })})||_{W^{\alpha ,2}} <||q_{\tau ,l}-q_{\tau }||_{W^{\alpha ,2}}. \end{aligned}$$

Step 3. \(q_\tau\) is converge to q in \(W^{\alpha ,2}({\mathbb {R}}^n)\) as \(\tau\) goes to infinity.

From (34),(13), and (19), since \(supp(q)\subset B(0,R)\), we have that

$$\begin{aligned} ||q_\tau -q||_{W^{\alpha ,2}}\le ||\phi \sum _{j=1}^{\tau }{(Q_j(q_{\tau })-Q_j(q))}||_{W^{\alpha ,2}}+||\phi q_\tau ^r||_{W^{\alpha ,2}}. \end{aligned}$$

Here,arguing as did to get (32),we obtain

$$\begin{aligned} ||q_\tau -q||_{W^{\alpha ,2}}\le D||q_\tau -q||_{W^{\alpha ,2}}+||\phi q_\tau ^r||_{W^{\alpha ,2}}, \end{aligned}$$
(35)

where

$$\begin{aligned} D=C \sum _{j=1}^{\tau }C_1^j\sum _{i=1}^{j+1}||q_ \tau ||_{W^{\alpha ,2}}^{i-1}||q||_{W^{\alpha ,2}}^{j+1-i}. \end{aligned}$$

Using (33), we have that

$$\begin{aligned} D\le C \sum _{j=1}^{\tau }(C_1||q||_{W^{\alpha ,2}})^j\sum _{i=1}^{j+1}(2CC_2)^{i-1}, \end{aligned}$$

and since \(||q||_{W^{\alpha ,2}}\) is small enough, we can get \(D\le 1/2\).

Thus, from (35) we have

$$\begin{aligned} ||q_\tau -q||_{W^{\alpha ,2}}\le 2||\phi q_\tau ^r||_{W^{\alpha ,2}}. \end{aligned}$$

Finally, using Lemma 1 and (27) we have that

$$\begin{aligned} ||q_\tau -q||_{W^{\alpha ,2}}\le 2CC_2^\tau ||q||_{W^{\alpha ,2}}^{\tau +1}. \end{aligned}$$
(36)

The result follows from here if \(||q||_{W^{\alpha ,2}}\) is small enough.

4 Proof of Propositions 1 and 2

In this section, we will prove Propositions 1 and 2. Before the proof, we introduce several known estimates for different operators in weighted Sobolev spaces.

The outgoing resolvent of the high order Laplacian denoted as \(R_0\)(defined in (8)), then the following lemma is true for small k.

Lemma 2

Let \(\alpha \in [0,2], k\in (0,b], b>0\), and \(\delta >1\). Then

$$\begin{aligned} ||R_0f||_{W_{-\delta }^{\alpha ,2}}\le C_{b,\delta }||f||_{L_\delta ^2}. \end{aligned}$$
(37)

For \(\theta _0\in S^{n-1}\) and \(k>0\) fixed, we introduce the following operator involving the outgoing resolvent of the high order Laplacian,

$$\begin{aligned} R_{k,\theta _0}f(x)=e^{-ik\theta _0\cdot x}R_0(f(\cdot )e^{ik\theta _0\cdot (\cdot )})(x). \end{aligned}$$
(38)

Lemma 3

Let \(\alpha \ge 0\), r and t be such that \(0\le \frac{1}{t}-\frac{1}{2}\le \frac{1}{n+1}\) and \(0\le \frac{1}{2}-\frac{1}{r}\le \frac{1}{n+1}\), \(\delta >1\), then

$$\begin{aligned} ||R_{k,\theta _0}f||_{W_{-\delta }^{\alpha ,r}}\le C k^{1-2m+\frac{n-1}{2}(\frac{1}{t}-\frac{1}{r})}||f||_{W_{\delta }^{\alpha ,t}}. \end{aligned}$$
(39)

We also introduce the restriction operator given by

$$\begin{aligned} S_{k,\theta _0}f(\theta )=\int _{R^n}e^{-ik(\theta -\theta _0)\cdot y}f(y)dy. \end{aligned}$$
(40)

The following lemma is a consequence of [23, Theorem 3(c)].

Lemma 4

Let \(\delta >1/2\), then

$$\begin{aligned} ||S_{k,\theta _0}f||_{L^2(S^{n-1})}\le C k^{-\frac{n-1}{2}}||f||_{L^2_\delta }. \end{aligned}$$
(41)

Using the Stein-Tomas restriction theorem we can get the following result which generalizes the previous one.

Lemma 5

( [24, Lemma 3.7]) Let \(\alpha >0\), \(0\le \frac{1}{t}-\frac{1}{2}\le \frac{1}{n+1}\), \(\omega (\theta )=|\theta -\theta _0|^\alpha\), then

$$\begin{aligned} ||\omega S_{k,\theta _0}f||_{L^2(S^{n-1})}\le C k^{\frac{n-1}{2}(\frac{1}{t}-\frac{3}{2})-\alpha }||f||_{W^{\alpha ,t}_{\delta (t)}}. \end{aligned}$$
(42)

Now we state the following result concerning the product of functions in weighted Sobolev spaces due to Zolesio [22].

Lemma 6

Let f be the compactly supported and \(\delta \in R\).

(i) For \(\alpha\) satisfying

$$\begin{aligned} 0 \le \alpha \ \ \ and \ \ \ \alpha > \frac{n}{2}-2 \end{aligned}$$
(43)

or

$$\begin{aligned} 0 < \alpha \ \ \ and \ \ \ \alpha \ge \frac{n}{2}-2, \end{aligned}$$
(44)

we have that there exists a constant \(C=C(supp(f))\) such that

$$\begin{aligned} ||fg||_{L^2_\delta } \le C ||f||_{W^{\alpha ,2}}||g||_{W^{2,2}_{-\delta }}, \ \ \ \forall \ g \in W^{2,2}_{-\delta }(R^n). \end{aligned}$$
(45)

(ii) For \(\alpha , t\) and r satisfying

$$\begin{aligned} 1 \le t <min (2,r) \ \ \ and \ \ \ 0 \le \frac{1}{2}+\frac{1}{r}-\frac{1}{t} \le \frac{\alpha }{n} \end{aligned}$$

or

$$\begin{aligned} 1 \le t \le min (2,r) \ \ \ and \ \ \ 0 \le \frac{1}{2}+\frac{1}{r}-\frac{1}{t} < \frac{\alpha }{n}, \end{aligned}$$

we have that there exists a constant \(C=C(supp(f))\) such that

$$\begin{aligned} ||fg||_{W^{\alpha ,t}_{\delta }} \le C ||f||_{W^{\alpha ,2}}||g||_{W^{\alpha ,r}_{-\delta }}, \ \ \ \forall \ g \in W^{\alpha ,r}_{-\delta }(R^n). \end{aligned}$$
(46)

(iii) For \(\alpha , \beta\) and p satisfying

$$\begin{aligned} 0 \le \beta \le min (\alpha ,2), \ \ \ 1\le p<2, \ \ \ and \ \ \ 0 \le n\left(1-\frac{1}{p}\right) \le \alpha +2-\beta , \end{aligned}$$

we have that there exists a constant \(C=C(supp(f))\) such that

$$\begin{aligned} ||fg||_{W^{\beta ,p}} \le C ||f||_{W^{\alpha ,2}}||g||_{W^{2,2}_{-\delta }}, \ \ \ \forall \ g \in W^{2,2}_{-\delta }(R^n). \end{aligned}$$
(47)

Proof of Proposition 1

From (11), for a fixed \(\theta _0\in S^{n-1}\) we have that

$$\begin{aligned} ||P_j(f)||_{W^{\alpha ,2}}^2 =(\int _{H_{\theta _0}}+\int _{H_{-\theta _0}})\langle \xi \rangle |\widehat{P_j(f)}(\xi )|^2d\xi =I+II. \end{aligned}$$

Let

$$\begin{aligned} \xi =k(\theta -\theta _0), \ \ \ d\xi =k^{n-1}|\theta -\theta _0|^2d\sigma (\theta )dk. \end{aligned}$$
(48)

Then

$$\begin{aligned} \begin{aligned} I\le&C\int _0^1 k^{n-1}\int _{S^{n-1}} |\widehat{P_j(f)(k(\theta -\theta _0))}|^2 d\sigma (\theta )dk\\&+C\int _1^{+\infty } k^{n-1+2\alpha }\int _{S^{n-1}}|\theta -\theta _0|^{2\alpha } |\widehat{P_j(f)(k(\theta -\theta _0))}|^2 d\sigma (\theta )dk\\&=C(I_1+I_2), \end{aligned} \end{aligned}$$
(49)

whenever \(0\le \alpha \le 1\).

Rewrite \(P_j\) as follows:

$$\begin{aligned} \widehat{P_j(f)}(k(\theta -\theta _0))=S_{k,\theta _0}(e^{-ik\theta _0\cdot (\cdot )}f_{j+1}R_0...f_2R_0(f_1 e^{ik\theta _0\cdot (\cdot )})(\theta ). \end{aligned}$$

Using this identity we have that

$$\begin{aligned} I_1=\int _0^1k^{n-1}||S_{k,\theta _0}(e^{-ik\theta _0\cdot (\cdot )}f_{j+1}R_0...f_2R_0(f_1 e^{ik\theta _0\cdot (\cdot )}) ||_{L^2(S^{n-1})}dk. \end{aligned}$$
$$\begin{aligned} \begin{aligned} I_1\le&C^{2j} ||f_{j+1}||_{W^{\alpha ,2}} ||f_{j}||_{W^{\alpha ,2}}...||f_{2}||_{W^{\alpha ,2}}||f_{1}||_{L^{2}_\delta }\\ \le&C^{2j} \prod _{l=1}^{j+1} ||f_{l}||_{W^{\alpha ,2}}. \end{aligned} \end{aligned}$$
(50)

For \(I_2\), we need to take more advantage of oscillations, so we write \(P_j\) in the terms of the operator \(R_{k,\theta _0}\) and the restriction operator \(S_{k,\theta _0}\), as follows:

$$\begin{aligned} \widehat{P_j(f)}(k(\theta -\theta _0))=S_{k,\theta _0}(f_{j+1} R_{k,\theta _0}...f_2 R_{k,\theta _0}f_1)(\theta ). \end{aligned}$$

Write \(\omega =|\theta -\theta _0|^\alpha\), we have that

$$\begin{aligned} I_2=\int _1^{+\infty }k^{n-1+2\alpha }||\omega S_{k,\theta _0}(f_{j+1} R_{k,\theta _0}...f_2 R_{k,\theta _0}f_1)||^2_{L^2}dk. \end{aligned}$$

In this case, we can bound the \(L^2(S^{n-1})\)-norm that appears in the last identity using (42), (46) and (39) as the following diagram illustrates:

$$\begin{aligned} W^{\alpha ,2}_{\delta } \ \underrightarrow{(j-1)-times } \ W^{\alpha ,t_j}_{\delta } \ \underrightarrow{\ R_{k,\theta _0} \ } \ \ W^{\alpha ,r_j}_{-\delta } \ \underrightarrow{\ Zolesio \ } \ W^{\alpha ,t_{j+1}}_{\delta } \ \underrightarrow{ \ S_{k,\theta _0} \ } \ L_\delta ^2(S^{n-1}), \end{aligned}$$

whenever there exist \(r_l\) and \(t_{l+1}\) satisfying for \(l=1,2...j\)

$$\begin{aligned} 0\le \frac{1}{t_{l+1}}-\frac{1}{2}\le \frac{1}{n+1} \ \ and \ \ 0\le \frac{1}{2}-\frac{1}{r_l}\le \frac{1}{n+1}, \end{aligned}$$
(51)
$$\begin{aligned} t_{l+1}< min(2,r_l) \ \ and \ \ 0\le \frac{1}{2}+\frac{1}{r_l}-\frac{1}{t_{l+1}}\le \frac{\alpha }{n}. \end{aligned}$$
(52)

Therefore, writing \(t_1=2\), we have

$$\begin{aligned} \begin{aligned} I_2\le&C^{2j} \prod _{l=1}^{j+1} ||f_{l}||^2_{W^{\alpha ,2}}\int _1^{+\infty }k^{(n-1)(\frac{1}{t_{j+1}}-\frac{1}{2})+ \sum _{l=1}^j(2(1-2m)+(n-1)(\frac{1}{t_{j}}-\frac{1}{r_j}))}dk \\&= C^{2j} \prod _{l=1}^{j+1} ||f_{l}||^2_{W^{\alpha ,2}}\int _1^{+\infty }\frac{1}{k^{(n-1) \sum _{l=1}^j(\frac{1}{t_{j}}-\frac{1}{r_j})+2(2m-1)j}}dk. \end{aligned} \end{aligned}$$

Since \(t_{l+1}\) and \(r_l\) have to satisfy (52), the best choice to get convergence of the previous integral is

$$\begin{aligned} \frac{1}{r_l}-\frac{1}{t_{l+1}}=\frac{\alpha }{n}-\frac{1}{2}. \end{aligned}$$
(53)

With this choice we get

$$\begin{aligned} \begin{aligned} I_2&\le C^{2j} \prod _{l=1}^{j+1} ||f_{l}||^2_{W^{\alpha ,2}}\int _1^{+\infty }\frac{1}{k^{(n-1)(\frac{\alpha }{n}-\frac{1}{2})j+2(2m-1)j}}dk\\&\le C^{2j} \prod _{l=1}^{j+1} ||f_{l}||^2_{W^{\alpha ,2}}, \end{aligned} \end{aligned}$$
(54)

for any \(j\in N\) fixed, whenever

$$\begin{aligned} \alpha >n\left(\frac{1}{2}-\frac{4m-3}{n-1}\right), \end{aligned}$$
(55)

and there exists \(r_l\) and \(t_{l+1}\) satisfying (51), (52) and (53) for \(l=0,1,2...,j\). Such \(r_l\) and \(t_{l+1}\) exist if

$$\begin{aligned} 0\le \alpha \le n/2. \end{aligned}$$
(56)

Finally, the result follows from (49), (50) and (54) if \(\alpha \le 1\) and satisfies (55), (56) and (43), (44), that is, if \(\alpha\) satisfies (7). \(\square\)

Before the proof of Proposition 2, we give the following Lemma,

Lemma 7

Let q satisfy Theorem 1, \(v(x,\theta _0,k)\) be the solution of (9), for \(\delta >1\),we have the following

$$\begin{aligned} ||v||_{W^{2,2}_{-\delta }}=O(k^{1-2m}),\ \ k\rightarrow 0. \end{aligned}$$
(57)

Proof of Proposition 2

Arguing as in the proof of Proposition 1, we split the norm to control into two pieces I and II, and we will just bound I, since II can be estimated in a similar way.

For I, making the change of variables given in (48). We write

$$\begin{aligned} \begin{aligned} I=&\int _{H_{\theta _0}}\langle \xi \rangle |\widehat{q_\tau ^r}(\xi )|^2d\xi \\ \le&C\int _0^{\frac{1}{2}}k^{n-1}\int _{S^{n-1}}| \widehat{q_\tau ^r}(k(\theta -\theta _0))|^2d\sigma (\theta _0)dk\\&+ C\int _{\frac{1}{2}}^{+\infty }k^{n-1+2\alpha }\int _{S^{n-1}}|\theta -\theta _0|^{2\alpha }|\widehat{q_\tau ^r}(k(\theta -\theta _0))|^2d\sigma (\theta _0)dk\\ =&C(I_1+I_2), \end{aligned} \end{aligned}$$
(58)

whenever \(0\le \alpha \le 1\).

To estimate \(I_1\), since q is compactly supported, use the definition of \(\widehat{q_\tau ^r}\) given in (15), and Cauchy-Schwarz inequality, we have

$$\begin{aligned} |\widehat{q_\tau ^r}(k(\theta -\theta _0)|\le ||(qR_0)^\tau (qv)||_{L_1}\le C||(qR_0)^\tau (qv)||_{L_\delta ^2}. \end{aligned}$$

We can bounded the \(L_\delta ^2\)-norm that appears in the last inequality using (45) and (37) as the following diagram illustrates:

$$\begin{aligned} W^{\alpha ,2}_{-\delta } \ \ \underrightarrow{\ \ Zolesio \ \ } \ L_\delta ^2 \ \ \underrightarrow{\ (\tau -1)-times \ } \ \ L_\delta ^2 \ \ \underrightarrow{\ \ R_0 \ \ } \ \ W^{\alpha ,2}_{-\delta } \ \ \underrightarrow{ \ Zolesio \ } \ \ L_\delta ^2. \end{aligned}$$

Therefore, if \(\delta >1\) and \(\alpha\) satisfies (43) or (44), we have

$$\begin{aligned} ||(qR_0)^\tau (qv)||_{L_\delta ^2} \le C^\tau ||q||^{\tau +1}_{W^{\alpha ,2}}||v||_{W^{2,2}_{-\delta }}, \end{aligned}$$

and thus

$$\begin{aligned} I_1 \le C^{2\tau } ||q||^{2(\tau +1)}_{W^{\alpha ,2}}\int _0^{\frac{1}{2}}k^{n-1}||v(\cdot ,\theta _0,k)||_{W^{2,2}_{-\delta }}dk. \end{aligned}$$

Since \(q\in W^{\alpha ,2}(R^n)\), the Sobolev embedding theorem guarantees that \(q\in L^2(R^n)\cap L^r(R^n)\) with \(r\in (\frac{n}{2},\frac{2n}{n-2\alpha }]\) whenever \(\alpha \ge 0\) and \(\frac{n}{2}-2\le \alpha \le \frac{n}{2}\), so we can use estimates (57) and \(n>2m-1\) in the previous inequality to obtain

$$\begin{aligned} I_1 \le C^{2\tau } ||q||^{2(\tau +1)}_{W^{\alpha ,2}}. \end{aligned}$$
(59)

On the other hand, denoting \(\omega =|\theta -\theta _0|^\alpha\), we can write

$$\begin{aligned} I_2=\int _{\frac{1}{2}}^\infty k^{n-1+2\alpha }||\omega \widehat{q_\tau ^r}(k(\theta -\theta _0))||^2_{L^2(S^{n-1})}dk. \end{aligned}$$
(60)

Arguing as in the proof of Proposition 1, to estimate \(I_2\), we write \(q^r_\tau\) in the terms of the operators \(R_{k,\theta _0}\) and \(S_{k,\theta _0}\) introduced in (38) and (40), respectively,

$$\begin{aligned} \widehat{q_\tau ^r}(k(\theta -\theta _0))=S_{k,\theta _0}((qR_{k,\theta _0}) ^m(e^{-ik\theta _0\cdot (\cdot )}qv))(\theta ). \end{aligned}$$

From here, using (42), (46) and (39) as the following diagram illustrates,

$$\begin{aligned} W^{\alpha ,t_1}_{\delta } \ \ \underrightarrow{\ (\tau -1)-times \ } \ \ W^{\alpha ,t_\tau }_{\delta } \ \ \underrightarrow{\ \ R_{k,\theta _0} \ \ } \ W^{\alpha ,r_\tau }_{-\delta } \ \ \underrightarrow{\ Zolesio \ } \ \ W^{\alpha ,t_{\tau +1}}_{-\delta } \ \ \underrightarrow{ \ S_{k,\theta _0} \ } \ \ L^2(S^{n-1}), \end{aligned}$$

where \(0\le \frac{1}{t_1}-\frac{1}{2}\le \frac{1}{n+1}\), \(r_l\) and \(t_{l+1}\) satisfy (51) and (52) for \(l=1,2...m\), we get

$$\begin{aligned} ||\omega \widehat{q_\tau ^r}(k(\theta -\theta _0))||^2_{L^2(S^{n-1})} \le C k^a ||q||^{2\tau }_{W^{\alpha ,2}} ||e^{-ik\theta _0\cdot (\cdot )}qv)||^2_{W^{\alpha ,t_1}_\delta } \end{aligned}$$
(61)

with

$$\begin{aligned} a=(n-1)\left(\frac{1}{t_{\tau +1}}-\frac{3}{2}\right)-2\alpha +\Sigma _{l=1}^\tau \left(2(1-2m)+(n-1)\left(\frac{1}{t_{l}}-\frac{1}{r_l}\right)\right). \end{aligned}$$

We can control the norm on the right-hand side of (61) multiplying (9) by \(q(x)e^{-ik\theta _0\cdot x}\), using the operator \(R_{k,\theta _0}\), the triangular inequality, and estimate (46) to write

$$\begin{aligned} ||e^{-ik\theta _0\cdot (\cdot )}qv)||_{W^{\alpha ,t_1}_\delta } \le C||q||_{W^{\alpha ,2}} (||R_{k,\theta _0}q||_{W^{\alpha ,r_0}_{-\delta }}+||R_{k,\theta _0}(e^{-ik\theta _0\cdot (\cdot )}qv))||_{W^{\alpha ,r_0}_{-\delta }}), \end{aligned}$$
(62)

whenever

$$\begin{aligned} \alpha \ge 0, \ \ t<min (2,r_0) \ \ and \ \ 0 \le \frac{1}{2}+\frac{1}{r_0}-\frac{1}{t_1} \le \frac{\alpha }{n}. \end{aligned}$$

Using (39) we get

$$\begin{aligned} ||R_{k,\theta _0}(e^{-ik\theta _0\cdot (\cdot )}qv))||_{W^{\alpha ,r_0}_{-\delta }} \le C k^b ||e^{-ik\theta _0\cdot (\cdot )}qv)||_{W^{\alpha ,t_1}_\delta } \end{aligned}$$

with

$$\begin{aligned} b=(1-2m)+\frac{(n-1)}{2}(\frac{1}{t_1}-\frac{1}{r_0}), \end{aligned}$$

whenever

$$\begin{aligned} \alpha \ge 0, \ \ 0 \le \frac{1}{t_1}-\frac{1}{2}< \frac{2m-1}{n-1}, \ \ and \ \ 0 \le \frac{1}{2}-\frac{1}{r_0} < \frac{2m-1}{n-1}. \end{aligned}$$
(63)

From here, using (1), since \(b<0\), then for \(k>1/2\) we have that

$$\begin{aligned} \begin{aligned} ||q||_{W^{\alpha ,2}}||R_{k,\theta _0}(e^{-ik\theta _0\cdot (\cdot )}qv))||_{W^{\alpha ,r_0}_{-\delta }}&\le C\frac{A}{2^b} ||e^{-ik\theta _0\cdot (\cdot )}qv)||_{W^{\alpha ,t_1}_\delta }\\&<\frac{1}{2}||e^{-ik\theta _0\cdot (\cdot )}qv)||_{W^{\alpha ,t_1}_\delta }, \end{aligned} \end{aligned}$$
(64)

whenever \(AC\le 2^{b-1}\).

Using (64) and (62) we obtain

$$\begin{aligned} ||e^{-ik\theta _0\cdot (\cdot )}qv)||_{W^{\alpha ,t_1}_\delta }\le 2C ||q||_{W^{\alpha ,2}}||R_{k,\theta _0}q||_{W^{\alpha ,r_0}}. \end{aligned}$$

From here, since (63), we can use (39) to get

$$\begin{aligned} ||e^{-ik\theta _0\cdot (\cdot )}qv)||_{W^{\alpha ,t_1}_\delta }\le C k^{(1-2m)+\frac{(n-1)}{2}(\frac{1}{2}-\frac{1}{r_0})} ||q||_{W^{\alpha ,2}}||q||_{W^{\alpha ,2}_\delta }. \end{aligned}$$

And since q has compact support, we can use (46) to obtain

$$\begin{aligned} ||e^{-ik\theta _0\cdot (\cdot )}qv)||_{W^{\alpha ,t_1}_\delta }\le C k^{(1-2m)+\frac{(n-1)}{2}(\frac{1}{2}-\frac{1}{r_0})} ||q||^2_{W^{\alpha ,2}}. \end{aligned}$$

Using this inequality and (61) in (60) we get

$$\begin{aligned} I_2 \le C^{2\tau } ||q||^{2\tau +2}_{W^{\alpha ,2}}\int _{\frac{1}{2}}^\infty k^{2(1-2m)(\tau +1)-(n-1)\Sigma _{l=0}^\tau (\frac{1}{r_l}-\frac{1}{t_{l+1}})}dk, \end{aligned}$$

for any \(r_l\) and \(t_{l+1}\) satisfying (51) and (52) for \(l=0,1...\tau .\)

If we choose \(r_l\) and \(t_{l+1}\) satisfying (53), the last integral is convergent if and only if

$$\begin{aligned} 2(2m-1)(\tau +1)+(n-1)(\tau +1)(\frac{\alpha }{n}-\frac{1}{2})>1, \end{aligned}$$

or

$$\begin{aligned} \alpha >\frac{n}{2}+\frac{n}{(\tau +1)(n-1)}-\frac{2(2m-1)n}{n-1}. \end{aligned}$$

Therefor, for any \(\tau\) in N, we have that

$$\begin{aligned} I_2 \le C^{2\tau } ||q||^{2\tau +2}_{W^{\alpha ,2}}, \end{aligned}$$
(65)

whenever

$$\begin{aligned} 0 \le \alpha <\frac{n}{2} \ \ \ and \ \ \ \alpha >\frac{n}{2}-\frac{2(2m-1)n}{n-1}. \end{aligned}$$

Finally, the result follows from (58), (59) and (61) for \(\alpha\) satisfying (7). \(\square\)