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Propagation property and its application to inverse scattering for fractional powers of the negative Laplacian

Enss (1983) proved a propagation estimate for the usual free Schroedinger operator that turned out later to be very useful for inverse scattering in the work of Enss--Weder (1995). Since then, this method has been called the Enss--Weder time-dependent method. We study the same type of propagation estimate for the fractional powers of the negative Laplacian and, as with the Enss--Weder method, we apply our estimate to inverse scattering. We find that the high-velocity limit of the scattering operator uniquely determines the short-range interactions.

Enss [5] proved the following estimate for the free Schrödinger operator (1.7) This estimate was proved not only for the spheres but more generally for the measurable subsets of R n (see Proposition 2.10 in Enss [5]). Before considering Theorem 1.1 further, we discuss the meaning of the estimate (1.7). From the perspective of classical mechanics, D x represents the momentum or equivalently the velocity of the particle of unit mass. On the left-hand side of (1.7), D x is localized to the neighborhood of v by the cut-off function f . Therefore, along the time evolution of the propagator e −itD 2 x /2 , the position of the particle behaves according to Because the behavior of the points on the sphere is the same, the center of the sphere moves toward vt from the origin We extract an interpretation of the estimate (1.7) from these observations. The behavior of the sphere (1.9) makes the characteristic functions on both sides of (1.7) disjoint. Thus, this gives rise to the decay associated with time and velocity. Theorem 1.1 is the fractional Laplacian version of (1.7). From (∇ ξ ω ρ )(v) = |v| 2ρ−2 v, the case where ρ = 1 in (1.6) is essentially equivalent to (1.7). Conversely, if ρ = 1/2 in (1.6), the decay on the right-hand side does not involve |v|. However, this does not conflict with the physical meaning. In the case where ρ = 1/2, the system is relativistic. In this system, the particle does not have a mass, and its velocity is the speed of light, which is normalized to 1. Therefore, the decay cannot include the velocity v. Spectral analysis for the relativistic Schrödinger operator was initiated by Weder [21], following which Umeda [16,17] studied the resolvent estimate and mapping properties associated with the Sobolev spaces. Wei [24] studied the generalized eigenfunctions. Weder [22] analyzed the spectral properties of the fractional Laplacian for the massive case, and Watanabe [19] studied the Katosmoothness. Giere [7] investigated the scattering theory and proved the asymptotic completeness of the wave operators for short-range perturbations. Recently, Kitada [11,12] constructed the long-range theory.
In Section 3, we assume that the dimension of the space satisfies n 2. As an application of Theorem 1.1, we consider a multidimensional inverse scattering. The high-velocity limit of the scattering operator uniquely determines the interaction potentials that satisfy the short-range condition below by using the Enss-Weder time-dependent method (Enss-Weder [6]).
where the bracket of x has the usual definition x = 1 + |x| 2 .
For the full Hamiltonian H ρ = H 0,ρ + V , where V belongs to the class stated above, the existence of the wave operators and their asymptotic completeness have already been proved in Kitada [11,12]. Thus, we can define the scattering operator S ρ = S ρ (V ) by Under these situations, the following uniqueness theorem can be proved.
We note that ρ = 1/2 is excluded in this theorem. As mentioned before, if ρ = 1/2, the system is relativistic and the speed of light is always equal to 1, that is, |v| ≡ 1. The Enss-Weder time-dependent method is also called the highvelocity method. As the name suggests, deriving the uniqueness of the interaction potentials requires the limit of |v|. Therefore, this method does not combine well with relativistic phenomena (see also Jung [10]).

Propagation Property
In this section, we prove Theorem 1.1. Regarding estimate (1.7), the idea of Enss [5] is very simple and understandable. The Galilean transformation in the direction of v enables a reduction to a static system, and iterations of the integration by parts, by taking the points of stationary phase into account, leads to (1.7). However, in our case, these ingredients do not work well because of the fractional powers. Instead, our main strategy is the asymptotic expansion of the symbolic calculus of pseudo-differential theory.
Here, we recall several basics of the calculus of pseudo-differential operators. They are recounted from standard textbooks. For m ∈ R, let S m 1,0 be the Hörmander symbol class, that is, we say p ∈ S m 1,0 if and only if p ∈ C ∞ (R n x × R n ξ ) and, for any multi-indices β and β ′ , If p 1 ∈ S m 1 1,0 and p 2 ∈ S m 2 1,0 , then the symbol of the product p 1 p 2 = q ∈ S m 1 +m 2 1,0 has the following asymptotic expansion for some M ∈ N (Chapter 8 in Wong [20]). Moreover, by the L 2 -boundedness theorem, if m 1 +m 2 −N 0, then there exists K ∈ N such that the operator-norm of r N is estimated by Proof of Theorem 1.1. The left-hand side of (1.6) is bounded uniformly in t and v. Therefore, it is sufficient to prove for |v| 2ρ−1 |t| 1. By using the unitary translations, we have the following relations We thus compute that The strategy of our proof is as follows. The momentum operator D x can move inside the compact region only because f is compactly supported. Therefore, (∇ ξ ω ρ )(D x + v) and (∇ ξ ω ρ )(v) almost cancel when |v| is sufficiently large, and the function χ in (2.10) behaves as though We now justify this strategy. Because |ξ| η on the support of f , we have This inequality implies Moreover, when 1/2 ρ < 1, and hold for |ξ| η, where ∇ 2 ξ ω ρ denotes the Hessian matrix of ω ρ . In the case where ρ = 1, it is clear that (2.16) We thus obtain, for 1/2 ρ 1 and |v| which satisfies (1.5), It follows from (2.17) that on the supports of f and χ. This means that because χ(x/(|v| 2ρ−1 |t|/16)) = 1 by (2.18). However, in the pseudo-differential calculus, the product of the symbols is not equal to the symbol of the product. The additional asymptotic error terms arise. By the product formula (2.4), the symbol of (2.19) becomes for any N ∈ N. All terms with |β| N − 1 vanish due to another characteristic function Next, we consider the remainder term R N . Because f is compactly supported, holds. Clearly, χ(x/(|v| 2ρ−1 |t|/16)) ∈ S 0 1,0 also holds. In particular, we see that for all β with |β| 1. Here, C β > 0 is independent of t and v. Therefore, it is sufficient to focus only on the derivative at x. By the estimate of the remainder (2.6), there exists N ′ ∈ N such that because |v| 2ρ−1 |t| 1. This completes the proof.

Uniqueness of Interactions
To apply the Enss-Weder time-dependent method, we have to assume that n 2 and that ρ > 1/2 from here on. The following Radon transformation-type reconstruction formula enables Theorem 1.3 to be proved. We devote ourselves to proving Theorem 3.1 in this section. Contrary to Enss-Weder [6], the key calculation in our proof is the pseudo-differential asymptotic expansion as in Theorem 1.1.
Theorem 3.1. Let v ∈ R n be given and letv = v/|v|. Suppose that η > 0, and that holds as |v| → ∞ for any V which satisfies Assumption 1.2, where (·, ·) is the scalar product of L 2 (R n ).
We first prepare the propagation estimate of the following integral form. In the proof of this proposition, we can see that Theorem 1.1 plays an important role. While · also indicates the norm in L 2 (R n ), for simplicity, we do not distinguish between the notations for the usual L 2 -norm and its operator norm in this paper.
holds as |v| → ∞ for any V which satisfies Assumption 1.2.
Proof. The original idea of this proof is given in Lemma 2.2 of Enss-Weder [6]. We extend it to the case of the fractional powers of the negative Laplacian. Choose follows. We compute where I 1 and I 2 are given by
Proof. The proof is similar to that of Corollary 2.3 in Enss-Weder [6], and therefore is sketched as follows. The difference between W ± ρ and 1 can be represented by the following integral form In the last equation, we changed the integral variable τ ′ = τ + t. By using Proposition 3.2, we have We are ready to prove the reconstruction theorem.
Proof of Theorem 3.1. As in the proof of Corollary 3.3, we represent the difference between W + and W − by the integral Recall the intertwining property e −itHρ W ± ρ = W ± ρ e −itH 0,ρ . We can then compute and where we defined I v (t) and R v in (3.21) by

Proposition 3.2 and Corollary 3.3 immediately give
Thus far, the proof has been roughly parallel to that in Enss-Weder [6]. However, the principal part of (3.21) demands further rigorous scrutiny. We first divide the integral as follows where σ > 2ρ − 1 is independent of t and v. We will later determine an upper bound on σ. Because I v (t) is uniformly bounded in t and v, the integral on |t| < |v| −σ is We next consider the integral on |t| |v| −σ , which is represented by We note that (∇ ξ ω ρ )(v) = |v| 2ρ−2 v. After the change of the integral variable τ = |v| 2ρ−1 t, the first term of the right-hand side of (3.27) converges as |v| → ∞ because we assumed that 2ρ − 1 − σ < 0. This also indicates that by the uniformly boundedness of (V (x +vt)Φ 0 , Ψ 0 ). Recall the relations (2.8) and (2.9). We then have Therefore, as in the proof of Theorem 1.1, we try to derive the order of decay in the second term on the right-hand side of (3.27) from the nearly cancellation of (∇ ξ ω ρ )(ξ + v) and (∇ ξ ω ρ )(v) on the support of F Φ 0 . In our assumptions, V belongs to C 1 (R n ), however we can compute as the pseudo-differential symbolic calculus in the Fourier integral. We particularly note that the second-and higher-order derivatives of V do not appear on the right-hand side of (3.31) because for 1 j n and, as in (2.14), (2.15) and (2.16) follows for any β. We also define the vector-valued function ψ v by to avoid complicated notation. Now, to estimate the second term in (3.27), we only have to consider the following norm, which includes the integrand of the j-th term on the right-hand side of (3.31), where J 1 and J 2 are given by between g j,v (D x ) and Φ 0 . Then J 1 is estimated so that J 1 J 1,1 + J 1,2 , where J 1,1 and J 1,2 are given by The estimate of J 1,1 is almost the same as (3.13). However, in this estimate, we choose ν ∈ R such that Therefore, for ν > 2, we obtain Although this estimate holds for any ν > 2, the exponent is better when ν is closer to 2 because and σ > 2ρ − 1. In the estimate of J 1,2 , we compute the following commutator by using the pseudo-differential product formula (2.4) for any N ∈ N. As in the proof of Theorem 1.1, the disjointness of two characteristic functions means that, for 0 |β| N − 1, Therefore, J 1,2 only has the remainder term R N . To estimate R N , we divide the integral again By using the L 2 -boundedness (2.6), when |t| |v| because |v| 2ρ−1 |t| 1 holds in this case, where |v| 2ρ−2 comes from (3.33). We therefore compute, for N 3 In contrast, when |v| −σ |t| < |v| 1−2ρ , there existsÑ N such that because |v| 2ρ−1 |t| < 1 holds, and is obtained. This estimate holds for anyÑ N ( 3). However, the best exponent is the smallestÑ because We next consider J 2 . On the supports of f 2 and 1 − χ, holds for large |v|, here we used (2.17). This says that x, ξ)/(|v| 2ρ−1 |t|/8)) = 1 by (3.52). However, symbolically, (3.53) is for any M ∈ N by using the asymptotic product formula (2.4) again. We note that for any β with |β| 1 and C β is independent of t. Therefore, for 0 |β| M − 1, it is sufficient to consider only. The term which includes (3.56) is estimated to be here we used the decay condition on V in (1.10) and the estimate of g j,v in (3.33). We then compute the following integral because γ > 1. The decay exponent in (3.58) is represented by Because σ > 2ρ − 1, the top term between 0 |β| M − 1 is |β| = M − 1. To estimate the term involving R M , we have to divide the integral into |v| −σ |t| < |v| 1−2ρ and |t| |v| 1−2ρ once more. By the same argument in (2.24), there exists M ′ ∈ N such that where, in the summation of 0 j M ′ , we used the following boundedness again From the Plancherel formula associated with the Radon transformation (see Helgason [8]), the proof of Theorem 1.3 can be performed in the same way as in Theorem 1.1 of Enss-Weder [6]. We thus omit the proof here.