Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions

We prove new global H\"older-logarithmic stability estimates for the near-field inverse scattering problem in dimension $d\geq 3$. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimension $d=2$ is also given.


Introduction
We consider the Schrödinger equation where We consider the resolvent R(E) of the Schrödinger operator L in L 2 (R d ): where σ(L) is the spectrum of L in L 2 (R d ).We assume that R(x, y, E) denotes the Schwartz kernel of R(E) as of an integral operator.We consider also We recall that in the framework of equation (1.1) the function R + (x, y, E) describes scattering of the spherical waves generated by a source at y (where H µ is the Hankel function of the first kind of order µ).We recall also that R + (x, y, E) is the Green function for L − E, E ∈ R + , with the Sommerfeld radiation condition at infinity.In addition, the function is considered as near-field scattering data for equation (1.1), where B r is the open ball of radius r centered at 0. We consider, in particular, the following near-field inverse scattering problem for equation (1.1): Problem 1.1.Given S + on ∂B r × ∂B r for some fixed r, E ∈ R + , find v on B r .This problem can be considered under the assumption that v is a priori known on R d \ B r .Actually, in the present paper we consider Problem 1.1 under the assumption that v ≡ 0 on R d \ B r for some fixed r ∈ R + .Below in this paper we always assume that this additional condition is fulfilled.
However, in some case it is much more optimal to deal with Problem 1.1 directly, see, for example, logarithmic stability results of [12] for Problem 1.1 in dimension d = 3.A principal improvement of estimates of [12] was given recently in [17]: stability of [17] efficiently increases with increasing regularity of v.
Problem 1.1 can be also considered as an example of ill-posed problem: see [20], [5] for an introduction to this theory.
In the present paper we continue studies of [12], [17].We give new global Hölder-logarithmic stability estimates for Problem 1.1 in dimension d ≥ 3, see Theorem 2.1.Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity.Results of such a type for the Gel'fand inverse problem were obtained recently in [16] for d ≥ 3 and in [35] for d = 2.
In addition, we give also global logarithmic stability estimates for Problem 1.1 in dimension d = 2, see Theorem 2.2.

Stability estimates
We recall that if v satisfies (1.2) and supp v ⊂ B r1 for some r 1 > 0, then where S + (E) is the near-field scattering data of v for equation (1.1) with E > 0, for more details see, for example, Section 2 of [12].

Estimates for d ≥ 3
In this subsection we assume for simplicity that where where Theorem 2.1.Let E > 0 and r > r 1 be given constants.Let dimension d ≥ 3 and potentials v 1 , v 2 satisfy (2.2).Let ||v j || m,1 ≤ N, j = 1, 2, for some N > 0. Let S + 1 (E) and S + 2 (E) denote the near-field scattering data for v 1 and v 2 , respectively.Then for τ ∈ (0, 1) and any s ∈ [0, s * ] the following estimate holds: where Proof of Theorem 2.1 is given in Section 5.This proof is based on results presented in Sections 3, 4.

Estimates for d = 2
In this subsection we assume for simplicity that ) Proof of Theorem 2.2 is given in Section 7.This proof is based on results presented in Sections 3, 6.
Remark 2.2.In the same way as in [12] and [17] for dimesnsion d = 3, using estimates (2.6) and (2.8), one can obtain logarithmic stability estimates for the reconstruction of a potential v from the inverse scattering amplitude f for any d ≥ 2.

Alessandrini-type identity for near-field scattering
In this section we always assume that assumptions of Theorems 2.1 and 2.2 are fulfilled (in the cases of dimension d ≥ 3 and d = 2, respectively).Consider the operators Rj , j = 1, 2, defined as follows We recall that (see [12]) for any functions φ 1 , φ 2 ∈ C(R d ), sufficiently regular in R d \ ∂B r and satisfying ) , respectively, the following identity holds: where where ν + and ν − are the outward and inward normals to ∂B r , respectively.
Remark 3.1.The identity (3.4) is similar to the Alessandrini identity (see Lemma 1 of [1]), where the Dirichlet-to-Neumann maps are considered instead of operators Rj .
To apply identity (3.4) to our considerations, we use also the following lemma: Lemma 3.1.Let E, r > 0 and d ≥ 2.Then, there is a positive constant C 7 (depending only on r and d) such that for any the following inequality holds: where H 1 (∂B r ) denotes the standart Sobolev space on ∂B r .
The proof of Lemma 3.1 is given in Section 8.
In addition, we have that: and, under assumtions of Theorem 2.1, where h j , ψ j denote h and ψ of (4.1) and (4.2) for v = v j , j = 1, 2.
This completes the proof of (2.8).
8 Proof of Lemma 3.1 In this section we assume for simplicity that r = 1 and therefore ∂B r = S d−1 .
We fix an orthonormal basis in L 2 (∂B r ): where p j is the dimension of the space of spherical harmonics of order j, where The precise choice of f jp is irrelevant for our purposes.Besides orthonormality, we only need f jp to be the restriction of a homogeneous harmonic polynomial of degree j to the sphere ∂B r and so |x| j f jp (x/|x|) is harmonic pn R d .In the Sobolev spaces H s (∂B r ) the norm is defined by The solution φ of the exterior Dirichlet problem can be expressed in the following form (see, for example, [4], [8]): where c jp are expansion coefficients of u in the basis {f jp : j ≥ 0; 1 ≤ p ≤ p j }, and φ jp denotes the solution of (8.6) with u = f jp , where H µ is the Hankel function of the first kind.Let Let consider the cases when j + d−3 2 ≤ 0. Case 1. j = 0, d = 2. Using the property dH  We recall that functions H (1) 0 and H (1) 1 have the following asymptotic forms (see, for example [38]): ∂Br ×∂Br) , and constants C 1 , C 2 > 0 depend only on N , m, d, r, τ . ) Br ) ≤ c 14 (E, r, σ) 3 + δ −1 β(8r 2 +8r) δ+ + c 12 (N E , r) ln 3β ln 3 + δ −1 2 )/dt = −H [38]Using the Green formula and the radiation condition for φ jp , φ 0 jp , we get that Using also the following property of the Hankel function of the first kind (see, for example,[38]):|H (1) µ (x)| is a decreasing function of x for x ∈ R + , µ ∈ R,(8.13) +∞ 1 t −j−d+2 h jp (t)t d−1 dt = Combining (8.5)-(8.8),(8.15),(8.19)and (8.21), we get that for some constant c ′ = c ′ (d) > 0