1.1. Outline of the article

The Calderón problem for a Schrödinger operator, also known as the Gel’fand-Calderón problem, asks for the reconstruction of a potential from the knowledge of certain boundary measurements of the solutions to the corresponding Schrödinger equation; see [Reference Calderón16, Reference Calderón17] and [Reference Gelfand29]. In this article, we are interested in the fixed-energy version of this problem: given an open, bounded domain $\Omega\subset\mathbb{R}^d$, $d\geq2$, with $\mathcal{C}^1$ boundary, an energy $\kappa\in\mathbb{C}$ and a real-valued potential q, consider the elliptic boundary-value problem for the Helmholtz equation

(1.1)\begin{equation} \left\{ \begin{array}{rll} -\Delta u(x) + q(x) u(x) -\kappa u(x) &=0, \quad x\in\Omega,\\ u|_{\partial\Omega} &= f. \end{array}\right. \end{equation}

The Dirichlet-to-Neumann (DtN) map associates the boundary value f with the normal derivative on $\partial\Omega$ of the corresponding solution u to (1.1). Whenever κ is not a Dirichlet eigenvalue of $-\Delta+q$, this produces a well-defined linear operator

(1.2)\begin{equation} \begin{array}{cccc} \Lambda_{q,\kappa}: & \mathcal{C}^\infty(\partial\Omega) &\longrightarrow&\mathcal{C}^\infty(\partial\Omega) \\ & f &\longmapsto & \partial_\nu u|_{\partial\Omega}, \end{array} \end{equation}

where ν is the vector field of exterior unit vectors normal to $\partial\Omega$.

The Gel’fand-Calderón problem, in its simplest form, consists in reconstructing q from the knowledge of $\Lambda_{q,\kappa}$ for some fixed $\kappa\in\mathbb{C}$. Since it is known that $\Lambda_{q,\kappa}-\Lambda_{0,\kappa}$ is always an operator that is bounded in $L^2(\partial\Omega)$ (see for instance Section 2), it is convenient to encode the inverse problem using the nonlinear map

(1.3)\begin{equation} \begin{array}{cccc} \Phi^{\kappa}: & \mathcal{X} &\longrightarrow& \mathcal{L}^2(L^2(\partial\Omega)) \\ & q &\longmapsto & \Lambda_{q,\kappa}-\Lambda_{0,\kappa}, \end{array} \end{equation}

where $\mathcal{X}$ is a class of admissible potentials for which the DtN map is defined. In this article $\mathcal{X}$ will be a subset of $L^p(\Omega)$ for p > 1 and $p \ge d/2$. The main issues one is interested in are:

1. i) Uniqueness. Is every $q\in \mathcal{X}$ uniquely determined by $\Lambda_{q,\kappa}$? This amounts to showing that $\Phi^{\kappa}$ is injective.

2. ii) Stability. Is the reconstruction process stable? That is, find a modulus of continuity for $\Phi^{\kappa}$. This is not possible in general, since $(\Phi^{\kappa})^{-1}$ is not continuous (see, for example, [Reference Alessandrini1, Reference Alessandrini and Cabib2, Reference Faraco, Kurylev and Ruiz26]). This is, therefore, an ill-posed inverse problem. Nevertheless, one can ask for conditional stability when additional requirements of regularity and boundedness are imposed on the class of admissible potentials.Footnote ¹

3. iii) Reconstruction. Find an effective procedure to reconstruct q from $\Lambda_{q,\kappa}$, in other words, compute $(\Phi^{\kappa})^{-1}$. This is related to the problem of characterization of the range $\Phi^{\kappa}(\mathcal{X})$.

The uniqueness for $d \ge 3$ was proved in [Reference Sylvester and Uhlmann59], and later in [Reference Nachman46, Reference Novikov48] with particular emphasis on the case of fixed energy (see also [Reference Caro and Rogers18, Reference Haberman31]). The two-dimensional case was solved in  [Reference Astala and Päivärinta4, Reference Nachman47] for conductivities and [Reference Blåsten, Imanuvilov and Yamamoto13, Reference Bukhgeim15] for potentials. These results are proved using the notion of Complex Geometric Optics solutions (CGO) from [Reference Faddeev25] ( $d\geq 3$) or different families of exponentially growing solutions of the equations when d = 2.

Conditional stability was proved in dimensions $d\ge 3$ by [Reference Alessandrini1] with a logarithmic-type modulus on continuity, which was shown to be optimal in [Reference Mandache43] (see also [Reference Koch, Rüland and Salo37] for a more detailed account of this issue). In dimension 2 it was proved in [Reference Barceló, Barceló and Ruiz7, Reference Clop, Faraco and Ruiz19] (see also [Reference Faraco and Prats27]). Improved stability at fixed energy was established in [Reference Isakov34, Reference Isakov, Lu and Xu35, Reference Novikov49] in dimension three and [Reference Santacesaria53, Reference Santacesaria54] in two dimensions. The instability estimates of [Reference Mandache43] have been improved in the κ ≠ 0 case in [Reference Isaev33, Reference Kow, Uhlmann and Wang39].

Reconstruction is a difficult issue in general, both from analytical and numerical points of view. The classical approach to uniqueness, based on exponentially growing families of solutions, leads to reconstruction strategies that involve analyzing a certain scattering transform (see, for instance, [Reference Astala, Faraco and Rogers3, Reference Nachman46, Reference Nachman47]), that can be transformed into numerical algorithms (see, among many others, [Reference Delbary, Hansen and Knudsen24, Reference Knudsen, Lassas, Mueller and Siltanen36, Reference Siltanen, Mueller and Isaacson56]). A different type of strategy, sometimes combined with the previous one, is based on linearization. This serves as the basis of one-step linearization methods (see [Reference Harrach and Seo32]), the so-called Calderón method (see, for example, [Reference Bikowski and Mueller11, Reference Shin and Mueller55]), the algorithm described in [Reference Bikowski, Knudsen and Mueller10], and has also been applied in deep learning approaches to the Calderón problem; see [Reference Martin and Choi45]. This kind of linearization methods are successful from the numerical point of view, but seem hard to justify rigorously. The main difficulty lies in proving the existence of a certain Born-type approximation for the inverse problem.

In this work, we address this question, showing the existence of a Born approximation for the inverse problem for radially symmetric potentials, and we analyze how this leads to interesting uniqueness, stability, and characterization results. This approach does not use any CGO-type construction.

This approach was initiated in [Reference Barceló, Castro, Macià and Meroño8, Reference Barceló, Castro, Macià and Meroño9] in the context of the Calderón problem, and is based on the notion of approximation in inverse scattering that can be traced back to the work of Born [Reference Born14]. It was successfully applied in [Reference Daudé, Macià, Meroño and Nicoleau23] to the Gel’fand-Calderón problem at zero energy; and a systematic exposition of this approach, which in principle applies to any inverse problem, was given in [Reference Maciá and Meroño41]. In the present context, it can be described as follows. The map $\Phi^{\kappa}$ is Fréchet differentiable and satisfies $\Phi^{\kappa}(0)=0$; if for every $q\in \mathcal{X}$ it is possible to find $q^{\mathrm{B}}_{\kappa}\in \mathcal{B}$ in some space $\mathcal{B}$ of functions or distributions solving

(1.4)\begin{equation} \mathrm{d}\Phi^{\kappa}_0(q^{\mathrm{B}}_{\kappa}) = \Phi^{\kappa}(q)= \Lambda_{q,\kappa}-\Lambda_{0,\kappa}, \end{equation}

then one has in fact obtained a factorization of $\Phi^{\kappa}$ into a linear and a nonlinear map:

The objective is to exploit this factorization to obtain uniqueness, stability, reconstruction and characterization results for the inverse problem. The main difficulty in implementing this strategy is to show that there exists a solution of equation (1.4). In this work we show that (1.4) can be solved and that the factorization (1.5) exists for the Gel’fand-Calderón problem at fixed energy $\kappa\in\mathbb{R}$, assuming radial symmetry on the potentials.

In the radial case, the existence of $q^{\mathrm{B}}_{\kappa}$ for κ = 0 has already been proved in [Reference Daudé, Macià, Meroño and Nicoleau23], a result that also implies a partial characterization of the set of DtN maps. A key step in the proof involves the tools introduced by Simon [Reference Simon57] in the context of inverse spectral theory of Schrödinger operators on the half-line, especially the notion of A-amplitude. In [Reference Maciá and Meroño41], the authors show that, in fact, the A-amplitude coincides with the notion of Born approximation for this one-dimensional inverse spectral problem.

Here, the proof of the existence of $q^{\mathrm{B}}_{\kappa}$ relies again on the analysis of a one-dimensional inverse spectral problem, the difficulty being that the corresponding Born approximation is no longer Simon’s A-amplitude. The existence of such a Born approximation, which we call the A_κ-amplitude, is an important part of this work. The main results of this article are:

1. i) Explicit description of $\mathrm{d}\Phi^{\kappa}_0$ and its inverse. We show that $\mathrm{d}\Phi^{\kappa}_0$ maps potentials to operators that are rotationally invariant, and whose eigenvalues are certain moments of the potential. Also, we give an explicit formula for $(\mathrm{d}\Phi^{\kappa}_0)^{-1}$. This is presented in Theorem 3.1 and Proposition 2.4.

2. ii) Existence of the Born approximation. This is stated in Theorem 1. Using the explicit formula for $(\mathrm{d}\Phi^{\kappa}_0)^{-1}$ and tools of spectral theory of Schrödinger operators on the half-line, most particularly the existence of an A_κ-amplitude (Theorem 4.4), we prove that $q^{\mathrm{B}}_{\kappa}$, the Born approximation at energy κ, exists as a radial distribution. We also present explicit formulas for $q^{\mathrm{B}}_{\kappa}$.

3. iii) Regularity of $q^{\mathrm{B}}_{\kappa}$. In Theorem 2 we prove that $q^{\mathrm{B}}_{\kappa}$ coincides with an integrable function outside the origin, that $q^{\mathrm{B}}_{\kappa}-q$ is one derivative smoother than q, and that $q^{\mathrm{B}}_{\kappa} =q$ at the boundary.

4. iv) Injectivity of $\Phi_{\mathrm{B}}^{\kappa}$ and stability of $(\Phi_{\mathrm{B}}^{\kappa})^{-1}$. The map $\Phi_{\mathrm{B}}^{\kappa}: q \mapsto q^{\mathrm{B}}_{\kappa}$ is injective in a strong sense: two potentials coincide in a neighborhood of the boundary if and only if their Born approximations at energy κ coincide in that same neighborhood. In addition, $(\Phi_{\mathrm{B}}^{\kappa})^{-1}$ is Hölder continuous, under mild a priori conditions on the potentials, with respect to the L ¹ norm in the complement of any arbitrarily small ball centered in the origin. This is the scope of Theorem 3, which is derived from its one-dimensional analogue Theorem 4.5.

5. v) High-energy/semiclassical limit. In Theorem 4 we show that the Born approximation at energy κ converges to the potential q in the high-energy limit $\kappa\to-\infty$. The analogous result for the direct problem, namely that $\Phi^{\kappa}(q)-\mathrm{d}\Phi^{\kappa}_0(q)$ converges to zero in the same regime, is proved in Proposition 2.5.

In (1.5), the maps $\mathrm{d}\Phi_0^{\kappa}$ and $\Phi^{\kappa}$ are continuous, but the ranges of each of them are not closed. Hence $(\mathrm{d}\Phi_0^{\kappa})^{-1}$ and $\Phi^{\kappa}$ are both discontinuous and only conditionally stable. Therefore, an important consequence of iv) is that the bad behavior of the inverse of $\Phi^{\kappa}$ is captured by the linear approximation $\mathrm{d}\Phi^{\kappa}_0$, since the non-linear map $(\Phi_{\mathrm{B}}^{\kappa})^{-1}$ is Hölder stable.

1.2. Statement of the main results

From now on, we denote by $L^{p}_{\mathrm{rad}}([{\mathbb{B}}^d,\mathbb{R})$ the closed subspace of radial functions that belong to $L^p({\mathbb{B}}^d,\mathbb{R})$. We will assume that $d\ge 2$ and that

(1.6)\begin{equation} p \in \mathbb{R} \text{ is admissible }\quad \iff \quad 1 \lt p \le \infty ,\; \text{and }\; p \ge d/2. \end{equation}

Also, we define

\begin{equation*} \quad \nu_d := \frac{d-2}{2},\qquad {\mathbb{N}}_0:=\mathbb{N}\cup\{0\}. \end{equation*}

Recall that the Dirichlet spectrum of $-\Delta$ on the ball $\mathbb{B}^d$ can be expressed in terms of the zeros of certain Bessel functions. The operator $\Lambda_{q,\kappa}$ is well defined as soon as $q\in L^{p}_{\mathrm{rad}}(\mathbb{B}^d,\mathbb{R})$, with p admissible, and $\kappa\in\mathbb{C}\setminus\operatorname{Spec}_{H^1_0}(-\Delta+q)$. Since q is radial, the DtN operator commutes with the action of $\operatorname{SO}(d)$ on $\mathbb{S}^{d-1}$; as a consequence, its eigenfunctions are spherical harmonics (see Appendix B for the definition and properties of spherical harmonics). More precisely, if $\mathfrak{H}^d_{\ell}$ denotes the subspace of spherical harmonics of degree $\ell\in\mathbb{N}_0$ on $\mathbb{S}^{d-1}$, then

(1.7)\begin{equation} \Lambda_{q,\kappa}|_{{\mathfrak{H}}^d_{\ell}}= \lambda_{\ell}[q,\kappa] \operatorname{\mathrm{Id}}_{{\mathfrak{H}}^d_{\ell}}. \end{equation}

When q = 0, the spectrum of $\Lambda_{0,\kappa}$ can be explicitly computed. If κ = 0 then $\lambda_{\ell}[0,0] = \ell$ for every $\ell\in\mathbb{N}_0$, and, if $\kappa\in\mathbb{C}\setminus\operatorname{Spec}_{H^1_0}(-\Delta)$ with κ ≠ 0, then

(1.8)\begin{equation} \lambda_{\ell}[0,\kappa]= \ell - \sqrt{\kappa}\dfrac{J_{\ell+1+\nu_d}(\sqrt{\kappa})}{J_{\ell+\nu_d}(\sqrt{\kappa})}, \qquad \forall \ell \in \mathbb{N}_0, \end{equation}

where J_ν is the Bessel function of the first kind of index $\nu\in\mathbb{C}$ (see Lemma 4.2).

Given $\ell\in\mathbb{N}_0$, the holomorphic function

(1.9)\begin{equation} \varphi_{\ell}(z):=\frac{J_{\ell+\nu_d}(z)}{z^{\nu_d}}=\frac{z^\ell}{2^{\ell+\nu_d}}\sum_{n=0}^\infty \frac{(-1)^n}{n!\Gamma(\ell+\nu_d+n+1)}\left(\frac{z}{2}\right)^{2n},\qquad z\in\mathbb{C}, \end{equation}

will play an important role throughout this work. For $\kappa\in\mathbb{C}$, the κ-moments of q are:

(1.10)\begin{equation} \sigma_{\ell}[q,\kappa] := \frac{1}{|\mathbb{S}^{d-1}|} \int_{\mathbb{B}^d} q(x) \varphi_{\ell}(\sqrt{\kappa}|x|)^2 \, \mathrm{d}x, \qquad \forall \ell \in \mathbb{N}_0. \end{equation}

Note that the functions $\kappa\mapsto\sigma_{\ell}[q,\kappa]$ are holomorphic in $\mathbb{C}$.

The κ-moments appear in the expression of the Fréchet differential of $\Phi^{\kappa}$ at q = 0. In Proposition 2.4 we prove that, as soon as κ is not a Dirichlet eigenvalue of $-\Delta$ in the ball, $\mathrm{d}\Phi^{\kappa}_0(q)$ is a bounded operator on $L^2(\mathbb{S}^{d-1})$ for every $q\in L^{\infty}_{\mathrm{rad}}(\mathbb{B}^d,\mathbb{R})$. These operators turn out to be invariant by the action of $\operatorname{SO}(d)$ and are characterized by their restriction on spherical harmonics, which is precisely given by

(1.11)\begin{equation} \varphi_{\ell}(\sqrt{\kappa})^{2}\mathrm{d}\Phi^{\kappa}_0 (q)|_{{\mathfrak{H}}^d_{\ell}} = \sigma_{\ell}[q,\kappa] \operatorname{\mathrm{Id}}_{{\mathfrak{H}}^d_{\ell}}, \qquad \forall \ell \in \mathbb{N}_0. \end{equation}

Therefore, by (1.7), that a function $q^{\mathrm{B}}_{\kappa}$ satisfies $\mathrm{d}\Phi^{\kappa}_0 (q^{\mathrm{B}}_{\kappa})=\Lambda_{q,\kappa}-\Lambda_{0,\kappa}$ is formally equivalent to the fact that $q^{\mathrm{B}}_{\kappa}$ solves the following moment problem:

(1.12)\begin{equation} \sigma_{\ell}[q^{\mathrm{B}}_{\kappa},\kappa] = (\lambda_{\ell}[q,\kappa]-\lambda_{\ell}[0,\kappa])\varphi_{\ell}(\sqrt{\kappa})^2, \qquad \qquad \forall \ell \in \mathbb{N}_0. \end{equation}

Note right away that, by (1.8) and (1.9), the left-hand side of the identity (1.12) is well defined as soon as $\kappa\in\mathbb{C}\setminus \operatorname{Spec}_{H^1_0}(-\Delta+q)$ even if $ \kappa\in \operatorname{Spec}_{H^1_0}(-\Delta)$.

The existence of a solution for (1.12) is not guaranteed. In order to prove that a solution exists, we need to allow for the possibility of $q^{\mathrm{B}}_{\kappa}$ being a distribution. Let $U\subseteq \mathbb{R}^d$ be an open set. We denote by $\mathcal{E}'(U)$ the space of distributions with compact support in U, and $\mathcal{E}'_{\mathrm{rad}}(U) \subset \mathcal{E}'(U)$ the subset of radially symmetric distributions.

The definition (1.10) can be extended from $L^{\infty}_{\mathrm{rad}}(\mathbb{B}^d,\mathbb{R})$ to the space $\mathcal{E}_{\mathrm{rad}}'(\mathbb{R}^d)$ as follows: for $f\in \mathcal{E}_{\mathrm{rad}}'(\mathbb{R}^d)$, $\kappa\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$ define

(1.13)\begin{equation} \sigma_{\ell}[f,\kappa]:= \frac{1}{\left\lvert \mathbb{S}^{d-1} \right\rvert}\left \langle f, (\varphi_{\kappa,\ell})^2 \right \rangle_{\mathcal{E}'\times\mathcal{C}^\infty},\quad \text{where }\varphi_{\kappa,\ell}(x):=\varphi_{\ell}(\sqrt{\kappa}|x|), \end{equation}

where $\left \langle \cdot, \cdot \right \rangle_{\mathcal{E}'\times\mathcal{C}^\infty}$ denotes the duality pairing in $\mathcal{E}'(\mathbb{R}^d)$. This makes sense since $(\varphi_{\kappa,\ell})^2\in\mathcal{C}^\infty(\mathbb{R}^d)$.

To state our first result, we define

\begin{equation*} \mathcal{B}_d := L^1_{\mathrm{rad}}(\mathbb{B}^d) + \mathcal{E}'_{\mathrm{rad}}(\mathbb{B}^d). \end{equation*}

The elements of $\mathcal{B}_d$ are radial distributions in $\mathbb{B}^d$ that coincide with L ¹ functions in a neighborhood of the boundary of $\mathbb{B}^d$. This implies that the extension by zero of a distribution in $\mathcal{B}_d $ is an element of $\mathcal{E}'_{\mathrm{rad}}(\mathbb{R}^d)$. In this way, when convenient, $\mathcal{B}_d$ can be identified with a subset of $\mathcal{E}'_{\mathrm{rad}}(\mathbb{R}^d)$. We adopt this perspective in a few points in this work, for example, to define the moments and the Fourier transform of elements of $\mathcal{B}_d$ using (1.13) and (1.14). Note that (1.11) allows us to extend $\mathrm{d}\Phi^{\kappa}_0$ to $\mathcal{B}_d$.

Let

\begin{equation*} X_{p,\kappa}(\mathbb{B}^d):=\{q\in L^p_{\mathrm{rad}}(\mathbb{B}^d,\mathbb{R})\,:\,\kappa\not\in\operatorname{Spec}_{H^1_0}(-\Delta+q)\}. \end{equation*}

We show, for potentials $X_{p,\kappa}$ with p satisfying (1.6), the existence of $q^{\mathrm{B}}_{\kappa}$ as an element of $\mathcal{B}_d$ and establish a formula for its Fourier transform that shows, in particular, that $q^{\mathrm{B}}_{\kappa}$ is uniquely determined by (1.12) in $\mathcal{B}_d$ (see also Theorem 3.1). The Fourier transform of $f\in\mathcal{E}_{\mathrm{rad}}'(\mathbb{R}^d)$ is defined as

(1.14)\begin{equation} \mathcal{F} (f)(\xi) := \left \langle f,e_{-i\xi} \right \rangle_{\mathcal{E}'\times\mathcal{C}^\infty}, \qquad e_\zeta(x):=e^{\zeta\cdot x},\qquad x,\xi \in \mathbb{R}^d,\; \zeta\in\mathbb{C}^d. \end{equation}

We denote by $Z_{\ell,d}$, $\ell\in\mathbb{N}_0$, the zonal functions, which are characterized in terms of $\mathcal{P}_{\ell,d}:L^2(\mathbb{S}^{d-1})\longrightarrow\mathfrak{H}_{\ell}^d$, the orthogonal projection onto $\mathfrak{H}^d_{\ell}$, by

\begin{equation*} \mathcal{P}_{\ell,d}u(x)=\int_{\mathbb{S}^{d-1}}Z_{\ell,d}(x\cdot y)u(y)\, \mathrm{d}y,\qquad u\in L^2(\mathbb{S}^{d-1}). \end{equation*}

The zonal functions are directly related to the Legendre polynomials; more details are given in Appendix B.

Theorem 1.

Let $\kappa\in\mathbb{R}$ and p be admissible. For every $q \in X_{p,\kappa}(\mathbb{B}^d)$ there exists a unique $q^{\mathrm{B}}_{\kappa} \in \mathcal{B}_d$ such that

(1.15)\begin{equation} \sigma_{\ell}[q^{\mathrm{B}}_{\kappa},\kappa] = (\lambda_{\ell}[q,\kappa]-\lambda_{\ell}[0,\kappa]) \varphi_{\ell}(\sqrt{\kappa})^2, \qquad \text{for all } \, \ell \in \mathbb{N}_0. \end{equation}

The Fourier transform of this distribution can be obtained from the expression:

(1.16)\begin{equation} \mathcal{F} (q^{\mathrm{B}}_{\kappa})(\xi) = (2\pi)^d \sum_{\ell=0}^\infty (\lambda_{\ell}[q,\kappa] - \lambda_{\ell}[0,\kappa])\varphi_{\ell}(\sqrt{\kappa})^2 Z_{\ell,d}\left(1-\frac{\left\lvert \xi \right\rvert^2}{2\kappa}\right). \end{equation}

If we further assume $\kappa \notin \operatorname{Spec}_{H^1_0}(-\Delta)$ then it also holds that

(1.17)\begin{equation} \mathcal{F} (q^{\mathrm{B}}_{\kappa})(\xi) = \left\langle \overline{e_{\zeta_1}} ,(\Lambda_{\kappa,q}-\Lambda_{\kappa,0})e_{\zeta_2}\right\rangle_{L^2(\mathbb{S}^{d-1})}, \end{equation}

with $\zeta_1,\zeta_2\in \mathbb{C}^d$ satisfying $\zeta_1+\zeta_2=-i\xi$ and $\zeta_1\cdot\zeta_1 =\zeta_2\cdot\zeta_2=-\kappa$.

We will refer to $q^{\mathrm{B}}_{\kappa}$ as the Born approximation of q at energy κ. Its explicit reconstruction formula (1.16) can be used to numerically approximate q; see Section 1.3 below. On the other hand, the identity (1.17) shows that $q^{\mathrm{B}}_{\kappa}$ can also be obtained by complex geometrical optics solutions as was done for κ = 0 in [Reference Barceló, Castro, Macià and Meroño8].

Our next result shows, among other properties, that $q^{\mathrm{B}}_{\kappa}$ always coincides with a locally integrable function outside the origin. In fact, the Born approximation is as singular as the potential is, except possibly at the origin.

We define the weighted L ¹-space $L^1_s(\mathbb{B}^d): = \{f\in L^1_{\mathrm{loc}}(\mathbb{B}^d\setminus\{0\}) : \; \left\lVert f \right\rVert_{L^1_s(\mathbb{B}^d)} \lt \infty\}$ where $s \in \mathbb{R}$ and

(1.18)\begin{equation} \left\lVert F \right\rVert_{L^1_s(\mathbb{B}^d)} := \int_{\mathbb{B}^d} |F(x)| |x|^{2(s-\nu_d)} \, \mathrm{d}x. \end{equation}

Theorem 2.

Let $\kappa \in \mathbb{R} $ and p be admissible. For every $q\in X_{p,\kappa}(\mathbb{B}^d)$, there exists an $\ell_q\geq0$ and a real and radial function $q^s_{\kappa} \in L^1_{\ell_q}(\mathbb{B}^d)$ such that $q^{\mathrm{B}}_{\kappa}|_{\mathbb{B}^d\setminus \{0\}} = q^s_{\kappa}$. In fact,

(1.19)\begin{equation} q^{\mathrm{B}}_{\kappa} - q\in \mathcal{C}(\overline{\mathbb{B}^d}\setminus \{0\}),\qquad (q^{\mathrm{B}}_{\kappa} - q)|_{\mathbb{S}^{d-1}} =0, \end{equation}

and $q^{\mathrm{B}}_{\kappa}- q$ belongs to $\mathcal{C}^{m+1}(\overline{\mathbb{B}^d} \setminus \{0\})$ if $q \in \mathcal{C}^{m}(\overline{\mathbb{B}^d} \setminus \{0\})$ for any $m\in \mathbb{N}_0$.

If in addition, $p \gt d/2$, $\kappa \lt (j_{0,1})^2$, and $q-\kappa \ge 0$ $a.e.$ in $\mathbb{B}^d$, then $q^{\mathrm{B}}_{\kappa} \in L^1_{\mathrm{rad}}(\mathbb{B}^d)$.

We prove two important properties of the map $(\Phi_{\rm B}^{\kappa})^{-1}$: a strong local form of injectivity and Hölder continuity under uniform $L^p(\mathbb{B}^d)$ bounds for the potentials.

We finally show that in the high-energy limit $\kappa\to-\infty$, the Born approximation at energy κ recovers the potential.

Theorem 4.

Let $q\in L^\infty(\mathbb{B}^d,\mathbb{R})$, $\kappa\in\mathbb{R}$, $\xi\in\mathbb{R}^d$, and $\zeta_1,\zeta_2\in \mathbb{C}^d$ be such that $\zeta_1+\zeta_2=-i\xi$, and $ \zeta_1\cdot\zeta_1 =\zeta_2\cdot\zeta_2=-\kappa$. Then, locally uniformly in ξ, we have

(1.21)\begin{equation} \lim_{\kappa\to-\infty}\left\langle \overline{e_{\zeta_1}} ,(\Lambda_{q,\kappa}-\Lambda_{0,\kappa})e_{\zeta_2}\right\rangle_{L^2(\mathbb{S}^{d-1})}=\mathcal{F} q(\xi). \end{equation}

In particular, whenever $q\in L^\infty_{\mathrm{rad}}(\mathbb{B}^d,\mathbb{R})$, in which case $q^{\mathrm{B}}_{\kappa}$ is always defined when $\kappa\leq -\left\lVert q \right\rVert_{L^\infty(\mathbb{B}^d)}$, we have

(1.22)\begin{equation} \lim_{\kappa\to-\infty}\mathcal{F} q^{\mathrm{B}}_{\kappa}(\xi)=\mathcal{F} q(\xi),\qquad \text{for all }\xi\in\mathbb{R}^d. \end{equation}

The local uniqueness phenomenon proved in the first part of Theorem 3 is illustrated in Fig. 4, whereas the phenomenon of recovery of singularities in Theorem 2 is illustrated in Figs. 1, 3 and 4. Evidence for the convergence of the Born approximations to the potential in physical space is presented in Fig. 3 below.

Figure 1.

Plots of $q(x)=2-\chi_{(\frac{1}{3},\frac{2}{3})}(\left\lvert x \right\rvert)$ (blue) and $(q+\kappa)^{\mathrm{B}}_{\kappa}-\kappa$ (orange). a) κ = 0, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.66066$. b) $\kappa=-2$, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.12155$. c) $\kappa=-1.5$, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.05551$. d) $\kappa=\kappa_*=-1.73614$, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.05084$.

Figure 2.

Plots of $q(x)=\cos(4\pi\left\lvert x \right\rvert)-5$ (blue) and $(q+\kappa)^{\mathrm{B}}_{\kappa}-\kappa$ (orange). a) κ = 0, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=16.4772$. b) κ = 4, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.51683$. c) κ = 5, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.07038$. d) $\kappa=\kappa_*=4.96727$, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.07200$.

Figure 3.

Plots of $q(x)=2-\chi_{(\frac{1}{3},\frac{2}{3})}(\left\lvert x \right\rvert)$ (blue) and $q^{\mathrm{B}}_{\kappa}$ (orange). a) κ = 0, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.66066$. b) $\kappa=-10$, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.56134$. c) $\kappa=-100$, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.35731$. d) $\kappa=-1000$, $\left\lVert \,\cdot\, \right\rVert_{L^1(\mathbb{B}^3)}=0.17218$.

Figure 4.

Plots of q_i (solid) of $q_3(x):= 3$, $q_2(x):= q_3(x)-\chi_{(0,\frac{2}{3})}(\left\lvert x \right\rvert)$, $q_1(x):= q_2(x)-\chi_{(0,\frac{1}{3})}(\left\lvert x \right\rvert)$ and their respective Born approximations $q_{i,\kappa}^{\mathrm{B}}$ (dashed) at $\kappa=-1$.

Theorems 1 to 4 are proved in Section 5. The analogues of Theorems 1, 2 and 3 in the case κ = 0 have been proved in [Reference Daudé, Macià, Meroño and Nicoleau23]. These results can be recovered from the ones in this work in the limit κ → 0. For instance, the classical Hausforff moments, which played the same role as the κ-moments here, can be obtained from $\sigma_{\ell}[q,\kappa]$ by

(1.23)\begin{equation} 4^{\ell+\nu_d} \Gamma(\ell+d/2)^2\lim_{\kappa\to 0}\frac{\sigma[q,\kappa]}{\kappa^\ell }=\frac{1}{|\mathbb{S}^{d-1}|} \int_{\mathbb{B}^d} q(x) |x|^{2\ell} \, \mathrm{d}x, \qquad \forall \ell \in \mathbb{N}_0. \end{equation}

The proofs of the main results in [Reference Daudé, Macià, Meroño and Nicoleau23] rely on tools from the inverse spectral theory of Schrödinger operators on the half-line, mainly results involving the notion of A-amplitude of a Schrödinger operator introduced by Simon in [Reference Simon57], and further developed in [Reference Avdonin, Mikhaylov and Rybkin6, Reference Gesztesy and Simon30, Reference Ramm and Simon51, Reference Remling52], among many other works. In our case, these tools are not directly applicable, which leads us to introduce and prove the existence of an A_κ-amplitude that encodes spectral information in a similar way as Simon’s A-amplitude does, but is better adapted to our setting. The main results on the A_κ-amplitude are stated in Section 4.2 and proved in Section 6.

We note that the approach based on one-dimensional inverse spectral theory has been applied in the context of the Steklov problem for warped product manifolds in [Reference Daudé, Helffer and Nicoleau20–Reference Daudé, Kamran and Nicoleau22]. In particular, the results in [Reference Daudé, Kamran and Nicoleau22] imply stability and uniqueness results for the radial Calderón problem, both for the conductivity and Schrödinger cases. We also mention that spectral theory methods had already been used in the context of the radial Calderón problem in [Reference Kohn and Vogelius38, Reference Sylvester58].

1.3. Numerical reconstruction

In this subsection, we show the capabilities of the Born approximation as an effective tool to approximate the potential; numerical methods based on this strategy are described in [Reference Barceló, Castro, Macià and Meroño9] in the case κ = 0. The key remark is that the Fréchet differential $\mathrm{d}\Phi^{\kappa}_0$ coincides with the differential of Φ⁰ around the constant function $-\kappa\in\mathbb{R}\setminus\operatorname{Spec}_{H^1_0}{\Delta}$, and that $\Lambda_{q,0}=\Lambda_{q+\kappa,\kappa}$. Therefore, for such κ, given $q\in L^\infty_{\mathrm{rad}}(\mathbb{B}^d,\mathbb{R})$ such that 0 is not a Dirichlet eigenvalue of $-\Delta+q$, one should have:

\begin{equation*} q\approx -\kappa + \mathrm{d}\Phi^0_{-\kappa}(\Lambda_{q,0} - \Lambda_{0,\kappa}) = -\kappa + (q+\kappa)^{\mathrm{B}}_{\kappa}. \end{equation*}

When κ is chosen appropriately, so that $q +\kappa$ is small in some norm, one can expect this function to be well-approximated by $(q+\kappa)^{\mathrm{B}}_{\kappa}$. Recall that, by Theorem 1, the Born approximation $(q+\kappa)^{\mathrm{B}}_{\kappa}$ is constructed solely in terms of $\operatorname{Spec}(\Lambda_{q,0})$ (the dependence on κ being explicit). A possible choice of κ that can be implemented numerically consists in ensuring that $\int_{\mathbb{B}^d}(q+\kappa)^{\mathrm{B}}_{\kappa} \mathrm{d}x=\mathcal{F}(q+\kappa)^{\mathrm{B}}_{\kappa}(0)$, which is a quantity that depends only on $\operatorname{Spec}(\Lambda_{q,0})$ and κ, is the smallest possible. It is therefore natural to choose $\kappa=\kappa_*$ defined by the equation

\begin{equation*}\mathcal{F}(q+\kappa_*)^{\mathrm{B}}_{\kappa_*}(0)=0.\end{equation*}

Numerical experiments suggest that if $\kappa_*$ exists, it can be obtained by the iterative fixed point scheme

\begin{equation*}\kappa_{n+1}=\kappa_n-\frac{\mathcal{F}(q+\kappa_{n})^{\mathrm{B}}_{\kappa_n}(0)}{|\mathbb{B}^d|}.\end{equation*}

We illustrate these ideas in Figs. 1 and 2 where we plot the radial profiles of $q:\mathbb{B}^3\to\mathbb{R}$ and the corresponding $(q+\kappa)^{\mathrm{B}}_{\kappa}-\kappa$ for

\begin{equation*}\kappa=0,\qquad\kappa=-q|_{\mathbb{S}^{d-1}},\qquad\kappa=-\frac{1}{2}\left(\sup_{\mathbb{B}^3} q+ \inf_{\mathbb{B}^3} q\right),\qquad\kappa=\kappa_*.\end{equation*}

In the caption of each plot, we include the $L^1(\mathbb{B}^3)$ norm of the difference of the functions plotted. Notice that in all eight plots, except for Fig. 2 (A) where the Gibbs phenomenon is too strong, $(q+\kappa)-(q+\kappa)^{\mathrm{B}}_{\kappa}$ is a continuous function outside x = 0 that vanishes at $\mathbb{S}^{2}$, as stated in Theorem 2.

The Julia code used to produce these figures can be found in [Reference Macià, Meroño and Sánchez-Mendoza42].