[Paper Review] On non-overdetermined inverse scattering at zero energy in three dimensions

Roman Novikov|ArXiv.org|May 31, 2006

Numerical methods in inverse problems4 citations

TL;DR

This paper develops a $ar{\partial}$-equation approach to non-overdetermined inverse scattering at zero energy in three dimensions, establishing uniqueness, reconstruction, stability, and approximate reconstruction for small potentials from backscattering-type data. The key contribution is a rigorous framework for recovering Schr"odinger potentials from restricted Faddeev scattering amplitudes in the complex domain, with direct applications to electrical impedance tomography.

ABSTRACT

We develop the d-bar -approach to inverse scattering at zero energy in dimensions d>=3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction procedure, stability estimate and approximate reconstruction for the problem of finding a sufficiently small potential v in the Schrodinger equation from a fixed non-overdetermined ("backscattering type") restriction h on $Γ$ of the Faddeev generalized scattering amplitude h in the complex domain at zero energy in dimension d=3. For sufficiently small potentials v we formulate also a characterization theorem for the aforementioned restriction h on $Γ$ and a new characterization theorem for the full Faddeev function h in the complex domain at zero energy in dimension d=3. We show that the results of the present work have direct applications to the electrical impedance tomography via a reduction given first in [Novikov, 1987, 1988].

Motivation & Objective

• To establish a uniqueness theorem for reconstructing small potentials in the 3D Schr"odinger equation from restricted Faddeev scattering amplitudes at zero energy.
• To develop a precise reconstruction procedure for the potential using the $ar{\partial}$-equation formalism applied to non-overdetermined (backscattering-type) data.
• To derive stability estimates and approximate reconstruction schemes for small potentials under $L_{\mu}^\infty$-regularity assumptions on the Fourier transform of the potential.
• To provide a characterization of the scattering data $h|_{\Gamma}$ and the full Faddeev function $h$ in the complex domain for small potentials.
• To demonstrate direct applications of the results to electrical impedance tomography via reduction from Novikov (1987, 1988).

Proposed method

• Utilizes the $ar{\partial}$-equation approach developed in Beals-Coifman (1985), Henkin-Novikov (1987), and Novikov (2002) for inverse scattering at zero energy.
• Applies the Faddeev generalized scattering amplitude $h(k,l)$ defined on the complex manifold $\Theta = \{k,l \in \mathbb{C}^3 : k^2 = l^2 = 0, \operatorname{Im}k = \operatorname{Im}l\}$, with restriction to $\Gamma \subset \Theta$.
• Employs the integral equation $H(k,p) = \hat{v}(p) - \int \frac{\hat{v}(p+\xi) H(k,-\xi)}{\xi^2 + 2k\xi} d\xi$ for $H(k,\cdot)$, with $k \in \Sigma = \{k \in \mathbb{C}^3 : k^2 = 0\}$, to relate the potential $v$ to the scattering data.
• Imposes the condition $\hat{v} \in L_\mu^\infty(\mathbb{R}^3)$ for $\mu \geq 2$ to ensure existence and uniqueness of solutions in the $L_\mu^\infty$-normed space.
• Uses contraction mapping arguments on the space $L_\mu^\infty((\mathbb{C} \setminus 0) \times (\mathbb{R}^3 \setminus \mathcal{L}_\nu))$ to solve the nonlinear integral equations arising in the reconstruction.
• Applies estimates on the kernel $1/\zeta$ in the complex plane and uses $L_\mu^\infty$-norm control to derive stability and convergence bounds for the reconstruction process.

Experimental results

Research questions

• RQ1 Can the potential $v$ in the 3D Schr"odinger equation be uniquely reconstructed from the restriction $h|_{\Gamma}$ of the Faddeev scattering amplitude at zero energy?
• RQ2 What is the precise reconstruction procedure for $v$ from $h|_{\Gamma}$, and what are the stability and convergence properties of this procedure?
• RQ3 What conditions on $\hat{v}$ ensure the solvability of the associated $ar{\partial}$-equation system and the validity of the reconstruction?
• RQ4 How does the restriction $h|_{\Gamma}$ relate to the classical backscattering amplitude, and what is its role in non-overdetermined inverse scattering?
• RQ5 What is the characterization of the scattering data $h|_{\Gamma}$ and the full $h$ in the complex domain for small potentials?

Key findings

• Uniqueness of the potential $v$ is established for $\hat{v} \in L_\mu^\infty(\mathbb{R}^3)$ with $\mu \geq 2$, under the condition $||\hat{v}||_\mu$ sufficiently small.
• A precise reconstruction procedure for $v$ is derived via the solution of a nonlinear integral equation in the $L_\mu^\infty$-space, using successive approximations.
• Stability estimates are obtained: the $L_\mu^\infty$-norm of the difference between reconstructed potentials is bounded by a constant multiple of the $L_\mu^\infty$-norm of the difference in scattering data.
• Approximate reconstruction is achieved with convergence rate $O(r)$, where $r = ||\hat{v}||_\mu$, under the contraction mapping argument with $\alpha = 4c_6(\mu)r < 1$.
• A characterization theorem is proven: $h|_{\Gamma}$ arises from a potential $v$ with $\hat{v} \in L_\mu^\infty(\mathbb{R}^3)$, $\mu \geq 2$, and $||\hat{v}||_\mu$ small, if and only if the associated solution to the $ar{\partial}$-equation satisfies the required integral equation.
• A new characterization theorem is established for the full Faddeev function $h$ in the complex domain at zero energy in $d=3$, under the same smallness and regularity assumptions on $\hat{v}$.