Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Born approximation in the three-dimensional Calderón problem

Juan A. Barceló, Carlos Castro, Fabricio Macià, Cristóbal J. Meroño

Abstract

Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrödinger operators $-Δ+q $. We study the Born approximation of $q$ in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension $d\ge 3$. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map associated to $-Δ+ q$, which is closely related to a particular moment problem. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis spherical harmonics. To show that the averaged Born approximation does not destroy information on the potential, we also study the high-energy behavior of the matrix elements of the Dirichlet to Neumann map.

The Born approximation in the three-dimensional Calderón problem

Abstract

Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrödinger operators . We study the Born approximation of in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension . We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map associated to , which is closely related to a particular moment problem. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis spherical harmonics. To show that the averaged Born approximation does not destroy information on the potential, we also study the high-energy behavior of the matrix elements of the Dirichlet to Neumann map.

Paper Structure

This paper contains 17 sections, 12 theorems, 221 equations.

Table of Contents

  1. Introduction and main results
  2. Motivations and setting of the results
  3. The Calderón problem on the ball. The radial case
  4. The Calderón problem on the ball. General Schrödinger operators
  5. Structure of the article
  6. Acknowledgments
  7. A few preliminary facts on spherical harmonics
  8. Spherical harmonics
  9. Representation of SO(d) on L2 and angular momentum
  10. Spherical harmonics and angular momentum when d=3
  11. The Born approximation: the radial case
  12. The Born approximation: the general case
  13. The structure of D-N map
  14. The radial case
  15. The general 3-dimensional case
  16. ...and 2 more sections

Key Result

Theorem 1

Let $d\ge 3$ and $q \in \mathcal{Q}_d$ be of the form $q = q_0(|\cdot|)$, where $q_0 \in L^\infty([0,1],\mathbb{R})$. Let $\xi\in\mathbb{R}^d$ and $\zeta_1^h,\zeta_2^h \in \mathcal{V}(d)$ such that $\zeta_1^h + \zeta_2^h = -ih\xi + ir_h$ for $h>0$ small enough, with $r_h\in\mathbb{R}^d$ and $|r_h|=o and If, in addition, $r_h=0$ then no limit needs to be taken in id:q_exp:

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 3
  • Lemma 2.1
  • proof
  • proof : Proof of \ref{['main_thm:Born_aprox_radial']}
  • ...and 18 more