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 −∆ + 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. We show that this approximation is well-defined and obtain a closed formula that involves the spectrum of the Dirichlet-to-Neumann map associated to −∆ + q. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the invariance properties of the exact 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. Motivated by these formulas we also study the high-energy behaviour of the matrix elements of the Dirichlet to Neumann map.1

The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case

“…In this section we show that formula (2.6) implies that the linearization of the radial Calderón problem is essentially a Hausdorff moment problem, a connection observed in [DKN21] and [BCMM21]. To see this, let f (x) = f 0 (|x|) be any radial L 1 (R d ) function with compact support.…”

“…A consequence of the results in [BCMM21] (see Section 2.3 for more details) is that for a radial conductivity γ as (1.4), and such that both γ − 1 and ∂ ν γ vanish at ∂B R , one has:…”

“…where ζ ∈ V ξ , as in the case of (1.3), see [BCMM21] for more details and references on the origin and motivation of this definition.…”

“…Recently, explicit formulas for the Born approximation of a closely related inverse problem -in which the elliptic operator in (1.2) is replaced by a Schrödinger operator −∆ + qhave been obtained in [BCMM21]. These formulas can be adapted to the conductivity inverse problem very easily, and show that the limit (1.3) exists when the conductivity is a sufficiently smooth radial function:…”

“…We denote by (λ R k [γ]) k∈N 0 the sequence of eigenvalues of the DtN map Λ γ in ∂B R . One can show that (see for instance [BCMM21]), when γ = 1 one has λ R k [1] = k/R and that the first eigenvalue always vanishes, that is λ R 0 [γ] = 0. From now on we will drop the super-index in the case R = 1 and use the notation…”