An inverse problem for the relativistic Schrödinger equation with partial boundary data

Abstract

We study the inverse problem of determining the vector and scalar potentials A(t, x) = (A0, A1, · · · , An) and q(t, x), respectively, in the relativistic Schrödinger equationwhere Ω is a C 2 bounded domain in R n for n ≥ 3 and T > diam(Ω) from partial data on the boundary ∂Q. We prove the unique determination of these potentials modulo a natural gauge invariance for the vector field term.

A partial data inverse problem for the convection-diffusion equation

“…Proof. The proof uses the arguments similar to the one used in [44,48,52] for the case of light ray transforms. We assume that t ∈ (0, T ) is arbitrary but fixed.…”

“…Our first aim is to show that h ij (ξ 0 ) = 0, for all 1 ≤ i, j ≤ n, then later we will prove that h ij (t, ξ) = 0 for 1 ≤ i, j ≤ n and ξ near ξ 0 . Following [44], consider a small perturbation ω 0 (a) of vector ω 0 = e 1 by ω 0 (a) := cos ae 1 + sin ae k where 3 ≤ k ≤ n.…”

“…For elliptic and hyperbolic inverse problems these kind of techniques have been used by several authors. Related to our work, we refer to [13,26] for the elliptic case and to [6,7,32,37,38,39,40,44] for the hyperbolic case.…”

“…is a formal L 2 adjoint of the operator L A,q . We construct these solutions by using a Carleman estimate in a Sobolev space of negative order as used in [26] for elliptic case and in [39,44] for hyperbolic case. Before going further following [39] we will give some definition and notation, which will be used later.…”

“…Proof of the above theorem is based on a Carleman estimate in a Sobolev space of negative order. To prove the Carleman estimate stated in Proposition 1, we follow the arguments similar to one used in [26,39,44] for elliptic and hyperbolic inverse problems. Proposition 1.…”