Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds

Abstract

In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderón problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set.

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

“…Our presentation of the construction follows closely [20,Section 4] to which we refer for omitted details. We mention here the recent work [31], which constructs related Gaussian beam quasimodes in a Riemannian setting by using more sophisticated methods, which lead to better estimates.…”

“…Next we insert the phase function Θ that we have constructed into the transport equations ( 30) and (31) to find an amplitude function a. To solve the transport equations, we write…”

Remarks on the anisotropic Calderón problem

“…A constructive counterpart to the uniqueness proofs in [3,5] is given in [6] for continuous potentials V . We also mention the works [4,14] that study a linearized version of (IP2)-(IP3) on transversally anisotropic manifolds (recall that the Dirichlet-to-Neumann map Λ g,V depends nonlinearly on V ), under the assumption that (M 0 , g 0 ) is real-analytic. Outside the category of CTA manifolds, we refer the reader to [32] that studies a variation of the anisotropic Calderón problem associated to the operator −∆ g + V − λ 2 for a fixed sufficiently large parameter |λ| ≫ V L ∞ (M ) .…”