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Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds

In this paper, we adapt the well-known \emph{local} uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schr{ö}dinger equation to prove \emph{local} uniqueness results in the setting of inverse \emph{metric} problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients $T(λ,n)$ and reflection coefficients $L(λ,n)$ and $R(λ,n)$ of a Dirac wave having a fixed energy $λ$ and angular momentum $n$. For instance, the reflection coefficients $L(λ,n)$ correspond to the scattering experiment in which a wave is sent from the \emph{left} end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed $λ\not=0$, the knowledge of the reflection coefficients $L(λ,n)$ (resp. $R(λ,n)$) - up to a precise error term of the form $O(e^{-2nB})$ with $B\textgreater{}0$ - determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude $B$ of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.