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In this paper, we adapt the well-known \emph{local} uniqueness results ofBorg-Marchenko type in the inverse problems for one dimensional Schr{\"o}dingerequation to prove \emph{local} uniqueness results in the setting of inverse\emph{metric} problems. More specifically, we consider a class of sphericallysymmetric manifolds having two asymptotically hyperbolic ends and study thescattering properties of massless Dirac waves evolving on such manifolds. Usingthe spherical symmetry of the model, the stationary scattering is encoded by acountable family of one-dimensional Dirac equations. This allows us to definethe corresponding transmission coefficients $T(\lambda,n)$ and reflectioncoefficients $L(\lambda,n)$ and $R(\lambda,n)$ of a Dirac wave having a fixedenergy $\lambda$ and angular momentum $n$. For instance, the reflectioncoefficients $L(\lambda,n)$ correspond to the scattering experiment in which awave is sent from the \emph{left} end in the remote past and measured in thesame left end in the future. The main result of this paper is an inverseuniqueness result local in nature. Namely, we prove that for a fixed $\lambda\not=0$, the knowledge of the reflection coefficients $L(\lambda,n)$ (resp.$R(\lambda,n)$) - 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 arethe Complex Angular Momentum method as well as some useful uniqueness resultsfor Laplace transforms.

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