2013/10/02 by Thierry Daudé, Daude, Thierry, Damien Gobin +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1310.0733
openalex publication_date 2013/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we adapt the well-known \local uniqueness results of\nBorg-Marchenko type in the inverse problems for one dimensional Schr "odinger\nequation to prove \local uniqueness results in the setting of inverse\n\metric problems. More specifically, we consider a class of spherically\nsymmetric manifolds having two asymptotically hyperbolic ends and study the\nscattering properties of massless Dirac waves evolving on such manifolds. Using\nthe spherical symmetry of the model, the stationary scattering is encoded by a\ncountable family of one-dimensional Dirac equations. This allows us to define\nthe corresponding transmission coefficients T(\λ,n) and reflection\ncoefficients L(\λ,n) and R(\λ,n) of a Dirac wave having a fixed\nenergy \λ and angular momentum n. For instance, the reflection\ncoefficients L(\λ,n) correspond to the scattering experiment in which a\nwave is sent from the \left end in the remote past and measured in the\nsame left end in the future. The main result of this paper is an inverse\nuniqueness result local in nature. Namely, we prove that for a fixed \λ\n not=0, the knowledge of the reflection coefficients L(\λ,n) (resp.\nR(\λ,n)) - up to a precise error term of the form O(e-2nB) with\nB textgreater0 - determines the manifold in a neighbourhood of the left\n(resp. right) end, the size of this neighbourhood depending on the magnitude\nB of the error term. The crucial ingredients in the proof of this result are\nthe Complex Angular Momentum method as well as some useful uniqueness results\nfor Laplace transforms.\n