Scattering maps of the time-dependent Schrödinger equation determine the metric up to diffeomorphism pullback when the maps differ by a compact operator.
The spectral projections and the resolvent for scattering metrics.Journal d’Analyse Mathematique, 79:241–298, 1999
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Determining metrics from the scattering map of the time-dependent Schr\"odinger equation
Scattering maps of the time-dependent Schrödinger equation determine the metric up to diffeomorphism pullback when the maps differ by a compact operator.