Scattering maps of the time-dependent Schrödinger equation determine the metric up to diffeomorphism pullback when the maps differ by a compact operator.
Scattering metrics and geodesic flow at infinity.Inventiones Mathematicae, 124(1):389–436
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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.