Determining metrics from the scattering map of the time-dependent Schr\"odinger equation

TL;DR

This paper investigates the relationship between scattering maps of time-dependent Schrödinger operators and the underlying metrics, showing they differ by a compact operator if and only if the metrics are related by a diffeomorphism.

Contribution

It establishes a characterization of metrics via their scattering maps, linking operator differences to geometric transformations.

Findings

Scattering maps differ by a compact operator if metrics are related by a diffeomorphism.

The result applies to certain classes of metrics in the Schrödinger equation.

Provides a geometric criterion for metric equivalence via scattering data.

Abstract

For a time dependent Schr\"odinger equation, the scattering map is the map sending the asymptotic profile of solution as $t\to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for certain class of metrics, the scattering maps associated to two Schr\"odinger operators with two time dependent metrics only differ by a compact operator if and only if these two metrics are related by a pull-back of a diffeomorphism.