| Abstract: |
| We study quantum inverse scattering for Schr\"{o}dinger equations with time-dependent quadratic Hamiltonians.The unperturbed system contains either harmonic or repulsive quadratic potentials whose strength decays at a critical rate in time. This decay produces scattering states that differ from both the free and time-independent cases. For real-valued interaction potentials satisfying suitable spatial decay conditions, the existence of wave operators has been established in previous work, and the associated scattering operator is well defined. The main result presented here proves uniqueness in inverse scattering: the interaction potential is uniquely determined by the scattering operator. In the harmonic case, scattering arises despite the confining nature of the quadratic potential. In the repulsive case, the time decay modifies the long-time behavior and allows decay comparable to the long-range class in the Stark effect, while remaining short-range within this framework. |
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