Inverse Scattering at a Fixed Quasi-Energy for Potentials Periodic in Time

TL;DR

This paper proves that the scattering matrix at a fixed quasi-energy uniquely determines certain time-periodic potentials with exponential decay, extending known results to more singular potentials in quantum scattering theory.

Contribution

It establishes uniqueness of the inverse scattering problem for time-periodic potentials with critical singularities, a case not previously addressed even in static scenarios.

Findings

Uniqueness of potential from fixed quasi-energy scattering matrix

Extension to potentials with critical spatial singularities

Results applicable to time-periodic quantum systems

Abstract

We prove that the scattering matrix at a fixed quasi--energy determines uniquely a time--periodic potential that decays exponentially at infinity. We consider potentials that for each fixed time belong to $L^{3/2}$ in space. The exponent 3/2 is critical for the singularities of the potential in space. For this singular class of potentials the result is new even in the time--independent case, where it was only known for bounded exponentially decreasing potentials.