An inverse scattering problem for short-range systems in a time-periodic electric field

TL;DR

This paper investigates inverse scattering problems for time-periodic short-range systems under electric fields, showing that high energy scattering data uniquely determine the electric potential, with specific conditions depending on the mean electric field and dimension.

Contribution

It demonstrates the unique determination of short-range electric potentials from high energy scattering data in time-periodic systems, extending previous results to cases with non-zero mean electric fields and different dimensions.

Findings

High energy scattering operators determine the potential when mean electric field is zero.

In dimensions n ≥ 3, the same holds for generic short-range potentials with non-zero mean.

In dimension 2, stronger decay conditions on the potential are required.

Abstract

We consider the time-dependent Hamiltonian $H(t)= {1 \over 2} p^2 -E(t) \cdot x + V(t,x)$ on $L^2(R^n)$, where the external electric field $E(t)$ and the short-range electric potential $V(t,x)$ are time-periodic with the same period. It is well-known that the short-range notion depends on the mean value $E\_0$ of the external field. When $E\_0=0$, we show that the high energy limit of the scattering operators determines uniquely $V(t,x)$. In the other case, the same result holds in dimension $n \geq 3$ for generic sghort-range potentials. In dimension 2, one has to assume a stronger decay on the electric potential.