Mathematical Research Letters

Volume 12 (2005)

Number 6

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

Pages: 885 – 896



François Nicoleau (Université de Nantes)


We consider a time-dependent Hamiltonian $H(t)= {1 \over 2} p^2 -E(t)\cdot x +V(t,x)$ on $L^2(\RR^n)$, 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 electric field. When $E_0 = 0$, we show that the high energy limit of the scattering operators determines uniquely $V(t,x)$. When $E_0 \not= 0$, the same result holds in dimension $n \geq 3$ for generic short-range potentials. In dimension $n= 2$, one has to assume a stronger decay on the electric potential.

Published 1 January 2005