An inverse scattering problem for the Schr\"{o}dinger equation in a semiclassical process

TL;DR

This paper investigates how the scattering operator for a semiclassical Schrödinger equation in dimensions three and higher can uniquely determine the potential at infinity, given precise knowledge of the scattering data near a fixed energy.

Contribution

It demonstrates that the scattering operator's behavior at small semiclassical parameters uniquely determines the potential at infinity in a high-dimensional setting.

Findings

Scattering operators determine the potential at infinity.

Results hold for dimensions n ≥ 3.

Precise knowledge of scattering operators up to O(h^∞) is sufficient.

Abstract

We study an inverse scattering problem for a pair of Hamiltonians $(H(h), H\_0 (h))$ on $L^2 (\r^n)$, where $H\_0 (h) = -h^2 \Delta$ and $H (h)= H\_0 (h) +V$, $V$ is a short-range potential with a regular behaviour at infinity and $h$ is the semiclassical parameter. We show that, in dimension $n \geq 3$, the knowledge of the scattering operators $S(h)$, $h \in ]0, 1]$, up to $O(h^\infty)$ in ${\cal{B}} (L^2(\r^n))$, and which are localized near a fixed energy $\lambda >0$, determine the potential $V$ at infinity.