The inverse backscattering for Schrödinger operators for potentials with noncompact support

Abstract

We consider the inverse backscattering problem for the Schrödinger operator H = −Δ + V on , n ≥ 3, as well as the higher‐order Schrödinger operator ( − Δ)m + V, m = 2,3,…. We show that in some suitable Banach spaces, the map from the potential to the backscattering amplitude is a local diffeomorphism. This kind of problem (for m = 1) was studied by Eskin and Ralston [Comm. Math. Phys., 124(2), 169‐215 (1989)], where they assumed that . In this paper, we replace the assumption on V with certain decay assumption at infinity.