Exponential instability in the inverse scattering problem on the energy interval

We consider the inverse scattering problem on the energy interval in three dimensions. We are focused on stability and instability questions for this problem. In particular, we prove an exponential instability estimate which shows optimality of the logarithmic stability result of [Stefanov, 1990] (up to the value of the exponent). 1. Introdution We consider the Schrödinger equation −∆ψ + v(x)ψ = Eψ, x ∈ R, (1.1) where v is real-valued, v ∈ L(R), v(x) = O(|x|), |x| → ∞, for some ǫ > 0. (1.2) Under conditions (1.2), for any k ∈ R \ 0 equation (1.1) with E = k has a unique continuous solution ψ(x, k) with asymptotics of the form ψ(x, k) = e − 2π e i|k||x| |x| f ( k |k| , x |x| , |k| )