Absence of reflection as a function of the coupling constant

Abstract

We consider solutions of the one-dimensional equation $-u'' +(Q+ λV) u = 0$ where $Q: \mathbb{R} \to \mathbb{R}$ is locally integrable, $V : \mathbb{R} \to \mathbb{R}$ is integrable with supp$(V) \subset [0,1]$, and $λ\in \mathbb{R}$ is a coupling constant. Given a family of solutions $\{u_λ \}_{λ\in \mathbb{R}}$ which satisfy $u_λ(x) = u_0(x)$ for all $x<0$, we prove that the zeros of $b(λ) := W[u_0, u_λ]$, the Wronskian of $u_0$ and $u_λ$, form a discrete set unless $V \equiv 0$. Setting $Q(x) := -E$, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment $-u'' + λV u = Eu$ gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then $V \equiv 0$.