Absence of reflection as a function of the coupling constant

TL;DR

This paper investigates how the zeros of the Wronskian between solutions of a one-dimensional Schrödinger-like equation depend on the coupling constant, showing they are discrete unless the potential is zero, with implications for reflectionless scattering.

Contribution

It proves that the zeros of the Wronskian as a function of the coupling constant are discrete unless the potential is identically zero, linking spectral properties to potential triviality.

Findings

Zeros of the Wronskian are discrete unless V ≡ 0.

Vanishing reflection coefficient on a set with an accumulation point implies V ≡ 0.

Results connect spectral data with the triviality of the potential.

Abstract

We consider solutions of the one-dimensional equation $-u'' +(Q+ \lambda V) u = 0$ 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 $\lambda \in \mathbb{R}$ is a coupling constant. Given a family of solutions $\{u_{\lambda} \}_{\lambda \in \mathbb{R}}$ which satisfy $u_{\lambda}(x) = u_0(x)$ for all $x<0$, we prove that the zeros of $b(\lambda) := W[u_0, u_{\lambda}]$, the Wronskian of $u_0$ and $u_{\lambda}$, 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 $-u'' + \lambda V u = Eu$ gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then $V \equiv 0$.