Absence of reflection as a function of the coupling constant

Abstract

We consider solutions of the one-dimensional equation $-u^{\prime \prime }+(Q+\lambda V)u=0$ 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 $-u^{\prime \prime }+\lambda Vu=Eu$ gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then $V\equiv 0$.

To appear in Journal of Mathematical Physics

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