Strictly convex corners scatter (Q1687704)

A quantum mechanical system has a non-scattering energy \(\lambda>0\) if the scattering matrix \(S(\lambda)\) corresponding to that energy has the eigenvalue 1. The system is modeled by the Schrödinger operator \(-\Delta+V\) on \(\mathbb{R}^n \) and the potentials \(V\) considered here are of the form \(V(x)=\chi_C(x)\phi(x)\) (\(x \in \mathbb{R}^n\)) where \(\chi_C\) is the characteristic function of a convex cone \(C\) with apex at the origin and opening angle strictly less than \(\pi\) and \(\phi(x)=O(e^{-\gamma x})\) for all \(\gamma>0\); also \(\phi(0) \neq 0\) and \(\phi\) satisfies a certain Hölder continuity condition. The main result is that, if \(n=2\), then \(-\Delta +V\) has no non-scattering energies. A somewhat weaker result holds when \(n=3\) and there is a commentary about the case \(n\geq 4\). The significance of these results in acoustic scattering is explained.