Hilbert Space Theory for Reflectionless Relativistic Potentials

Abstract

We study Hilbert space aspects of explicit eigenfunctions for analytic difference operators that arise in the context of relativistic two-particle Calogero–Moser systems. We restrict attention to integer coupling constants $g/\hslash$, for which no reflection occurs. It is proved that the eigenfunction transforms are isometric, provided a certain dimensionless parameter $a$ varies over a bounded interval $(0,a_{\max })$, whereas isometry is shown to be violated for generic $a$ larger than $a_{\max }$. The anomaly is encoded in an explicit finite-rank operator, whose rank increases to $\infty$ as $a$ goes to $\infty$.