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/ℏ, 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 ∞ as a goes to ∞.