JournalsprimsVol. 36 , No. 6DOI 10.2977/prims/1195139643

Hilbert Space Theory for Reflectionless Relativistic Potentials

  • Simon N. M. Ruijsenaars

    Centre for Mathematics and Computer Science, Amsterdam, Netherlands
Hilbert Space Theory for Reflectionless Relativistic Potentials cover

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 ∞.