# Hilbert Space Theory for Reflectionless Relativistic Potentials

### Simon N. M. Ruijsenaars

Centre for Mathematics and Computer Science, Amsterdam, Netherlands

## 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 $∞$.

## Cite this article

Simon N. M. Ruijsenaars, Hilbert Space Theory for Reflectionless Relativistic Potentials. Publ. Res. Inst. Math. Sci. 36 (2000), no. 6, pp. 707–753

DOI 10.2977/PRIMS/1195139643