Lacunae of Hyperbolic Riesz Kernels and Commutative Rings of Partial Differential Operators

The purpose of this Letter is to demonstrate a close connection between the problem of describing supercomplete commutative rings of partial differential operators (in the sense of Krichever–Veselov) and the theory of lacunae for hyperbolic Riesz kernels. As an application, we give a simple and explicit construction of additional quantum integrals for the generalized Calogero–Moser problem associated with a finite reflection group W and W-invariant integer root multiplicities mα. These quantum integrals are not W-invariant and cannot be constructed directly using the standard technique of Dunkl operators.