Baxter $Q$-operator for the hyperbolic Calogero--Moser system

We introduce a $Q$-operator ${\mathcal{Q}}_z$ for the hyperbolic Calogero–Moser system as a one-parameter family of explicit integral operators. We establish the standard properties of a $Q$-operator, i.e.~invariance of Hamiltonians, commutativity for different parameter values and that its eigenvalues satisfy an explicitly given first order ordinary difference equation in the parameter $z$.