Coupling and particle number intertwiners in the Calogero model

It is long known that quantum Calogero models feature intertwining operators, which increase or decrease the coupling constant by an integer amount, for any fixed number of particles. We name these as ``horizontal'' and construct new ``vertical'' intertwiners, which \emph{change the number of interacting particles} for a fixed but integer value of the coupling constant. The emerging structure of a grid of intertwiners exists only in the algebraically integrable situation (integer coupling) and allows one to obtain each Liouville charge from the free power sum in the particle momenta by iterated intertwining either horizontally or vertically. We present recursion formul\ae\ for the intertwiners as a factorization problem for partial differential operators and prove their existence for small values of particle number and coupling. As a byproduct, a new basis of non-symmetric Liouville integrals appears, algebraically related to the standard symmetric one.