Coupling and particle number intertwiners in the Calogero model

TL;DR

The paper introduces vertical intertwiners $W_n(g)$ for the Calogero model, which connect spectra across different particle numbers at integral coupling $g$, complementing the well-known horizontal intertwiners $M_n(g)$ that shift the coupling. It develops a recursion/factorization framework to construct these vertical intertwiners, proving existence in several cases (notably for $n=2$ up to $g=7$ and for some higher $n$) and showing their compatibility with all Liouville charges via $I_r^{+1}(n,g)=p_{n+1}^r+I_r(n,g)$. A key outcome is the emergence of non-symmetric Liouville charges $oldsymbol{S}_{n+1}^k(g)$ that commute with the Hamiltonian and with each other, providing a richer integrable structure that cannot be replicated solely by symmetric Liouville charges; these charges relate, through symmetric combinations, to the standard Liouville ring and the antisymmetric charge $Q$. The work also connects to the Chalykh–Feigin–Veselov models and suggests a broader network of intertwiners enabling computations and generalizations to other root systems, offering new tools for algebraic integrability and multi-particle quantum systems.

Abstract

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 formulæ 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.