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 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.

💡 Research Summary

The paper investigates a novel two‑dimensional network of intertwining operators in the quantum Calogero–Sutherland model. Traditionally, “horizontal” intertwiners (M_n(g)) have been known: for a fixed particle number (n) they shift the coupling constant (g) by an integer, satisfying (M_n(g) H_n(g)=H_n(g+1) M_n(g)) and similarly for all Liouville charges. These operators are built from antisymmetric products of Dunkl operators and have differential order (n(n-1)/2), independent of (g). Their adjoints implement the inverse shift (g\to g-1), and the combination of forward and backward shifts yields an extra Hermitian charge (Q_n(g)) that exists only for integer (g); this charge is totally antisymmetric under particle exchange and squares to a polynomial in the symmetric charges, thereby extending the Liouville ring (algebraic integrability).

The authors introduce a complementary family, termed “vertical” intertwiners (W_n(g)). For a fixed integer coupling (g) they increase the particle number: (W_n(g) H_{n+1}(g)=H_n(g) W_n(g)). The differential order of (W_n(g)) is (n(g-1)). The construction is motivated by earlier work of Chalykh, Feigin, and Veselov, who exhibited a special case (g=2) where an extra particle with a modified Calogero interaction can be removed by a first‑order intertwiner. The present work generalizes this to arbitrary integer (g) and to higher particle numbers.

Explicit formulas are derived for low‑dimensional cases: (W_2(g)) for (g=2,3,4,5,6,7) and (W_3(g)) for (g=2,3). The authors prove existence up to (g=7) by solving a factorization problem: the vertical intertwiner must satisfy commutation relations with the horizontal ones, e.g. (M_{n+1}(g) W_n(g)=W_n(g+1) M_n(g)). This relation ensures that the horizontal and vertical operators form a lattice (grid) on the integer lattice ((n,g)). Consequently any Hamiltonian (H_n(g)) with integer (n,g) can be reached either by moving horizontally from the free system (H_n(1)) or vertically from the single‑particle free system (H_1(g)), or by any mixed path.

A further by‑product of the vertical intertwiners is the discovery of a new basis of Liouville integrals, denoted (S_k^{(n)}) with (k=1,\dots,n). While the standard integrals (I_r(n,g)) are fully symmetric under the Coxeter group (S_n), the new integrals are invariant only under the subgroup (S_{n-1}) and transform antisymmetrically under the full group. They can be expressed algebraically as polynomials in the symmetric charges and the odd charge (Q_n(g)). The authors provide explicit examples for (n=2,3) and discuss the algebraic relations between the two bases.

The paper concludes that the existence of both horizontal and vertical intertwiners is a hallmark of algebraic integrability at integer coupling. The grid of intertwiners not only offers a constructive method to generate the full spectrum of Calogero models from elementary free systems but also enriches the algebraic structure by introducing non‑symmetric conserved quantities. This dual‑direction intertwining framework opens new avenues for studying multi‑particle quantum integrable systems, their spectral properties, and underlying quantum algebras.