Completing the theory of vertical (particle-number-changing) intertwiners in the rational Calogero model

The rational quantum Calogero model at integer coupling admits 'horizontal' intertwining operators, which change the coupling constant at fixed particle number, and — as constructed by Correa, Inzunza and Lechtenfeld (JHEP 2025) — 'vertical' intertwiners $W_n(g)$ of differential order $n(g-1)$, which add one interacting particle at fixed coupling, generating a two-dimensional grid of intertwiners and a new non-symmetric basis of Liouville integrals. The source formulates a cluster of open problems around these operators. The existence of $W_n(g)$ for arbitrary particle number and integer coupling follows from the Berest–Chalykh theory of deformed Calogero–Moser operators and generalized locus configurations (Commun. Math. Phys. 400 (2023) 133); the open core is the constructive existence problem recorded in the source's Section 3.3: an explicit construction of $W_n(g)$ for arbitrary particle number $n$ and integer coupling $g\ge 2$, for which the paper proposes the factorization recursion $M_{n+1}(g)\,W_n(g)=W_n(g{+}1)\,M_n(g)$. Section 5 of the source additionally raises further questions around these operators: the relation of a constructive $W_n(g)$ to the Cherednik-algebraic shift operators; the kernel and image structure of $W_n(g)$ on finite-dimensional eigenspaces under harmonic confinement; the behavior of the non-involutive (conformal, $\mathrm{sl}(2,\mathbb{R})$-related) conserved charges under vertical intertwining; the precise overlap between the standard symmetric Liouville ring and the new non-symmetric charge ring; and vertical intertwiners beyond the rational $A$-type case, where abstract existence for other nested rational Coxeter pairs follows in principle from the Berest–Chalykh theory, while explicit constructions, and the trigonometric, hyperbolic and elliptic potentials, angular reductions and non-integer coupling, remain open.

Background

The Calogero model (Calogero, J. Math. Phys. 12 (1971) 419) is the paradigmatic exactly-solvable quantum many-body system: $n$ particles on a line with inversely quadratic pair interactions, governed (in units $\hbar=1$) by

$$H_n(g)=\sum_{i=1}^{n}p_i^{2}+\sum_{i<j}\frac{2g(g-1)}{(x_i-x_j)^{2}},$$

where $g$ is the coupling constant. The model is Liouville integrable: it possesses $n$ mutually commuting conserved charges $I_r$ polynomially generated from the free power sums in the momenta. Separating the free center-of-mass motion leaves reduced charges $\bar I_r$ ($r=2,\dots,n$) built only from coordinate and momentum differences. For integer $g$ the model is moreover algebraically integrable: there is an additional conserved charge $Q$, antisymmetric under particle exchange, of differential order $\frac{1}{2}n(n-1)(2g-1)$.

A central structural tool is intertwining operators: nonzero differential operators $A$ with $A\,H=H'\,A$, which map the spectral problem of one Hamiltonian to another. The horizontal intertwiners $M_n(g)$ — of differential order $\frac{1}{2}n(n-1)$, constructed as a Vandermonde-type antisymmetrized product of Dunkl operators $\pi_i(g)$ (differential-difference operators combining derivatives with coupling-weighted transpositions of the particle coordinates) — satisfy $M_n(g)H_n(g)=H_n(g{+}1)M_n(g)$, extend to all Liouville charges, and have adjoints reversing the coupling step; their composites $M_n(g)^\dagger M_n(g)$ are polynomials $R_n(\bar I)$ in the reduced charges with coupling-independent coefficients. These are the physics counterparts of the shift operators for Calogero–Moser systems long established in the mathematics literature via Dunkl operators (see Chalykh's survey, arXiv:2409.09005). Horizontal intertwiners change the coupling $g\to g{+}1$ at fixed particle number $n$.

Correa, Inzunza and Lechtenfeld (arXiv:2504.01177; JHEP 07 (2025) 128) constructed the complementary vertical intertwiners, which change the particle number at fixed integer coupling. Their starting point is a one-parameter family of integrable Hamiltonians $\widetilde H_{n+1}(g)$ due to Chalykh, Feigin and Veselov (J. Math. Phys. 39 (1998) 695), in which the $(n{+}1)$st particle couples to the others through shifted inverse-square terms; at $g=2$ this degenerates to the standard $(n{+}1)$-particle Calogero Hamiltonian, so the known intertwiner $\widetilde W_n(2)$ adds one interacting particle. CIL define, for general integer $g$, vertical intertwiners $W_n(g)$ of differential order $n(g-1)$ through

$$W_n(g)\,H_n^{+1}(g)=H_{n+1}(g)\,W_n(g),\qquad W_n(g)^\dagger\,H_{n+1}(g)=H_n^{+1}(g)\,W_n(g)^\dagger,$$

where $H_n^{+1}(g)$ is the $(n{+}1)$-particle Hamiltonian in which only the first $n$ particles interact (interaction graph $A_{n-1}$ plus a disconnected node), and $H_{n+1}(g)$ is the fully interacting one (graph $A_n$). They prove existence only for small cases — $W_2(g)$ up to $g=7$ (explicitly constructed to $g=5$) and $W_3(3)$, with $W_3(2)$ known previously — state the general case as a conjecture, and propose a recursion: $W_n(g{+}1)$ should follow from the factorization $M_{n+1}(g)\,W_n(g)=W_n(g{+}1)\,M_n(g)$, i.e. by splitting the right factor $M_n(g)$ off the product $M_{n+1}(g)W_n(g)$. Horizontal and vertical intertwiners organize into a grid in which any $H_n(g)$ at integer coupling is reached from free particles by iterated intertwining. As a byproduct, the composites $\mathcal S_{n+1}^{k}(g)=W_n^{k}(g)W_n^{k}(g)^\dagger$ ($k=1,\dots,n{+}1$, one for each choice of the particle added last) form a non-symmetric basis of Liouville integrals: $n{+}1$ mutually commuting charges, all of the same differential order $2n(g-1)$, commuting also with the standard reduced charges and with $Q$, yet not expressible as polynomials in them.

The mathematical backdrop is the theory of deformed Calogero–Moser operators of Berest and Chalykh (Commun. Math. Phys. 400 (2023) 133). A generalized locus configuration of type $W$ is a finite hyperplane arrangement $\mathcal A$ in $V=\mathbb C^{N}$ containing the root hyperplanes of a finite Coxeter group $W$ (a finite group generated by reflections), such that the hyperplanes in $\mathcal A\setminus R$ carry non-negative integer multiplicities $k_\alpha$ and the potential $u_{\mathcal A}=\sum_{\alpha\in\mathcal A_+}k_\alpha(k_\alpha{+}1)(\alpha,\alpha)/(\alpha,x)^2$ obeys a divisibility ('locus') condition at those extra hyperplanes. Associated to it are the deformed operator $L_{\mathcal A}=\Delta-u_{\mathcal A}$ and the standard operator $L_W$ for the root-system part. Their Theorem 3.3 proves that $L_{\mathcal A}$ is completely integrable and that a nonzero differential shift operator $S$ with $L_{\mathcal A}S=SL_W$ exists; $S$ is constructed in the spherical subalgebra (the $W$-invariant part) of the rational Cherednik algebra (the algebra extending the group algebra of $W$ by the coordinates and Dunkl operators, deforming the differential operators on the quotient $V/W$), and has finite order $N$. Relatedly, the quasi-invariants — polynomials whose difference under reflection in each extra hyperplane is divisible by a high power of the defining linear form — form a ring $Q_{\mathcal A}$ realized as a maximal commutative algebra of deformed charges. Theorem 8.1 extends this to harmonically confined operators, with spectral shift $2\omega N$, and shows that $S^\omega(S^\omega)^*$ commutes with the confined deformed Hamiltonian. For the nested arrangement relevant here — the full $A_n$ arrangement in $\mathbb C^{n+1}$ viewed as a configuration of type $S_n$ with uniform multiplicity $g-1$ — one has $L_W=H_n^{+1}(g)$, $L_{\mathcal A}=H_{n+1}(g)$ and $N=n(g-1)$; as recorded in the Note added to the CIL paper (communicated by O. Chalykh), the existence of the vertical intertwiners is a special case of this theorem. Existence is thus settled; what is not settled is everything the horizontal theory possesses beyond existence: the constructive factorization scheme, the finite-dimensional eigenspace analysis, the fate of the conformal (non-involutive) charges, and the precise algebraic relation between the symmetric and non-symmetric charge rings.

Problem Statement

Establish a constructive existence proof for the vertical (particle-number-changing) intertwiners of the rational Calogero model, taking their existence for arbitrary particle number and integer coupling as given by the Berest–Chalykh shift-operator theorem. The objects are the differential operators $W_n(g)$ of order $n(g-1)$ defined by

$$W_n(g)\,H_n^{+1}(g)=H_{n+1}(g)\,W_n(g),\qquad W_n(g)^{\dagger}\,H_{n+1}(g)=H_n^{+1}(g)\,W_n(g)^{\dagger},$$

where $H_n^{+1}(g)$ is the $(n{+}1)$-particle Hamiltonian with only the first $n$ particles interacting and $H_{n+1}(g)$ the fully interacting one, at the same integer coupling $g\ge 2$ (the case $g=1$ is trivial, $W_n(1)=1$; non-integer or negative coupling is addressed only under item (5) below). The core objective is:

(1) Construction. Give a constructive algorithm that produces $W_n(g)$ explicitly as a partial differential operator with rational coefficients for arbitrary $n$ and integer $g\ge 2$. The source's proposed — but not mandatory — route is the Correa–Inzunza–Lechtenfeld recursion $M_{n+1}(g)\,W_n(g)=W_n(g{+}1)\,M_n(g)$, obtaining $W_n(g{+}1)$ by splitting the right factor $M_n(g)$ off the product $M_{n+1}(g)W_n(g)$; any constructive method producing $W_n(g)$ explicitly for arbitrary $n$ and integer $g\ge 2$ addresses the problem. As an optional clarification, one may also determine in what sense (proportionality, normalization, polynomial prefactors) the resulting operators coincide with the Berest–Chalykh shift operators for the nested $A_{n-1}\subset A_n$ arrangement.

Beyond this core constructive problem, the source (its Section 5) raises the following further questions. They are the source's own suggested directions for the wider program, not conjunctive completion requirements; results on any of them constitute progress:

(2) Eigenspace structure. Under a confining harmonic potential, where the joint eigenspaces of the symmetric Liouville charges are finite-dimensional, determine the action of $W_n(g)$ and its adjoint on these eigenspaces: the dimensions, bases and physical character of the kernels and images, and the resulting isospectrality pattern — in parallel with the known analysis for the horizontal intertwiners. (The spectral shift $2\omega N$ with $N=n(g-1)$ under harmonic confinement is already established by Berest–Chalykh's Theorem 8.1.)

(3) Conformal charges. Determine the behavior of the non-involutive conserved charges — conserved quantities that are not mutually commuting, tied to the conformal invariance of the rational Calogero model and generating its $\mathrm{sl}(2,\mathbb{R})$ conformal algebra (the dynamical symmetry algebra spanned by the Hamiltonian, the dilatation generator and the special-conformal generator) inside the full non-commutative algebra of conserved quantities — under vertical intertwining, clarifying the role of the conformal algebra in the vertical sector.

(4) Charge rings. Give a precise algebraic characterization of the overlap subring between the totally symmetric reduced-Liouville ring $\{\bar I_2,\dots,\bar I_{n+1}\}$ (together with the antisymmetric charge $Q$ at integer coupling) and the non-symmetric ring generated by the commuting charges $\mathcal S_{n+1}^{k}(g)=W_n^{k}(g)W_n^{k}(g)^{\dagger}$. Here the permutation variants $W_n^{k}(g)$ ($k=1,\dots,n{+}1$) are obtained from $W_n(g)$ by conjugation with the permutation that makes particle $k$ the one whose interaction is turned on last, followed by the symmetric restriction $\mathrm{res}[\cdot]$ of the source (its eq. (52), e.g. $W_2^{3}(2)\equiv W_2(2)$, $W_2^{2}(2)=\mathrm{res}\,s_{23}W_2(2)$, $W_2^{1}(2)=\mathrm{res}\,s_{13}W_2(2)$). In particular: which polynomials in the elementary symmetric charges can be recovered from the $\mathcal S$-basis, and what the minimal polynomial 'dressings' are that produce $Q$ from antisymmetric combinations.

(5) Scope of the construction. Determine for which generalizations vertical intertwiners exist and can be constructed. For rational Calogero models based on other Coxeter reflection groups, abstract existence follows in principle from the Berest–Chalykh theory of Coxeter configurations for nested pairs of finite Coxeter groups, so the open part there is the explicit construction; open in addition are trigonometric, hyperbolic or elliptic pair potentials; angular (radial) reductions; a direct construction via Dunkl operators or the Dunkl-deformed symplectic algebra; and whether the framework admits any analytic or representation-theoretic continuation to non-integer or negative coupling, where the differential order $n(g-1)$ fails to be a non-negative integer.

References

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