Dynamical symmetries of the Calogero-Coulomb model

Dynamical symmetries of the Calogero–Coulom b mo del Tigran Hak oby an ∗ Y er evan State University, A lex Mano o gian 1, 0025, Y er evan, Armenia and Alihkanyan National Lab or atory, Y er evan Physics Institute, A likhanian Br. 2, 0036, Y er evan, Armenia (Dated: Marc h 24, 2026) W e construct the dynamical symmetry of the quantum Calogero mo del with particle exchange in a confining Coulom b field. This symmetry is gov erned b y the algebra so ( N + 1 , 2), deformed b y exc hange (Dunkl) op erators, with its inv arian t sector generated by the Dunkl angular momentum tensor and the modified Laplace–Runge–Lenz v ector. The equidistan t analogue of the Hamiltonian, with a linear sp ectrum, is expressed in terms of the conformal subalgebra so (1 , 2). In addition, the w av e functions of the Calogero–Coulom b Hamiltonian are classified in to infinite-dimensional lo west-w eigh t so (1 , 2) multiplets. I. INTR ODUCTION Exactly solv able systems play a fundamental role in quantum physics, providing explicit realizations of hidden symmetries and serving as b enc hmarks for nonp erturbative methods. Among these systems, the Calogero mo del, whic h describ es one-dimensional iden tical particles with in verse-square in teractions, is a prominent example [ 1 – 3 ]. More general inv erse-square p oten tials based on finite reflection groups, as well as h yp erb olic, trigonometric, and elliptic functions, spin-exc hange in teractions, and related extensions, are also integrable (see the reviews [ 4 – 7 ] and references therein). Moreov er, the rational Calogero models p ossess a complete set of constants of motion, rendering them maximally sup erin tegrable [ 8 – 10 ]. The (sup er)integrabilit y is robust under the inclusion of certain external p oten tials and p ersists for the angular parts of the asso ciated Hamiltonians, as well as for systems defined on surfaces of constant curv ature [ 11 – 15 ]. On the other hand, the inv erse-square interaction p oten tial complicates the structure of Calogero mo dels but do es not affect the (sup er)in tegrability of the system [ 11 ]. F or instance, the full symmetry of the Calogero mo del — originally formulated with an oscillator confining p otential — can b e regarded as a g deformation of the u ( N ) unitary algebra, which enco des the in v ariants of the N -dimensional isotropic oscillator [ 14 , 16 ]; see also Refs. [ 17 , 18 ] for a quan tum-group extension. Here, the parameter g sp ecifies the coupling constan t of the in verse-square in teraction. This corresp ondence extends to the dynamical symmetry , which encompasses also ladder op erators that raise or lo wer states along the energy scale [ 19 ]. It is well known that the N -dimensional confining Coulom b mo del is also sup erin tegrable characterized by the symmetry group S O ( N + 1), generated b y the angular momen tum and the Laplace–Runge–Lenz v ector. The spectrum of the quantum system, how ever, is not equidistan t. At first glance, this appears to preclude the existence of spectrum- generating op erators. Nevertheless, a mo dified system can b e constructed that shares the same stationary states while exhibiting a linear sp ectrum. The equidistant Hamiltonian p ossesses a dynamical symmetry describ ed by the conformal group S O ( N + 1 , 2) in ( N + 1)-dimensional Minko wski space. This structure was first demonstrated in the case of the h ydrogen atom [ 20 , 21 ]. The energy sp ectrum of the Coulomb mo del with a rational Calogero p oten tial, is known to p ossess a huge degen- eracy [ 22 ]. At the same time, the Calogero–Coulomb system, including its extensions to constant-curv ature spaces, is superintegrable [ 11 ]. The corresponding constan ts of motion, including an analogue of the Laplace–Runge–Lenz v ector, hav e b een constructed [ 12 , 23 , 24 ]. Moreov er, the complete set of conserved quantities of the (confining) quan tum Calogero–Coulom b problem forms a g -deformation of the so ( N + 1) algebra [ 15 ]. In this article, we construct and analyze the dynamical symmetries of the quantum Calogero–Coulom b system. F ollo wing many of the earlier developmen ts in Calogero-type mo dels, our approach emplo ys the exc hange (Dunkl) op- erator formalism, which incorporates particle-exchange effects with coupling constan t g into co v ariant differen tial [ 25 ]. This formulation provides a p o w erful and algebraically transparent framework for systematically deriving solutions, in tegrals of motion, symmetry algebras, and ladder op erators. W e demonstrate that the dynamical symmetry (sp ectrum-generating symmetry) of the quan tum Calogero–Coulomb system is described by an exchange-operator deformation of the conformal algebra so ( N + 1 , 2). Within this structure, a three-dimensional conformal subalgebra so (1 , 2) emerges, which con tains the modified Calogero–Coulom b Hamilto- nian with linear sp ectrum. F or the mo dified system, w e construct a deformed Laplace–Runge–Lenz vector, distinct ∗ tigran.hakob yan@ysu.am ; hakob@y erphi.am 2 from the analogous vector obtained for the original mo del [ 12 ]. T ogether with the Dunkl angular momentum, this v ector generates a complete symmetry algebra of the equidistant Calogero–Coulom b mo del. This algebra represents an exchange-operator deformation of the so ( N + 1) algebra, with its Casimir element expressed in terms of the Hamiltonian, and it retains a structure closely related to that of the original system [ 15 ]. The commutation relations among the dynamical symmetry generators are derived. They incorp orate particle- exc hange op erators directly into the structure constants and are expressed compactly in matrix form. F urthermore, w e establish the equiv alence b et w een the quadratic Casimir elements of the so (1 , 2) and Dunkl angular momentum algebras, and we calculate the numerical v alue of the Casimir element of the dynamical symmetry algebra. Finally , we construct the w a ve functions of the Calogero–Coulom b Hamiltonian, both with and without particle exc hange. These wa ve functions are classified according to the infinite-dimensional low est-weigh t irreducible represen- tation of the so (1 , 2) conformal algebra. The conformal spin of the m ultiplet is parameterized by the orbital quantum n umber, while the states are lab eled b y the radial quan tum n um b er. The pap er is organized as follows. In Sect. I I , we briefly review the Calogero–Coulom b mo del and its symmetries within the Dunkl-operator formalism. In Sect. I I I , we construct and analyze the so (1 , 2) conformal subalgebra together with an equidistant analogue of the Calogero–Coulomb Hamiltonian. In Sect. IV , w e develop a Dunkl-deformed analogue of the full conformal algebra so ( N + 1 , 2), whose rotated realization describ es the dynamical symmetry of the Calogero–Coulomb Hamiltonian. In Sect. V , using deformed spherical harmonics, we construct the eigenfunctions of the Calogero–Coulom b system and study their so (1 , 2) m ultiplet structure. The results are summarized and discussed in Sect. VI . App endix A presents the deriv ation of the commutation relations among the generators of the dynamical symmetry algebra, while App endix B contains explicit calculations of the squares of three vectors, including the Laplace–Runge–Lenz v ector, that enter the algebra. I I. CALOGER O–COULOMB SYSTEM In this section, we briefly review the exchange-operator formalism and its application to the symmetries of the Calogero–Coulom b model with particle exchange. A. Hamiltonian via exchange (Dunkl) op erators The Hamiltonian of the N -particle quantum Calogero mo del in external Coulomb p oten tial [ 22 ] has the following form: ˜ H γ = N X i =1 p 2 i 2 m + X i 0, and set the mass and Plank’s constan t to unity , m = ℏ = 1. An elegant approach to quantum Calogero-type models is based on the Dunkl operators, whose comp onents mutually comm ute [ 25 ]; see Refs. [ 26 – 29 ] for reviews: ∇ i = ∂ i − X j  = i g x i − x j s ij , (2) [ ∇ i , ∇ j ] = 0 , [ ∇ i , x j ] = S ij := ( − g s ij i  = j, 1 + g P k  = i s ik i = j. (3) In the ab o v e equations, the op erator s ij p erm utes the i -th and j -th particles. The Dunkl op erator can b e view ed as a cov ariant deriv ativ e with a flat connection and a nonlo cal imaginary field with particle exchanges. Its comm utation relations with the co ordinates also in volv e exchange op erators and reduce to the canonical commutation relations in the free-coupling limit, lim g → 0 S ij = δ ij . The algebra ( 3 ) is often referred to in the literature as the S N -extended Heisenberg algebra [ 30 ]. The corresp onding abstract algebraic structure is known as the rational Cherednik algebra asso ciated with the symmetric group [ 31 – 33 ]. The Dunkl op erator together with the co ordinate generate a particular representation of this algebra. 3 Pursuing further the analogy with particle motion in an external field, one can define a co v arian t momentum op erator π . The corresp onding Hamiltonian includes a Calogero-type p otential that, how ever, inv olves particle exchanges [ 34 , 35 ]. In the presence of a Coulomb p oten tial, this leads to the Calogero–Coulomb Hamiltonian with particle exc hange [ 11 , 12 ]: H γ = π 2 2 − γ r = p 2 2 + X i