The tetrahexahedric angular Calogero model

The spherical reduction of the rational Calogero model (of type An−1 and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.


Introduction and summary
The Calogero (or Calogero-Moser) model [1][2][3] 1 is the paradigmatical n-particle integrable system in one space dimension. Originally defined for the root system of A 1 ⊕ A n−1 , the Calogero model was quickly generalized for any finite Coxeter group of rank n [4]. In particular the rational version is remarkable for its conformal properties and its maximal superintegrability [8,9] (see also [10]). Moreover, when the coupling constant g is integral, the quantum model enjoys additional and algebraically independent conserved quantities, which make it 'analytically integrable' [11,12]. In this case, intertwining operators relate the energy spectrum with that of the free theory (at g=0 or g=1). This structure was worked out in the 90s by a Russian group [11,12] using Darboux dressing and by Dutch mathematicians [13][14][15] employing Dunkl operators [16,17].
Maximal superintegrability for a quantum system of phase-space dimension 2n has been characterized in the literature; for a recent account see [18] where it is termed 'maximal degenerate integrability'. The Hamiltonian H is part of a rank-n abelian algebra of Liouville charges (commuting Hamiltonians), but this is embedded in a larger commutant of H, which is spanned by the Liouville charges plus n−1 additional integrals of motion (Wojciechowski charges). The latter do not commute with one another but form a polynomial algebra [9] together with the Liouville charges. Denoting the commutant generators by C and distinguishing between Liouville charges C ′ and Wojciechowski charges C ′′ so that {C} = {C ′ , C ′′ }, one has (see, e.g. [19])

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where A and B are polynomial in C ′ and linear in C ′′ , and we have suppressed the counting labels. This implies that each H eigenspace splits into simultaneous eigenspaces of the other n−1 Liouville charges. Hence, we can find an H eigenbasis labelled by n quantum numbers (ℓ α ) with α = 1, . . . , n. Furthermore, these simultaneous eigenspaces carry a representation of the algebra (1.1), which governs their degeneracies.
Since the rational Calogero Hamiltonian is part of an SL(2,R) conformal algebra, it is natural to reduce the rational model to an integrable system on a sphere. Restricting ourselves to the customary A 1 ⊕ A n−1 model, one may employ the translational invariance and firstly separate the A 1 part representing the center-of-mass motion. The relative-motion Hamiltonian may be interpreted as ruling a single particle in R n−1 , subject to a potential which is singular at the reflection hyperplanes and decays like r −2 with distance r from the origin. Secondly, this radial scaling suggests a further codimension-one reduction to S n−2 , which defines the 'angular (relative) rational Calogero model'. 2 This system describes a particle moving on S n−2 in a collection of 1 2 n(n−1) Higgs-oscillator potentials [20,21], each one centered at a positive root of A n−1 (and its antipode) and blowing up at the intersection of the reflection hyperplane with the unit sphere. These singular loci tessellate the (n−1)-sphere into n! hyperspherical simplices in which the particle is trapped.
A number of classical features of angular rational Calogero model (center-of-massreduced or not) have been recently analyzed by an Armenian group [22][23][24][25][26][27]. Intertwining relations for the quantum model were investigated by M. Feigin already in [28], but more recently the energy spectra and eigenstates have been constructed [29], and the algebra of angular Dunkl operators has been studied [30]. These partial results leave open the following questions: • What is a complete set of algebraically independent conserved charges?
• Can one construct a subset of Liouville charges (in mutual involution)?
• What is a complete set of algebraically independent Hamiltonian intertwiners?
• Can one find additional conserved quantities which get intertwined?
• What kind of algebra is generated by the conserved charges and intertwiners?
• Are these models maximally quantum super-or even analytically integrable?
The answers are not easily obtained for the angular Calogero models because they are linked to nontrivial mathematical problems. For instance, the commutant of H is the ring of all Weyl-invariant polynomials in the angular Dunkl operators (for a definition see the following section), and it is not obvious how to construct a minimal set of algebraically independent generators. Moreover, to establish the superintegrability one should find the maximal abelian subalgebras (i.e. the Liouville charges) in this ring.
Lacking a general strategy to attack these issues, we take an explicit look at the simplest examples, n=3 and n=4. The first one, leading to a rank-one angular system, is JHEP10(2015)191 quite trivial but well known, while the second one is still tractable and serves to illustrate most of the general concepts. Therefore, in this paper, we exemplarily analyze the ranktwo quantum system which is obtained from the four-particle rational Calogero model, via translational reduction to R 3 and then radial reduction to S 2 . It describes a particle on the two-sphere trapped in one of 24 right isosceles spherical triangles which make up the spherical projection of a so-called tetrahexahedron.
After presenting general (and mostly previous) results on the A n−1 angular models in section 2, the essentially trivial n=3 case, known as the Pöschl-Teller model, is reworked in section 3 as a warmup example. section 4 introduces the n=4 model and recapitulates its energy spectrum and eigenstates [29], before our results concerning conserved charges and intertwiners are collected in section 5. There, we establish that any conserved charge of the tetrahexahedric angular Calogero model is built from two independent operators, say J 4 and J 6 (besides the Hamiltonian), and that this system features two independent Hamiltonian intertwiners, which we denote by M 3 and M 6 . We provide explicit expressions for both and work out their (non-standard) intertwining relations with the elementary conserved charges. Since one cannot produce two commuting words with J 4 and J 6 , the model enjoys only two charges in involution but, given three independent conserved quantities, it is maximally superintegrable. For integer value of the coupling, an additional independent conserved charge introduces a Z 2 grading (in the spirit of 'nonlinear supersymmetry' without fermionlike degrees of freedom [31,32]) and renders the model analytically integrable. Various interesting algebraic relations are collected in the appendix.

The angular (relative) rational Calogero model
We parametrize the n-particle quantum phase space with coordinates x µ and momenta p ν , subject to the canonical commutation relations (setting = 1) [ x µ , p ν ] = i δ µ ν with µ, ν = 1, . . . , n . (2.1) The center-of-mass coordinate and momentum, decouple in the rational Calogero Hamiltonian and will be ignored in our starting Hamiltonian which describes the relative motion of (A n−1 ) Calogero particles on the real line, or of a single quantum particle in R n−1 subject to a potential singular at the A n−1 Weyl chamber

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walls. The Hamiltonian is invariant under the reflection g → 1−g of the real coupling g, but higher conserved charges and intertwiners will not share this property. Since this Hamiltonian is scale covariant, it pays to introduce on R n−1 polar coordinates (r, θ) with a (relative) radius r and n−2 angles θ together with their momenta p r and L θ , (2.5) The (relative) angular momentum squared L 2 is best expressed by parametrizing the relative particle motion in R n−1 with coordinates y i where i = 1, . . . , n−1, Jacobi coordinates are a suitable choice for {y i }. Together with a dilatation generator D and a special conformal generator K, the Hamiltonian H forms an SL(2,R) commutator algebra, as is easily seen in the polar representation Here, the angular (relative) Hamiltonian H Ω is independent of the radial coordinate or momentum, and it coincides (up to a constant shift) with the Casimir operator of the conformal SL(2,R), This defines our angular Calogero system (for A n−1 ), which describes a quantum particle on the sphere S n−2 under the influence of the potential where R + is the set of positive roots for A n−1 and θ α denotes the angle (geodetic arc length on S n−2 ) between the point θ and the intersection of the ray in direction α with the unit sphere. Each of the 1 2 n(n−1) contributions of cos −2 form is also known as a 'Higgs oscillator' [20,21]. This potential is a very special one as it is tied to the root system. Its singular walls are of codimension one and tesselate the sphere into n! hyperspherical simplices.
For the spectral problem, let us pass to the position representation and encode momenta via differental operators, H Ω → r 2 H + 1 2 (r∂ r + n−3) r∂ r . (2.11)

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It is customary to remove the first-order radial derivative term via a similarity transformation, Ψ(r, θ) = r − n−2 2 u(r, θ) and so that the Hamiltonian on R n−1 acts as follows, As the potential is repulsive, the spectrum is positive semi-definite and continuous, where the wave function depends on further (suppressed) quantum numbers, but q parametrizes the angular Hamiltonian eigenvalues [29] (see also the appendices of [2]), For vanishing potential (g=0 or 1), H Ω = 1 2 L 2 , and q = ℓ ∈ N 0 is the familiar total angular momentum for a free particle on S n−2 , with a degeneracy of ℓ+n−2 n−2 − ℓ+n−4 n−2 . The interaction greatly reduces the degeneracy and shifts this quantum number [29], where the decomposition ℓ → (ℓ 3 , ℓ 4 , . . . , ℓ n ) into additional quantum numbers yields the correct degeneracy, with p n (ℓ) denoting the number of partitions of ℓ into integers not larger than n or, equivalently, the number of partitions of ℓ into at most n integers. Its generating function reads 18) and the large-ℓ asymptotics is p n (ℓ) = ℓ n−1 n! (n−1)! + O(ℓ n−2 ), implying deg n (ε q ) ∼ ℓ n−2 for ℓ → ∞. Note that the degeneracy does not depend on g but only on ℓ (for any fixed n). Up to n=5, fairly explicit formulae may be found [33]:

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The wave functions may also be given explicitly. From revealing a radial wave function of Bessel type. The angular wave function v q ( θ) depends on all additional quantum numbers (ℓ 3 , ℓ 4 , . . . , ℓ n ) and is a bit harder to find [29]. It can be constructed in the following way, is the Vandermonde of the original R n coordinates and D ν denotes the so-called Dunkl operator [16,17], with s νρ permuting the position and momenta of the νth with those of the ρth particle. The Dunkl operators play a central role in asserting the integrability of the Calogero model, because deforming ∂ ν → D ν essentially creates the interacting system from the free one. Their Newton sums yield the Liouville charges of the Calogero system, and their mutual commutativity guarantees that of the charges. It is convenient to switch again to the relative coordinates {y i }, in which the Dunkldeformed momenta and angular momenta take the form (up to a factor of −i) respectively, where s α is the Weyl reflection on the hyperplane orthogonal to the root α. Polynomials in the Dunkl operators are not just differential operators but also act by Weyl reflections. Their restriction to functions totally symmetric under the Weyl group is called 'residue' (denoted by 'res') and produces a purely differential operator. In this way, the Hamiltonian can be obtained from the homogeneous symmetric polynomial of degree two, Analogously, the symmetric quadratic polynomial in the angular Dunkl operators L ij shifted by a certain pure-reflection term, We remark that the residue of the pure-reflection term simply produces the ground-state energy ε q (ℓ=0). Clearly, H and H Ω may be considered as 'Dunkl deformations' of (− 1 2 times) the Laplacian on R n−1 and S n−2 , respectively. To express the angular wave function v q ( θ) in terms of the relative coordinates, we must rewrite ∆ in terms of {y i } and the Newton sums of the D ν in (2.22) in terms of the D i , which depends on our choice for {y i }. Phrased differently, we need to identify the totally Weyl-symmetric homogeneous polynomials σ µ {D i } of orders µ = 3, 4, . . . , n, to go inside the large brackets of (2 .22), From the form of (2.29) one can infer that v q ( θ) is a ratio of a degree-q homogeneous polynomial h q (y) in {y i } to the qth power of the radial coordinate r and hence may be viewed as a Dunkl-deformed harmonic polynomial on R n−1 . Another property visible from (2.29) is that h q (y) contains a factor of ∆ g , which may be split off to define h (g) which then is in the kernel of the Hamiltonian 'in the potential-free frame', (2.32) The potential-free Dunkl-deformed harmonic polynomial h (g) ℓ is homogeneous of degree ℓ only, for any value of g. In particular, for the ground state one has and hence the full ground-state wave function is totally symmetric (antisymmetric) under particle permutations or Weyl reflections for even (odd) integer values of g. Since all other ingredients besides ∆ g in (2.22) are completely symmetric, this symmetry property of the integer-g ground state extends to all excited states above it. Furthermore, the reflection symmetry g+1 ↔ −g is broken since one tower of states is Weyl symmetric while the other one is antisymmetric.

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It is well known [13][14][15] that there exists an intertwining operator M = M (g) which establishes an isospectrality of H (g) and H (g+1) . This differential operators of order 1 2 n(n−1) has the simple form and using the Weyl antisymmetry of M it is straightforward to verify that (see e.g. [28]) simultaneously acts on the radial wave function √ rJ q+(n−3)/2 and on the angular wave ℓ . Up to n=4 the expressions for M (g) were worked out explicitly in [19]. However, the angular system H Ω comes with its own intertwiners M s = M (g) s of some order s ∈ N, which then provides additional intertwiners for H. This is suggested by the form (2.16) of the quantum number q appearing in (2.15), which reveals a partial 4 isospectrality for H Ω , Indeed, as shown in [28], one can more or less copy the strategy from R n−1 to S n−2 and obtain the lowest-order angular intertwiner, with s = 1 2 (n−1)(n−2), which obeys (2.39). By the same token, any Weyl-invariant function of {L ij } will yield, via its residue, a conserved angular quantity C t of some order t. We remark that the angular Dunkl operators do not commute with each other but form a subalgebra of a rational Cherednik algebra [30]. It is a deformation of the so(n−1) Lie algebra generated by {L ij }. The general problem of identifying all Liouville charges for the angular system reduces to constructing a maximal abelian subalgebra

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in the algebra of Weyl-symmetric polynomials in {L ij }, which is not an easy task. Likewise, identifying the minimal independent set of Weyl-antiinvariants and hence intertwiners is nontrivial. Now, since any such M s and C t is scale invariant, i.e. r independent, it does not touch the radial wave function, and hence it is also true that (2.40) To summarize, we have the connection In general, the angular conserved charges C t will not be in involution with one another, but some combinations may be. The angular intertwiners M s are differential operators of order s and have, in contrast to M , a sizeable kernel, with dim ker n (ℓ) = deg n (ℓ) − deg n (ℓ − 1 2 n(n−1)) .  which provide particular cases of some C t for t=2s. In the full Calogero model, R = M † M turned out to be a specific polynomial in the Liouville charges, which was evaluated by infinitely separating the particles thus making contact with the free case. This move is not possible for R 2s due to the compactness of S n−2 . Neither is it true that R , and so these charges intertwine as follows,

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and similarly in the opposite direction. Here, ρ 2s is a particular expression in other conserved charges. Quite generally, we cannot expect conserved charges other than the angular Hamiltonian to be invariant under g → 1−g, and in fact one finds that (2.47) Therefore, higher conserved quantities C t intertwine in a more complicated fashion, where Γ s ′ t ′ st (g) are certain polynomials in g of order s+t−s ′ −t ′ −1 at the most. The sum runs over all linearly independent conserved quantities including When the coupling g is an integer, repreated intertwining relates all quantities with their analogs in the free theory (g=0 or 1), which allows one to generate analytic expressions for all wave functions. Moreover, in this case additional conserved charges make the model 'analytically integrable' and produce a 'supercomplete' graded ring of commuting differential operators [11,12,19].

Warmup: the hexagonal or Pöschl-Teller model
Let us work out and test the general assertions of the previous section on the simplest nontrivial case, which occurs for n = 3. The three-particle (or A 2 ) rational Calogero model is long known to be completely separable, the ensuing angular (relative) model being the famed Pöschl-Teller model of a particle on a circle with a cos −2 (3φ) potential.
We start by introducing Jacobi coordinates (y 1 , y 2 ) for the relative motion (and the center of mass X), and then polar (r, φ) and complex (w,w) coordinates on R 2 , y 1 = r cos φ and y 2 = r sin φ =⇒ w := y 1 + iy 2 = r e iφ .

(3.2)
After the standard similarity transformation, the (reduced) Hamiltonian

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and its angular cousin take the following form as differential operators on R 2 and S 1 , respectively, where the angular potential receives contributions from the three positive roots of A 2 , The poles separate the circle into six disjoint arcs, pertaining to the hexagonal symmetry of the A 2 root system. We display the angular Hamiltonian in complex coordinates as well, Specializing (2.15)-(2.17) to n=3, one finds that the angular momentum ℓ = 3ℓ 3 is a multiple of three and The √ r factor in (2.21) gets cancelled in the similarity transformation (2.12), giving

8) and the angular wave function reads
where we have rewritten in complex coordinates. 5 Owing to the S 3 Weyl group of A 2 , we take profit from the complex cubic roots of unity to cast the holomorphic Dunkl operator D w = 1 2 (D 1 − iD 2 ) into the simple form where the three basic Weyl reflections act as follows in the complex plane, 12) or on the polar angle,

16) and in the polar representation the intertwiner becomes
where the angular intertwiner M Ω (to be given shortly) has been used.
Reducing to the angular subsystem, i.e. the Pöschl-Teller model, the formula (2.38) yields (3.18) which simplifies to (3.19) or to Taking the residue, we arrive at x 12 x 23 x 31 (3.21) or, in complex and polar parametrizations, . (3.22) 6 We remark that D 3 w + D 3 w changes sign under any Weyl reflection while D 3 w − D 3 w is totally symmetric.

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The penultimate expression is the familiar first-order operator which enjoys the following remarkable properties: • it adds zeros at φ = 0, 2π 3 , 4π 3 and poles at φ = π 3 , π, 5π 3 • it intertwines between symmetric and antisymmetric Weyl group singlets • it is a Vandermonde dressing of the angular momentum • it is the unique angular intertwiner For g=0, the expression (3.9) is readily evaluated (with r 0 → ln r and ℓ = 3ℓ cos ℓφ for ℓ even sin ℓφ for ℓ odd , (3.23) which are all Weyl symmetric. Acting repeatedly with M 1 on these wave functions, one alternates between Weyl symmetric and Weyl antisymmetric and generates the wave functions at the same value of ℓ for any integer coupling. The low-lying states for g = 0, 1, 2, 3 are tabulated in table 1. The action of the intertwiners is illustrated in figure 1. The only exception is the one-dimensional kernel of M 1 , consisting of the ground state v (g) 0 ∼ r −3g ∆ g ∼ cos g 3φ . which provides ρ (g+1) 2 = 9. We read off the intertwining relation (3.26) For generic values of g, there can be no further independent conserved quantity. However, when g is integer, one additional algebraically independent conserved charge can be constructed, which is a Weyl-antiinvariant differential operator of order 2g−1 that squares to

Tetrahexahedric model: the spectrum
The main purpose of this paper is to work out the angular (relative) rational 4-particle (or A 3 ) Calogero model as the simplest non-separable case, in order to demonstrate the viability of the general observations of section 2. We take advantage of the isometry A 3 ≃ D 3 and employ D 3 -type relative coordinates. 7 To construct the model and find its spectrum, we introduce not Jacobi but Walsh-Hadamard coordinates {y i } and their derivatives together with the center of mass X via (4.1) The polar (or spherical) parametrization of R 3 defines the radius r and two angles θ and φ, y 1 = r sin θ cos φ , y 2 = r sin θ sin φ , y 3 = r cos θ .
(4.8) This has been named 'cuboctahedric Higgs oscillator' potential [22] since it is the superposition of six Higgs oscillators placed at the positive roots of A 3 ≃ D 3 , and those (together with their antipodes) form the vertices of a cuboctahedron [35]. However, a better name is 'tetrahexahedral' potential, because the intersections of the Weyl chamber walls with the two-sphere form six great circles, which trace the 36 edges of a tetrahexahedron (or tetrakis hexahedron) [36]. Its 24 identical faces are isosceles triangles. The spherical projection tessellates the two-sphere into 24 spherically triangular domains in which the particle is trapped. This sequence of geometrical configurations, starting from the A 3 root system (cuboctahedron) and ending at the walls of the angular potential (tetrahexahedron) is displayed in figure 2, and color and contour plots of the potential are shown in figure 3.
The angular energy eigenvalues become The interaction reduces the 2ℓ+1 dimensional free eigenspace (spanned by the spherical harmonics Y ℓm ) to the subspace invariant under the S 4 Weyl group action generated e.g. by   have a spherical Bessel function as their radial part and involve two specific Weyl-symmetric homogeneous Dunkl polynomials, of degree three and four, as well as the coordinate Vandermonde, ∼ r 6 sin 2 θ cos 4 θ cos 2 2φ (tan 2 θ cos 2 φ − 1)(tan 2 θ sin 2 φ − 1) .
For completeness, we here display some quantities in the potential-free frame. The potential-free Hamiltonian so that the complete wave function may be written as

Tetrahexahedric model: intertwiners & integrability
According to (2.34), the four-particle intertwiner is and it acts on the wave functions as The rather complicated explicit form of this six-order partial differential operator has been constructed in [19]. Let us turn our attention to the angular intertwiners M s . Their building blocks are the Dunkl-deformed angular momenta or their complex combinations L ± = L x ± iL y . Employing (4.4) and (4.14), they read The chemistry literature seems to be unaware of the straightforward construction scheme (4.12).

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and transform under the Weyl reflections in the following fashion, For convenience, we also note their transformation under certain even Weyl group elements, The construction (2.38) then yields a simple cubic polynomial in the angular Dunkl operators, which indeed is an anti-invariant under the Weyl group. It remains to evaluate the residue M Ω = res M Ω for arriving at x 6 +y 6 +z 6 (x 2 −y 2 )(x 2 −z 2 )(y 2 −z 2 ) + cyclic permutations . (5.8) Inserting the free angular momentum differential operators (4.4), we finally obtain x 6 +y 6 +z 6 (x 2 −y 2 )(x 2 −z 2 )(y 2 −z 2 ) + cyclic permutations . (5.9) This is not the only elementary intertwiner however. At order six, one finds the next Weyl-antiinvariant function of {L x , L y , L z }, namely Its residue then produces a second intertwiner,  The complete expression is too lengthy to display in this paper. We have checked that all higher-order intertwiners, generated from further Weyl-antiinvariants, can be decomposed into linear combinations of M 3 and M 6 with Weyl-symmetric functions of {L x , L y , L z } as coefficients. In this sense, M 3 and M 6 algebraically generate the set of all intertwiners for H Ω . As a curiosity, we report a remarkable (non-local) example from this set, In all cases, the action on the angular states is as expected, ℓ−6 (θ, φ) , (5.13) and the size of the kernels dim ker M s = dim ker 4 (ℓ) is independent of g as was given in (2.43). The action of the intertwiners M 3 and M 6 as well as M (5.2) is illustrated in figure 4. What about additional conserved charges in our tetrahexahedral angular Calogero system? The S 4 Weyl group action (5.5) on the Dunkl angular momenta is sufficiently simple to characterize the algebra of Weyl-symmetric polynomials. It is generated by J k := res L k x + L k y + L k z for k = 0, 2, 4, 6 , (5.14)

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Assuming superintegrability, we conjecture that the number of ring generators equals 2r−3 and r−1 for the first and second item, respectively, but one has to go to n=5 (or r=4) to really test the second assertion. Especially useful will be the answer to the third item, as different intertwiners can be employed to partially lift the growing degeneracy of the energy levels.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.