Symmetry, Integrability and Geometry: Methods and Applications From Quantum AN (Calogero) to H4 (Rational) Model ⋆

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate and, hence, the existence of the 2nd order integral, (v) an algebraic form in invariants of a discrete symmetry group (in space of orbits).


Introduction
In this paper we will make an attempt to overview our constructive knowledge about (quasi)exactly-solvable potentials having a form of a meromorphic function in Cartesian coordinates. All these models have a discrete group of symmetry, admit separation of variable(s), possess an (in)finite set of polynomial eigenfunctions. They have an infinite discrete spectrum which is linear in the quantum numbers. All of them are characterized by the presence of a hidden (Lie) algebraic structure. Each of them is a type of isospectral deformation of the isotropic harmonic oscillator.
Let us consider the Hamiltonian = the Schrödinger operator The main problem of quantum mechanics is to solve the Schrödinger equation finding the spectrum (the energies E and eigenfunctions Ψ). Since the Hamiltonian is an infinitedimensional matrix, solving the Schrödinger equation is equivalent to diagonalizing the infinitedimensional matrix. It is a transcendental problem: the characteristic polynomial is of infinite order and it has infinitely-many roots. Usually, we do not know how to make such a diagonalization exactly (explicitly) but we can ask: Do models exist for which the roots (energies), some of the them or all, can be found explicitly (exactly)? Such models do exist and we call them solvable. If all energies are known they are called exactly-solvable (ES), if only some number of them is known we call them quasi-exactly-solvable (QES). Surprisingly, all such models the present author familiar with, are provided by integrable systems. The Hamiltonians of these models are of the form in the exactly-solvable case and in the quasi-exactly-solvable case, where ω,ω k , Γ are parameters, W (Ω) is a function on unit sphere and r is the radial coordinate. In both cases there exists the integral where L is the angular momentum operator, due to the separation of variables in spherical coordinates. Now we consider some examples from the ones known so far.

Case O(N )
The Hamiltonian reads or, in spherical coordinates, The Hamiltonian (2) where E 0 is the lowest energy and the parameterl is chosen in such a way as to remove singular term ∝ 1 r 2 in the potential in (3). (5) is the algebraic form of the Hamiltonian (3). The gauge-rotated Hamiltonian h O(N ) (5) is sl(2)-Lie-algebraic (see below), it has infinitely-many finite-dimensional invariant subspaces in polynomials P n , n = 0, 1, . . . forming the infinite flag (see below), its eigenfunctions P n (r 2 = t) are nothing but the associated Laguerre polynomials.
By adding to h O(N ) (5) the operator we get the operator h O(N ) + δh (qes) which has a single finite-dimensional invariant subspace whereγ, Γ,ω are parameters and γ is replaced by the operator F. In (7) a finite number of the eigenfunctions is of the form they can be found algebraically. It is worth noting that at a = 0 the operator h O(N ) + δh (qes) remains exactly-solvable, it preserves the infinite flag of polynomials P and the emerging Hamiltonian has a form of (3).

Case (Z 2 ) N
The Hamiltonian reads or, in spherical coordinates, and F is given by (4). The Hamiltonian (8) is (Z 2 ) N symmetric. It defines the so-called Smorodinsky-Winternitz integrable system [1] which is in reality the maximally-superintegrable (there exist (2N − 1) integrals including the Hamiltonian) and exactly-solvable. Gauging away in (8) the ground state, , and changing variables to t i = x 2 i we arrive at the algebraic form. Also it admits QES extension. The system described by the Hamiltonian (8) at ν i = ν is a particular case of the BC N -rational system (see below).

Case A N −1
This is the celebrated Calogero model (A N −1 rational model) which was found in [3]. It describes N identical particles on a line (see Fig. 1) with singular pairwise interaction.
The Hamiltonian is where the singular part of the potential can be written as Here r is the radial coordinate in the space of relative coordinates (see below for a definition) and W A N−1 (Ω) is a function on the unit sphere.
The ground state of the Hamiltonian (9) reads Let us make the gauge rotation and introduce center-of-mass variables and then permutationally-symmetric, translationally-invariant variables are elementary symmetric polynomials, and hence, the variable t 2 , which plays fundamental role, is defined by radius in space of relative coordinates. After the center-of-mass separation, the gauge rotated Hamiltonian takes the algebraic form [4] h Cal = A ij (t) where Eigenvalues of (12) are hence, the spectrum is linear in the quantum numbers p 2,3,...,N = 0, 1, . . ., it corresponds to anisotropic harmonic oscillator with frequency ratios 1 : 2 : 3 : · · · : (N − 1). It is easy to check that the gauge-rotated Hamiltonian h Cal has infinitely many finitedimensional invariant subspaces where n = 0, 1, 2, . . .. As a function of n the spaces P (N −1) n form the infinite flag (see below).

2.3.1
The gl d+1 -algebra acting by 1st order dif ferential operators in R d It can be checked by the direct calculation that the gl d+1 algebra realized by the first order differential operators acting in R d in the representation given by the Young tableaux as a row (n, 0, 0, . . . , 0 ) has a form where n is an arbitrary number. The total number of generators is (d + 1) 2 . If n takes the integer values, n = 0, 1, 2, . . ., the finite-dimensional irreps occur It is a common invariant subspace for (13). The spaces P n at n = 0, 1, 2, . . . can be ordered P 0 ⊂ P 1 ⊂ P 2 ⊂ · · · ⊂ P n ⊂ · · · ⊂ P.
Such a nested construction is called infinite flag (filtration) P. It is worth noting that the flag P (d) is made out of finite-dimensional irreducible representation spaces P (d) n of the algebra gl d+1 taken in realization (13). It is evident that any operator made out of generators (13) has finite-dimensional invariant subspace which is finite-dimensional irreducible representation space.

Algebraic properties of the Calogero model
It seems evident that the Hamiltonian (12) has to have a representation as a second order polynomial in generators (13) where the raising generators J + i are absent. Thus, gl(N ) (or, strictly speaking, its maximal affine subalgebra) is the hidden algebra of the N -body Calogero model. Hence, h Cal is an element of the universal enveloping algebra U gl(N ) . The eigenfunctions of the N -body Calogero model are elements of the flag of polynomials P (N −1) . Each subspace P (N −1) n is represented by the Newton polytope (pyramid). It contains C N −1 n+N −1 eigenfunctions, which is equal to the volume of the Newton polytope.
Making the gauge rotation of the integral (1) with W A N−1 (Ω) given by (10) where F 0 is the lowest eigenvalue of the integral, F Cal Ψ 0 = F 0 Ψ 0 , the integral gets the algebraic form where f ij is 2nd degree polynomial in t, f 2j = 0, and g i is 1st degree polynomial in t, g 2 = 0. It also can be rewritten as the second degree polynomial in the gl(N ) generators,

sl(2)-quasi-exactly-solvable generalization of the Calogero model
By adding to h Cal (12), the operator we get the operator h Cal + δh (qes) having finite-dimensional invariant subspace By making a gauge rotation of h Cal + δh (qes) and changing of variables to Cartesian ones we arrive at the Hamiltonian [5] H (qes) For the Hamiltonian, (k + 1) eigenfunctions are of the form where Ψ 0 is given by (11), P k is a polynomial of degree k in All remaining eigenfunctions can be represented in the same form but P k 's are not polynomials anymore being functions depending on all variables x i . It is worth noting that at a = 0 the operator h Cal + δh (qes) remains exactly-solvable, it preserves the flag of polynomials P (N −1) and the emerging Hamiltonian has a form of (9) with the extra term Γ r 2 in the potential. Its ground state eigenfunction is (r 2 ) γ Ψ 0 . It is the exactly-solvable generalization of the Calogero model (9) with the Weyl group W(A N −1 ) as the discrete symmetry group,

Case: Hamiltonian reduction method
In this method 1 a family of integrable and exactly-solvable Hamiltonians associated with Weyl (Coxeter) symmetry was found with the Calogero model as one of its representatives. The idea of the method is beautiful and sufficient transparent, • Take a simple group G, • Define the Laplace-Beltrami (invariant) operator on its symmetric space (free/harmonic oscillator motion), • Radial part of Laplace-Beltrami operator is the Olshanetsky-Perelomov Hamiltonian relevant from physical point of view. The emerging Hamiltonian is the Weyl-symmetric, it can be associated with root system, it is integrable with integrals given by the invariant operators of higher than two orders with a property of solvability.
Rational case. This case appears when the coordinates of the symmetric space are introduced in such a way that the zero-curvature surface occurs. Emerging the Calogero-Moser-Sutherland-Olshanetsky-Perelomov Hamiltonian in the Cartesian coordinates has the form, where R + is a set of positive roots, x is a position vector and ν |α| are coupling constants (parameters) which depend on the root length. If roots are of the same length, then ν |α| have to be equal, if all roots are of the same length like for A n , then all ν |α| = ν. In the Hamiltonian Reduction the parameters ν |α| take a set of discrete values, however, they can be generalized to any real value without loosing a property of integrability as well as of solvability with the only constraint of the existence of L 2 -solutions of the corresponding Schrödinger equation. The configuration space for (14) is the Weyl chamber. The ground state wave function is written explicitly, The Hamiltonian (14) is completely-integrable: there exists a commutative algebra of integrals (including the Hamiltonian) of dimension which is equal to the dimension of the configuration space (for integrals, see Oshima [7] with explicit forms of those). For each Hamiltonian (14) after separation of center-of-mass coordinate (if applicable) the radial coordinate (in the space of relative coordinates) can be also separated. It gives rise to the existence of one more integral of the second order (1). Hence, the Hamiltonian (14) is super-integrable. The Hamiltonian (14) is invariant with respect to the Weyl (Coxeter) group transformation, which is the discrete symmetry group of the corresponding root space.
The Hamiltonian (14) has a hidden (Lie)-algebraic structure. In order to reveal it we need to • Gauge away the ground state eigenfunction making similarity transformation • Consider the Hamiltonian in the space of orbits of Weyl (Coxeter) group by taking the Weyl (Coxeter) polynomial invariants as new coordinates, these invariants are where a's are the degrees of the Weyl (Coxeter) group, Ω is an orbit.
The invariants t are defined ambiguously, up to invariants of lower degrees, they depend on chosen orbit.

Case BC N
The BC N -rational model is defined by the Hamiltonian, where ω, ν, ν 2 are parameters. If ν = 0, the Hamiltonian (16) is reduced to (8). The symmetry of the system is S N ⊕ (Z 2 ) N (permutations x i → x j and x i → −x i ). The ground state function for (16) reads (15)). Making the gauge rotation and changing variables where r is radius, we arrive at [8] This is the algebraic form of the BC N Hamiltonian. Assuming polynomiality of the eigenfunctions we find the eigenvalues: hence, the spectrum is equidistant, linear in the quantum numbers and corresponds to anisotropic harmonic oscillator with frequency ratios 1 : 2 : 3 : · · · : N . The Hamiltonian h BC N has infinitely many finite-dimensional invariant subspaces of the form where n = 0, 1, 2, . . .. They naturally form the flag P (N ) . The Hamiltonian can be immediately rewritten in terms of generators (13) as a polynomial of the second degree, where the raising generators J + i are absent. Hence, gl(N + 1) is the hidden algebra of the BC N rational model, the same algebra as for the A N -rational model. The eigenfunctions of the BC Nrational model are elements of the flag of polynomials P (N ) . Each subspace P Evidently, the commutator Gauge-rotated integral where f ij is 2nd degree polynomial, f 1j = 0, and g i is 1st degree polynomial, g 1 = 0, in terms of the gl(N + 1) generators. It is worth mentioning that the commutator of [h, f ] vanishes only in the realization (13), otherwise, 2.5.1 sl(2)-quasi-exactly-solvable generalization of the BC N rational model By adding to h BC N the operator which is the similar to one for the Calogero model, we get the operator h BC N + δh (qes) which has the finite-dimensional invariant subspace Making a gauge rotation of h BC N + δh (qes) and changing the variables σ's back to the Cartesian ones the Hamiltonian becomes H (qes) for which (k + 1) eigenfunctions are of the form where P k is a polynomial of degree k in It is worth noting that at a = 0 the operator h BC N + δh (qes) remains exactly-solvable, it preserves the flag of polynomials P (N ) and the emerging Hamiltonian has a form of (16) with the extra term Γ r 2 in the potential. Its ground state eigenfunction is (r 2 ) γ Ψ 0 . It is the exactlysolvable generalization of the BC N -rational model (16) with the Weyl group W(BC N ) as the discrete symmetry group, Now we are in a position to draw an intermediate conclusion about A N and BC N rational models.
• Both A N -and BC N -rational (and trigonometric) models possess algebraic forms associated with preservation of the same flag of polynomials P (N ) . The flag is invariant w.r.t. linear transformations in space of orbits t → t+A. It preserves the algebraic form of Hamiltonian.
• Their Hamiltonians (as well as higher integrals) can be written in the Lie-algebraic form where Pol 2 is a polynomial of 2nd degree in generators J of the maximal affine subalgebra of the algebra b of the algebra gl N +1 in realization (13). Hence, gl N +1 is their hidden algebra. From this viewpoint all four models are different faces of a single model.
• Supersymmetric A N -and BC N -rational (and trigonometric) models possess algebraic forms, preserve the same flag of (super)polynomials and their hidden algebra is the superalgebra gl(N + 1|N ) (see [8]).
In a connection to flags of polynomials we introduce a notion 'characteristic vector'. Let us consider a flag made out of "triangular" linear space of polynomials where the "grades" f 's are positive integer numbers and n = 0, 1, 2, . . .. In lattice space P

Case G 2
Take the Hamiltonian where ω, ν, µ are parameters. It describes the Wolfes model of three-body interacting system [9] or, in the Hamiltonian reduction nomenclature, the G 2 -rational model. The symmetry of the model is dihedral group D 6 . The ground state function is

Making the gauge rotation
and changing variables and separating the center-of-mass coordinate we arrive at which is the algebraic form of the Wolfes model. The eigenvalues of h G 2 are It coincides to the spectrum of anisotropic harmonic oscillator with frequency ratio 1 : 3.
Separating the center-of-mass in (17) and introducing the polar coordinates (̺, ϕ) in the space of relative coordinates we arrive at the Hamiltoniañ It is evident that the integral of motion which appears due to separation of variables in polar coordinates (cf. (1)) has the form It is evident that after gauge rotation with Ψ 0 and change of variables to (λ 1 , λ 2 ) the integral F takes algebraic form. The Hamiltonian h G 2 has infinitely many finite-dimensional invariant subspaces hence the flag P (1,2) with the characteristic vector f = (1, 2) is preserved by h G 2 . The eigenfunctions of h G 2 are are elements of the flag of polynomials P (2) (1,2) . Each subspace P (2) n,(1,2) − P (2) n−1, (1,2) contains ∼n eigenfunctions which is equal to length of the Newton line L n n = n λ 1 p 1 λ 2 p 2 | p 1 + 2p 2 = n . A natural question to ask: What about hidden algebra? Namely: Does algebra exist for which P (2) n, (1,2) is the space of (irreducible) representation? Surprisingly, this algebra exists and it is, in fact, known.
Let us consider the Lie algebra spanned by seven generators It is non-semi-simple algebra gl(2, R) ⋉ R (2) (S. Lie [10] at n = 0 and A. González-Lopéz et al. [11] at n = 0 (case 24)). If the parameter n in (19) is a non-negative integer, it has as common (reducible) invariant subspace. By adding to gl(2, R) ⋉ R (2) (see (19)), the action on P n, (1,2) , where J 0 = t∂ t +2u∂ u −n, and all of them are of degree 2. These new generators have a property of nilpotency, and commutativity: (19) plus (20) span a linear space with a property of decomposition: g (2) . Fig. 2). Eventually, infinite-dimensional, eleven-generated algebra (by (19) and J 0 plus (20), so that the eight generators are the 1st order and three generators are of the 2nd order differential operators) occurs. The Hamiltonian h G 2 can be rewritten in terms of the generators (19), (20) with the absence of the highest weight generator J 4 n , where J 2,3 = J 2,3 0 . Hence, gl(2, R) ⋉ R (2) is the hidden algebra of the Wolfes model. (i) G 2 Hamiltonian admits two mutually-non-commuting integrals: of 2nd order as the result of the separation of radial variable r 2 (see (18)) and of the 6th order. If ω = 0 the latter integral degenerates to the 3rd order integral (the square root can be calculated in closed form).
(ii) Both integrals after gauge rotation with Ψ 0 take in variables λ 1,2 the algebraic form. Both preserve the same flag P (1,2) . (iii) Both integrals can be rewritten in term of generators of the algebra g (2) : integral of 2nd order in terms of gl(2, R) ⋉ R (2) generators only and while one of the 6th order contains generators from L as well [13].

sl(2)-quasi-exactly-solvable generalization
By adding to h G 2 , the operator (the same as for the Calogero and the BC N models) we get the operator h G 2 + δh (qes) having single finite-dimensional invariant subspace Making a gauge rotation of h G 2 + δh (qes) , changing of variables (Y, λ 1,2 ) back to the Cartesian coordinates and adding the center-of-mass the Hamiltonian becomes H (qes) for which (k + 1) eigenfunctions are of the form where P k is a polynomial of degree k in r 2 . It is worth noting that at a = 0 the operator h G 2 + δh (qes) remains exactly-solvable, it preserves the flag of polynomials P (2) (1,2) and the emerging Hamiltonian has a form of (17) with the extra term Γ r 2 in the potential. Its ground state eigenfunction is (r 2 ) γ Ψ 0 . It is the exactlysolvable generalization of the G 2 -rational model (17) with the Weyl group W(G 2 ) as the discrete symmetry group, 2.7 Cases F 4 and E 6,7,8 In some details these four cases are described in [12].

Case I 2 (k)
In some details this case is described in [13]. It is worth noting that although the Hamiltonian Reduction nomenclature is assigned to this case the parameter k takes any real value. Discrete symmetry group D 2k of the Hamiltonian appears for integer k.

Case H 3
The H 3 rational Hamiltonian reads where {i, j, k} = {1, 2, 3} and all even permutations, ω, ν are parameters and the golden ratio and its algebraic conjugate. Symmetry of the Hamiltonian (21) is the H 3 Coxeter group (the full symmetry group of the icosahedron). It has the order 120. In total, the Hamiltonian (21) is symmetric with respect to the transformation The ground state is given by Making the gauge rotation we arrive at new spectral problem After changing variables (x 1,2,3 → τ 1,2,3 ): in the gauge-rotated Hamiltonian, it emerges in the algebraic form [14] h which is amazingly simple comparing the quite complicated and lengthy form of the original Hamiltonian (21). The Hamiltonian h H 3 preserves infinitely-many spaces P (1,2,3) n = τ n 1 1 τ n 2 2 τ n 3 3 | 0 ≤ n 1 + 2n 2 + 3n 3 ≤ n , n ∈ N, with characteristic vector is (1, 2, 3), they form an infinite flag. The spectrum of h H 3 is given by is finite-dimensional representation space of a Lie algebra of differential operators which we call the h (3) algebra. It is infinite-dimensional but finitely generated algebra of differential operators with 30 generating elements of 1 st (14), 2 nd (10) and 3 rd (5) orders, respectively, plus one of zeroth order. They span 5 + 5 Abelian (conjugated) subalgebras of lowering and raising generators L and U 2 and one of the Cartan type algebra B (for details see [14]). The algebra B = gℓ 2 ⊕ I 2 , where I 2 is two-dimensional ideal. The Hamiltonian h H 3 can be rewritten in terms of the generators of the h (3) -algebra.
By adding to h H 3 (21) the operator (6) in the variable τ 1 (of the same type as for the Calogero, BC N and G 2 models) we get the operator h H 3 +δh (qes) which has the finite-dimensional invariant subspace of the dimension (k + 1). Hence, this operator is quasi-exactly-solvable. Making the gauge rotation of this operator and changing variables τ back to Cartesian ones we arrive at the quasiexactly-solvable Hamiltonian of a similar type as for the Calogero, BC N and G 2 models [14]. By adding to h H 3 (21) the operator 4γ ∂ ∂τ 1 we preserve the property of exact-solvability. This operator preserves the flag P (1,2,3) and the emerging Hamiltonian has a form of (21) with the extra term Γ r 2 in the potential. Its ground state eigenfunction is (r 2 ) γ Ψ 0 . It is the exactlysolvable generalization of the H 3 -rational model (21) with the Coxeter group H 3 as the discrete symmetry group,

Case H 4
The H 4 rational Hamiltonian reads where {i, j, k, l} = {1, 2, 3, 4} and all even permutations, ω, ν are parameters and the golden ratio and it algebraic conjugate. Symmetry of the Hamiltonian (22) is the H 4 Coxeter group (the symmetry group of the 600-cell). It has order 14400. In total, the Hamiltonian (22) is symmetric with respect to the transformation  (14) orders, respectively, plus one of zeroth order. They span 26 + 26 Abelian (conjugated) subalgebras of lowering and raising generators L and U 3 and one of the Cartan type algebra B (for details see [16]). The algebra B = gℓ 2 ⊕ I 3 , where I 3 is three-dimensional ideal. The Hamiltonian h H 4 can be rewritten in terms of the generators of the h (4) -algebra.
By adding to h H 4 (22) the operator (6) in the variable τ 1 (of the same type as for the Calogero, BC N and G 2 , H 3 models) we get the operator h H 4 + δh (qes) , which has the finite-dimensional invariant subspace of dimension (k + 1). Hence, this operator is quasi-exactly-solvable. Making the gauge rotation of this operator and changing variables τ back to Cartesian ones we arrive at the quasi-exactlysolvable Hamiltonian of a similar type as for the Calogero, BC N and G 2 models [15]. By adding to h H 4 (22) the operator 4γ ∂ ∂τ 1 we preserve the property of exact-solvability. This operator preserves the flag P (1,5,8,12) and the emerging Hamiltonian has a form of (22) with the extra term Γ r 2 in the potential. Its ground state eigenfunction is (r 2 ) γ Ψ 0 . It is the exactly-solvable generalization of the H 4 -rational model (22) with the Coxeter group H 4 as the discrete symmetry group,

Conclusions
• For rational Hamiltonians for all classical A N , BC N and exceptional root spaces G 2 , F 4 , E 6,7,8 (also trigonometric) and non-crystallographic H 3,4 , I 2 (k) there exists an algebraic form after gauging away the ground state eigenfunction, and changing variables to symmetric (invariant) variables. Their eigenfunctions are polynomials in these variables. They are orthogonal with the squared ground state eigenfunction as the weight factor. • Their hidden algebras are gl(N + 1) for the case of classical A N , BC N and new infinitedimensional but finite-generated algebras of differential operators for all other cases. All these algebras have finite-dimensional invariant subspace(s) in polynomials. • The generating elements of any such hidden algebra can be grouped into an even number of (conjugated) Abelian algebras L i , U i and one Lie algebra B. They obey a (generalized) Gauss decomposition (see Fig. 3). A description of all these algebras will be given elsewhere.
There are two solvable potentials in 1D in [0, ∞) which can be generalized to D.
In conclusion I must emphasize that the algebraic nature of all above-considered systems was revealed when: The invariants of the discrete group of symmetry of a system are taken as variables (space of orbits).