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Quantum Calogero-Moser Models: Integrability for all Root Systems

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.


Introduction
Calogero-Moser models are one-dimensional dynamical systems with long range interactions having a remarkable property that they are integrable at both classical and quantum levels.
In fact the integrability or more precisely the triangularity of the quantum Hamiltonian was first discovered by Calogero [1] for the model with inverse square potential plus a confining harmonic force and by Sutherland [2] for the particles on a circle with the inverse square potential. Later classical integrability of the models in terms of Lax pairs was proved by Moser [3]. Olshanetsky and Perelomov [4] showed that these models were based on A r root systems and generalisations of the models based on other root systems including the non-crystallographic ones were introduced [5].
In this paper we discuss quantum Calogero-Moser models with degenerate potentials, that is the rational with/without harmonic force, the hyperbolic and the trigonometric potentials based on all root systems. We demonstrate, based on previous works of universal Lax pairs for classical [6,7,8] and quantum models [9], that various results known for the quantum A r models can be generalised to the models based on any root systems as well. They are (i) Construction of a complete set of quantum conserved quantities in terms of quantum Lax pairs and other methods. (ii) Universal proof of Liouville integrability for the rational, hyperbolic and trigonometric potential models. Namely, the quantum conserved quantities commute among themselves. (iii) Triangularity of the quantum Hamiltonian is demonstrated explicitly for all the models. In other words, the Hamiltonian is shown, in certain bases, to be decomposed into a sum of finite dimensional triangular matrices. Thus any eigenvalue equation can be solved by finite steps of linear algebraic processes only. This also gives the entire discrete spectrum of the models. As unique eigenfunctions of the Hamiltonian, generalisation of Jack polynomials and multivariable Laguerre (Hermite) polynomials are defined for all root systems. (iv) Equivalence of the quantum Lax pair method and that of so-called differential-reflection (Dunkl) operators [10] is demonstrated. (v) For rational models with harmonic confining force, an algebraic construction of all excited states in terms of creation (annihilation) operators is achieved.
For the A r models, the Lax pairs, conserved quantities and their involution were discussed by many authors with varied degrees of completeness and rigour, see for example, [5], [11]- [20]. The point (iv) was shown by Wadati and collaborators [18] and point (v) was initiated by Perelomov [12] and developed by Brink and collaborators [17] and Wadati and collaborators [18]. A rather different approach by Heckman and Opdam [21,22] to Calogero-Moser models with degenerate potentials based on any root systems should also be mentioned in this connection.
For the general background and the motivations of this series of papers and the physical applications of the Calogero-Moser models with various potentials to lower dimensional physics, ranging from solid state to particle physics and supersymmetric Yang-Mills theory, we refer to our previous papers [6,7] and references therein. This paper is organised as follows. In section two, quantum Calogero-Moser Hamiltonian with degenerate potentials is introduced as a factorised form (2.5). Connection with root systems and the Coxeter invariance is emphasised. Some rudimentary facts of the root systems and reflections are summarised in Appendix A. A universal Coxeter invariant ground state wavefunction together with its energy eigenvalue are presented. In section three we show that all the excited states are also Coxeter invariant and that the Hamiltonian is triangular in certain bases. Complete sets of quantum conserved quantities are derived from quantum Lax operator L in section four. Instead of the trace, the total sum of L n is conserved.
That is Ts(L n ) = µ,ν∈R (L n ) µν , in which R is a representation space of the Coxeter group.
The details of the complete set for each root system are given in Appendix B. In section five the creation and annihilation operators for the rational models with harmonic force are derived. In section six, the equivalence of the Lax pair operator formalism and the socalled differential-reflection (Dunkl) operators is demonstrated and the quantum conserved quantities are expressed in terms of the latter. In section seven an algebraic construction of excited states in terms of the differential-reflection (Dunkl) operators for rational models with harmonic force is presented. The complete sets of explicit eigenfunctions for the rank two models are derived in terms of separation of variables based on the Coxeter invariant variables. Section eight gives a universal proof of the Liouville integrability for models with rational (without the confining force), hyperbolic and trigonometric potentials. For rational models with harmonic force, the involution is demonstrated for those based on classical root systems. A simple use of the quantum Lax pair with spectral parameter is mentioned. The final section is for summary and comments.

Quantum Calogero-Moser Models
In this section we briefly introduce the quantum Calogero-Moser models along with appropriate notation and background for the main body of this paper. A Calogero-Moser model is a Hamiltonian system associated with a root system ∆ of rank r, which is a set of vectors in R r with its standard inner product. A brief review of the properties of the root systems and the associated reflections will be found in the Appendix A.

Factorised Hamiltonian
The dynamical variables of the Calogero-Moser model are the coordinates {q j } and their canonically conjugate momenta {p j }, with the canonical commutation relations: These will be denoted by vectors in R r q = (q 1 , . . . , q r ), p = (p 1 , . . . , p r ). (2. 2) The momentum operator p j acts as As for the interactions we consider only the degenerate potentials, that is the rational (with/without harmonic force), hyperbolic and trigonometric potentials: in which a is an arbitrary real positive constant, determining the period of the trigonometric potentials. They imply integrability for all of the Calogero-Moser models based on the crystallographic root systems. Those models based on the non-crystallographic root systems, the dihedral group I 2 (m), H 3 , and H 4 , are integrable only for the rational potential. The rational potential models are also integrable if a confining harmonic potential is added to the Hamiltonian. Since we will discuss the universal properties and solutions applicable to all the interaction types as well as those for specific interaction potentials, let us adopt the conventional nomenclature for them. We call the models with rational, hyperbolic, trigonometric and rational with harmonic force the type I, II, III and V models, respectively.
(Type IV models have elliptic potentials which we will not discuss in this paper.) The Hamiltonian for the quantum Calogero-Moser model can be written in a 'factorised form': It should be noted that the above factorised Hamiltonian (2.7) consists of an operator part H, which is the Hamiltonian in the usual definition, and a constant E 0 which is the ground state energy to be discussed later: The real positive coupling constants g |ρ| are defined on orbits of the corresponding Coxeter group, i.e. they are identical for roots in the same orbit. That is, for the simple Lie algebra cases one coupling constant g |ρ| = g for all roots in simply-laced models and two independent coupling constants, g |ρ| = g L for long roots and g |ρ| = g S for short roots in non-simply laced models. For the I 2 (m) models, there is one coupling if m is odd, and two independent ones if m is even. Let us call them g e for even roots and g o for odd roots. Throughout this paper we consider the coupling constants at generic values. We parametrise the positive roots of I 2 (m) as ρ j = (cos((j − 1)π/m), sin((j − 1)π/m)), j = 1, . . . , m. (2.10) The H 3 and H 4 models have one coupling constant g |ρ| = g, since these root systems are simply-laced. It should be noted that the operator part of the HamiltonianĤ is strictly positive for g |ρ| ≥ 1.
The simplest way to introduce the factorised form is through supersymmetry [9,23], in which function W is called a superpotential: The potential V (u) (2.3) and the function w(u) are related by (2.13) The following Table I gives these functions for each potential: II sinh au a coth au -a 2 / sinh 2 au trigonometric III sin au a cot au -a 2 / sin 2 au For proofs that the factorised Hamiltonian (2.6) actually gives the quantum Hamiltonian (2.7) for all the root systems and potentials see [5,11,9]. It is easy to verify that for any potential V (u), the Hamiltonian is invariant under reflection of the phase space variables in the hyperplane perpendicular to any root H(s α (p), s α (q)) = H(p, q), ∀α ∈ ∆ (2.14) with s α defined by (A.2).
Some remarks are in order. For all of the root systems and for any choice of potential (2.3), the Calogero-Moser model has a hard repulsive potential ∼ 1/(α · q) 2 near the reflection hyperplane H α = {q ∈ R r , α·q = 0}. The strength of the singularity is given by the coupling constant g |α| (g |α| − 1) which is independent of the choice of the normalisation of the roots.
(Thus for rational models with/without harmonic force there is equivalence, A 2 ∼ = I 2 (3), without loss of generality. In the case of the trigonometric potential, the configuration space is further limited due to the periodicity of the potential to where α h is the highest root.
The fact that the classical motions are confined in the Weyl chamber (alcove) P W (P W T ) does not necessarily mean that the corresponding quantum wavefunctions vanish identically outside of the region. On the contrary, as we will see soon, the ground state wavefunction (subsection 2.2) and all the other excited states wavefunctions (section 3) are Coxeter invariant, reflecting the Coxeter invariance of the Hamiltonian (2.14). In the early years of Calogero-Moser models in which those based on the A r root system were mainly discussed, these Coxeter invariant solutions were considered as totally symmetric states of bosonic systems. We will not, however, adopt this interpretation, for in the models based on the other root systems the reflection is not the same as particle interchange. The quantum theory we are discussing is the so-called first quantised theory. That is, the notions of identical particles and the associated statistics are non-existent.

Ground state wavefunction and energy
One merit of the factorised Hamiltonian (2.5) is the ease of derivation of the ground state wavefunction and of the Hamiltonian (3.3) derived by the similarity transformation in terms of the ground state wavefunction. Supersymmetric formulation of the Calogero-Moser models [23,16,9] provides a natural setting for the introduction of the factorised Hamiltonian. The universal ground state wavefunction is The exponential factor e − ω 2 q 2 exists only for the rational potential case with the harmonic confining force. It is easy to see that it is an eigenstate of the Hamiltonian (2.5) with zero eigenvalue: In other words, the above solution (2.17) provides an eigenstate of the Hamiltonian operator H with energy E 0 . The fact that it is a ground state (for type I, III and V) can be easily shown within the framework of the supersymmetric model [9] thanks to the positivity of the supersymmetric Hamiltonian. It should be stressed that E 0 is determined purely algebraically [9], without really applying the operator on the left hand side of (2.20) to the wavefunction.
This type of ground states has been known for some time. It is derived by various methods, see for example [3,5], and also by using supersymmetric quantum mechanics for the models based on classical root systems [23,16].
The ground state energy for the rational potential cases are The same for the hyperbolic and trigonometric potential cases are E 0 = 2a 2 ̺ 2 × −1 hyperbolic, 1 trigonometric, (2.22) in which can be considered as a 'deformed Weyl vector' [5,22]. Again these formulas are universal.
That is they apply to all of the Calogero-Moser models based on any root systems. A negative E 0 for the obviously positive Hamiltonian of the hyperbolic potential model indicates that the interpretation of e W as an eigenfunction is not correct. This function diverges as |q| → ∞ for the hyperbolic and the rational potential cases, destroying the hermiticity of the Hamiltonian.
Obviously we have P W (P W T ) e 2W (q) dq = ∞ : type I and II, finite : type III and V, (2.24) in which P W and P W T denote that the integration is over the regions defined in (2.15) and (2.16). Naturally, most existing results in quantum Calogero-Moser models are for the models with trigonometric potential and the rational potential with harmonic force which have normalisable states and discrete spectra.
It should be remarked that the domain of the universal ground state wavefunction Φ 0 could be considered as the entire R r space except for the points on the reflection hyperplanes, that is the disjoint union of all the Weyl chambers (alcoves), instead of the initial Weyl chamber/alcove (P W , P W T ) in which the classical motions are restricted due to the singular potential. In fact, Φ 0 and W are characterised as Coxeter invariant: (2.25) in whichš ρ is the representation of the reflection in the function space. For an arbitrary function f of q, its action is defined by This definition can be generalised to the entire Coxeter group G ∆ : for an arbitrary element In the rest of this paper we discuss mainly the type III and V models which have normalisable states and discrete spectra.

Coxeter invariant excited states, triangularity and spectrum
The Coxeter invariance of W implies those of H andH:

Rational potential with harmonic force
First, let us determine the structure of the set of eigenfunctions of the transformed Hamil-tonianH, for the type V models: Obviously a constant and ωq 2 − E 0 /ω are its eigenfunctions with eigenvalue 0 and 2ω, respectively. Let us suppose that a polynomial P (q) is an eigenfunction ofH: Due to the Coxeter invariance ofH (3.4), we know thatš ρ P together with the difference are also eigenfunctions with the same eigenvalue: if the latter is not identically zero. Since Q is a polynomial which is odd under reflectionš ρ it can be factorised as with a non-negative integer n and a polynomialQ. By substituting (3.8) into (3.7) and using the explicit form ofH near the reflection hyperplane ρ · q = 0, we obtain  Table II: in which {e j }, j = 1, . . . , r, are the exponents of ∆. Let us denote them by  Table II: The degrees f j in which independent Coxeter invariant polynomials exist.
Thus we arrive at: the quantum Calogero-Moser models with the rational potential and the harmonic confining force is algebraically solvable for any (crystallographic and non-crystallographic) root system ∆. The spectrum of the operator HamiltonianĤ is with a non-negative integer N which can be expressed as As shown above, there exists a unique eigenstate ψ N (q) for each φ N (q): It satisfies the orthogonality relation with respect to the inner product in P W : These polynomials {ψ N (q)} are generalisations of the multivariable Laguerre (Hermite) polynomials [14] known for the A r (B r , D r ) root systems to arbitrary root systems.
Some remarks are in order.
1. There is no Coxeter invariant linear function in q. The quadratic invariant polynomial q 2 = q · q exists in all the root systems. This corresponds to the universal fact that f 1 = 2, (e 1 = 1) for all the root systems. Moreover, this is related to the fact that a special sub-series of the excited states with N = 2n 1 = n 1 f 1 , n j = 0, j ≥ 2, can be expressed universally in terms of Laguerre polynomials in q 2 . This will be discussed at the end of section 5 and in subsection 7.2.
2. The other Coxeter invariants corresponding to the degrees f 2 , . . . , f r could be interpreted as special 'angular' variables of a unit sphere S r−1 (q 2 = 1), with the first Coxeter invariant √ q 2 being the radial coordinate. These would provide proper variables for describing solutions. Solutions in terms of separation of variables are in general possible only for the simplest cases, that is the rank two models, which will be demonstrated in subsection 7.3.
3. For ∆ = A 1 , the simplest root system of rank one, the HamiltonianH can be rewritten in terms of a Coxeter invariant variable u = ωq 2 as: (3.20) The Laguerre polynomial satisfying the differential equation provides an eigenfunction with eigenvalue 2ωn, which corresponds to the eigenvalue 2ωn + E 0 ofĤ. This is a well-known result.
4. Triangularity of the type I models is also obvious from the above argument.

Trigonometric potential
Here we consider those root systems associated with Lie algebras. In order to determine the excited states of the type III models, we have to consider the periodicity. The superpotential W and the Hamiltonian H are invariant under the following translation: 22) in which l ∨ is an element of the dual weight lattice, that is As is well known in quantum mechanics with periodic potentials, the wavefunctions diagonalising the translation operators are expressed as in which a vector µ ∈ R r is as yet unspecified. In other words, up to the overall phase factor e 2iaµ·q , this is a Fourier expansion in terms of the simple roots.
Let P T (q) be a polynomial in e ±2iaα j ·q , α j ∈ Π and suppose that a function φ(q) is an eigenfunction ofH:H φ(q) = λφ(q), (3.26) in which the explicit form ofH is given bỹ As above, due to the Coxeter invariance ofH (3.4), we know thatš ρ φ together with the difference ϕ = (1 −š ρ )φ are also eigenfunctions with the same eigenvalue: it can be expressed as in a neighbourhood of the reflection hyperplane ρ · q = 0. In this neighbourhood, the singularity structure ofH for the trigonometric potential is the same as that ofH for the rational potential discussed in the previous subsection. Thus we obtain, as before, a contradictioñ ϕ| ρ·q=0 = 0. In other words, the eigenfunction φ (3.25) must be Coxeter invariant. This in turn requires that the unspecified vector µ in (3.25) to be an element of the weight lattice the following condition is necessary, but not sufficient, for the Coxeter invariance of φ. Thus we arrive at (3.30).
Let us introduce a basis for the Coxeter invariant functions of the form (3.25). Let λ be and W λ be the orbit of λ by the action of the Weyl group: which is Coxeter invariant. The set of functions {φ λ } has an order >: Next we show thatH is lower triangular in this basis. By using (3.27) we obtaiñ First let us fix one positive root ρ and a weight µ in W λ such that ρ · µ = 0. Then Without loss of generality we may assume The contribution of the pair (µ, µ ′ ) in the summation of (3.36) reads which is the generalisation of Sutherland's fundamental identity eq (15) in [2] to arbitrary root systems. The summation in the expression correspond to φ λ ′ with λ ′ being lower than λ. Thus (3.36) reads which is independent of µ. Thus we have demonstrated the triangularity ofH: with the eigenvalue In other words, for each dominant weight λ there exists an eigenstate ofH with eigenvalue proportional to λ(λ + 2̺). Let us denote this eigenfunction by ψ λ (q): and call it a generalised Jack polynomial [25]- [28]. It satisfies the orthogonality relation with respect to the inner product in P W T : In the A r model, specifying a dominant weight λ is the same as giving a Young diagram which designates a Jack polynomial. It should be emphasised, however, that {ψ λ } are not identical to the Jack polynomials even for the A r root systems, because of different treatments of the center of mass coordinates. Detailed properties of these polynomials for various root systems will be published elsewhere.
Thus we arrive at: the quantum Calogero-Moser models with the trigonometric potential are algebraically solvable for any crystallographic root system ∆. The spectrum of the HamiltonianĤ is given by (3.44) in which λ is an arbitrary dominant weight. This is generalisation of Sutherland's original argument [2] to the models based on arbitrary root systems.
Some remarks are in order.
1. The weights µ appearing in the lower order terms {φ λ ′ }'s are those weights contained in the Lie algebra representation belonging to the highest weight λ.
2. As a simple corollary we find that for a minimal weight λ, is an eigenfunction ofH. A minimal representation [6] consists of a single Weyl orbit and all of its weights µ satisfy ρ ∨ · µ = 0, ±1, ∀ρ ∈ ∆.
3. If λ = α h , the highest root of a simply-laced root system, W λ is the set of roots itself.
Then the lower order terms are constants only. We find that is an eigenstate ofH.
4. If λ = α Sh , the highest short root of a non-simply laced root system, W λ is the set of short roots itself. The lower order terms are constants, too. Similarly as above, we find that is an eigenstate ofH. Here ∆ L(S) is the set of long (short) roots.
5. If −λ / ∈ W λ then there is another set of functions containing the weight −λ which belongs to the same eigenvalue.
8. Triangularity of type II models follows from the same algebraic reasoning.

Quantum Lax pair and quantum conserved quantities
Historically, Lax pairs for Calogero-Moser models were presented in terms of Lie algebra representations [3,5], in particular, the vector representation of the A r models. However, the invariance of Calogero-Moser models is that of Coxeter group but not that of the asso-

General case
Here we recapitulate the essence of the quantum Lax pair operators for the Calogero-Moser models with degenerate potentials and without spectral parameter. The case with spectral parameter will be discussed briefly in subsection 8.3 in connection with the proof of involution of the quantum conserved quantities for A r model [15]. The quantum Lax pair in this subsection applies to all the degenerate potential cases except for the case of the rational potential with the harmonic force, which will be treated separately in subsection 4.2. For details and a full exposition, see [9]. The Lax operators without spectral parameter are which is essential for deriving quantum conserved quantities. The matrix elements of the operators {ŝ α , α ∈ ∆} and {Ĥ j , j = 1, . . . , r} are defined as follows: The form of the function x depends on the chosen potential, and the function y are defined by (2.12), (2.13). Note that these relations are only valid for the degenerate potentials (2.3).
The underlying idea of the Lax operator L, (4.1), is quite simple. As seen from (4.9), L is a "square root" of the Hamiltonian. Thus one part of L contains p which is not associated with roots and another part contains x(ρ · q), a "square root" of the potential V (ρ · q), which being associated with a root ρ is therefore accompanied by the reflection operatorŝ ρ .
Another explanation is the factorised Hamiltonian H (2.5). We obtain, roughly speaking, L ∼ √ H ∼ p + i ∂W ∂qŝ and the property of reflectionŝ 2 = 1 explains the sign change in the first term in (2.5).
It is straightforward to show that the quantum Lax equation Independent conserved quantities appear at such power n that n = 1 + exponent (4.8) of each root system. These are the degrees at which independent Coxeter invariant polynomials exist. There are r exponents for each root system ∆ of rank r. Thus we have r independent conserved quantities in Calogero-Moser models. We list in Table II these powers for each root system. In particular, the power 2 is universal to all the root systems and the quantum Hamiltonian (2.7) is given by where the constant C R is the quadratic Casimir invariant, which depends on the representation. It is defined by Some remarks are in order.

The Lax pair is Coxeter covariant:
which ensures the Coxeter invariance of the conserved quantities.

Lax pairs can be written down in various representations and the quantum conserved
quantities Q n do depend on the representations, in general. If necessary, we denote by Q R n the explicit representation dependence.

The availability of plural representations of the Lax pair and the conserved quantities
is essential for the completeness of the set of conserved quantities as polynomials of the momentum operators. For example, let us consider the case of D r with even r, which has two independent conserved quantities at power r, see Table II

Rational potential with harmonic force
The quantum Lax pair for the type V models needs a separate formulation. The explicit form of the Hamiltonian is (4.14) The canonical equations of motion are equivalent to the following Lax equations for L ± : in which (see section 4 of [7]) M is the same as before (4.2), and L ± and Q are defined by with L,Ĥ as earlier (4.1), (4.4). If we define hermitian operators L 1 and L 2 by as before. Such quantum conserved quantities have been previously reported for models based on A r root systems [16,18]. It should be remarked that Ts(L n 2 ) is no longer the same as Ts(L n 1 ) due to quantum corrections. It is elementary to check that the first conserved quantities give the Hamiltonian (4.14) H ∝ Ts(L 1 ) = Ts(L 2 ) + const.  They are hermitian conjugate to each other with respect to the standard hermitian inner product of the states defined in P W : We will show later in section 6, (6.15) that the creation (annihilation) operators commute among themselves: so that the state (5.1) does not depend on the order of the creation.
The proof is very simple. By using (4.15) we obtain follows after taking the total sum. This simply says that B ± n creates (annihilates) a state having energy nω. In other words we havê Moreover, it is trivial to show that which implies that the ground state is annihilated by all the annihilation operators Some remarks are in order.
1. In most cases the energy levels are highly degenerate. The above basis is neither orthogonal nor normalised.

The independence of the creation-annihilation operators can also be shown in a similar
way to that of the conserved quantities. As with the conserved quantities, plural representations are necessary to define the full set of creation-annihilation operators in some models. This aspect will be discussed in later sections in connection with the ℓ operators.
3. Reflecting the universality of the first exponent, f 1 = 2, the creation and annihilation operators of the least quanta, 2ω, exist in all the models. They form an sl(2, R) algebra together with the HamiltonianĤ: The sl(2, R) algebra was discussed by many authors (see, for example, [12,30,17,21] and others) in connection with the models based on classical root systems. We will show later in subsection 7.2 that the states created by B + 2 (b + 2 ) only can be expressed by the Laguerre polynomial: It is trivial to verify that L (Ẽ 0 −1) n (ωq 2 ) is an eigenfunction ofH (3.5) The normalisation of the state is also dictated by the sl(2, R) relations. The Laguerre polynomial wavefunctions appear as 'radial' wavefunctions in all the cases [31]. This will be shown explicitly for for the rank two models given in subsection 7.3.

4.
As is emphasised by Perelomov [12] and Gambardella [30] the sl(2, R) algebra and the corresponding Laguerre wavefunctions are more universal than Calogero-Moser models.
They arise when the potentials are homogeneous functions in q of degree −2 with the confining harmonic force.
5. The operators {Q n } and {B ± n } do not form a Lie algebra. They satisfy interesting non-linear relations, for example, This tells, for example, that although B + n and b + 2 create different units of quanta n and 2, they are not independent Clarification of the algebraic structure (5.15) for each root system is wanted.

ℓ operators
In this section we will show the equivalence of the quantum conserved quantities obtained in the Lax operator formalism of section 4 and those derived in the 'commuting differential operators' formalism initiated by Dunkl [10] and followed by many authors. Again the equivalence is universal, applicable to the models based on any root systems. We propose to call the operators in the latter approach simply 'ℓ operators', since they are essentially the same as the L operator in the Lax pair formalism and that they are not mutually commuting, as we will show presently, when the interaction potentials are trigonometric (hyperbolic), (6.14). Although these two formalisms are formally equivalent, the ℓ operator formalism has many advantages over the Lax pair one. Roughly speaking, the 'vector-like' objects ℓ µ 's are easier to handle than the matrix L µν .
Let us fix a representation R of the Coxeter group G ∆ and define for each element µ ∈ R the following differential-reflection operator It is linear in µ and Coxeter covariant They are hermitian operators, ℓ † µ = ℓ µ , with respect to the standard inner product for the states (5.4).
Commutation relations among ℓ operators can be evaluated in a similar manner as those appearing in the Lax pair [7,9], that is, by decomposing the roots into two-dimensional sub-root systems. We obtain One important use of the ℓ operators is the proof of involution of quantum conserved quantities. For type I models Heckman [21] gave a universal proof based on the commutation relation (6.7): [ℓ n µ , ℓ m ν ]ψ = 0, rational model. (6.8) This was the motivation for the introduction of the commuting differential-reflection operators by Dunkl, [10]. In fact, Dunkl's and Heckman's operators were the similarity transformation of ℓ µ by the ground state wavefunction e W : As for type V models, we define ℓ ± corresponding to L ± (4.16): The conserved quantities are expressed as polynomials in ℓ ± operators: Likewise the creation and annihilation operators B ± n (5.2) are expressed as The commutation relations among ℓ ± operators are easy to evaluate, since ℓ operators commute in the rational potential models (6.7): From these it follows that the creation (annihilation) operators B ± n do commute among themselves: It is also clear that ℓ ± µ / √ 2ω are the 'deformation' of the creation (annihilation) operators of the ordinary multicomponent harmonic oscillators. In fact we have ℓ + µ e W = 2iω(µ · q) e W and ℓ − µ e W = 0. (6.16) In the next section we present an alternative scheme of algebraic construction of excited states of type V models by pursuing the analogy that ℓ ± are the creation and annihilation operators of the unit quantum. This method was applied to the A r models by Brink et. al and others [17,15,18].
7 Algebraic construction of excited states II

Operator solution of the triangular Hamiltonian
In subsection 3.1, we have shown that an eigenfunction of H with eigenvalue Nω is given by in which P N (q) is a Coxeter invariant polynomial in q of homogeneous degree N andP N −2 (q) is a Coxeter invariant polynomial in q of degree N −2 and lower. The non-leading polynomial P N −2 (q) is completely determined by the leading one P N (q) due to the triangularity. This solution can be written in an operator form as follows.
Suppose P N (q) is expressed as We obtain a Coxeter invariant polynomial in the creation operators ℓ + by replacing q · µ by ℓ + µ /(2iω): P N (q) ⇒ 1 (2iω) N P N (ℓ + ). This creates the above eigenfunction of H from the ground state: The proof is again elementary. By using the commutation relations among ℓ ± operators it is straightforward to derive the explicit expression of the Hamiltonian in terms of ℓ ± : (7.4) in which the second term vanishes upon acting on a Coxeter invariant state. Next we obtain which is an ℓ operator version of (4.15). Since a commutator is a derivation, we obtain (7.6) in which the first term in r.h.s. vanishes due to the Coxeter invariance of P N . Thus we arrive at the desired commutation relation and the eigenvalue equation Since the action of the creation operators on the ground state is our assertion (7.3) is proved. It should be stressed that in this formalism the Coxeter invariance of the polynomial P is important but not how it is obtained.
Like the above Hamiltonian (7.4), the ℓ operator formulas of higher conserved quantities (6.12) contain extra terms: Here V T stands for vanishing terms when they act on a Coxeter invariant state. The same is true for most formulas derived in section 6.

States Created by B + 2
Here we derive the explicit forms of the subseries of eigenstates obtained by multiple applications of the least quanta creation operator B + 2 (5.2), or its normalised form b + 2 (5.11). It is convenient to work with the similarity transformed operator Let f (u) be an arbitrary function of u ≡ ωq 2 , then it is Coxeter invariant. We find andb By using the Laguerre differential equation (3.21) and the recurrence formulas of the Laguerre polynomial L (α) n (u), Thus the induction is proved and we arrive at (5.12). The orthogonality of the states can be easily understood as the du part of the measure e 2W d r q = e −u uẼ 0 −1 dudΩ, dΩ : angular part, is the proper weight function for the Laguerre polynomial L (Ẽ 0 −1) n (u). (v j · q), (7.20) where {v j } is a set of vectors given in (B.3). If we introduce the two-dimensional polar coordinates system 4 for q q = r(sin θ, cos θ), (7.21) then the principal Weyl chamber is

Explicit solutions of the rank two models
The two Coxeter invariant variables read: (v j · q) = 2( r 2 ) m cos mθ, (7.23) and the latter variable varies the full range, −1 < cos mθ < 1 in the P W . Thus solving the eigenvalue equation forH (3.6) by separation of variables in the polar coordinate system is compatible with Coxeter invariance. We adopt as two independent variables u ≡ ωr 2 , z ≡ cos mθ.
The z part admits polynomial solutions in which the indices m(ℓ + g) and m(ℓ + (g o + g e )/2) can be written in a unified way as mℓ +Ẽ 0 − 1. Thus the eigenstates of the Hamiltonian are obtained:
8 Involution of conserved quantities 8.1 Universal proof of involution of quantum conserved quantities for type I, II and III models Here we present a proof of involution of quantum conserved quantities {Q n } derived from the universal Lax pair in subsection 4.1 for type I, II and III models. The proof is applicable to all models based on any root systems. Though a universal proof of involution for type I models is given by Heckman [21] as recapitulated in section 6, we believe the universal proof applicable to type II and III models as well is new. It depends on a theorem by Olshanetsky and Perelomov [11]. Our own contribution is that we have provided a universal Lax pair and conserved quantities satisfying all the requirements of the theorem.
Liouville's theorem states the complete integrability as the existence of an involutive set of conserved quantities as many as the degrees of freedom. We have already given conserved quantities {Q n } (4.7) independent and as many as the degrees of freedom (see Appendix B).
3. Scaling property for those of type I models: as a consequence of the above point.
4. For type II and III models, the asymptotic behaviour near the origin: We need to show the vanishing of Let us decompose J lm into the leading part and the rest: where the sum is taken over all permutations of indices σ(t, k 1 , . . . , k s ) = (j 1 , . . . , j s+1 ). In [33] it is proved (Lemma 2.5, p. 407) that the system (8.8) has only polynomial solutions.
Then Olshanetsky and Perelomov argue that for type I models the scaling property tells that c k 1 ,...,ks (κq) = κ s−l−m c k 1 ,...,ks (q). Since s < l + m (8.5), it follows that the only polynomial solution satisfying the condition is the null polynomial. Thus we obtain c j 1 ,...,js (q) = 0 ⇒ J 0 lm = 0 and J lm = 0. The same results follow for type II and III models by considering the asymptotic behaviour for |q| → 0. Thus the involution of all the conserved quantities {Q n } is proved. This result also implies the involution of classical conserved quantities by taking the classical limit (h → 0).

Rational models with the harmonic confining force
In this subsection we show the involution of quantum conserved quantities for the type V models based on the root systems of classical Lie algebras. The method is a straightforward generalisation of the one developed by Polychronakos on the A r model. This is made possible by the availability of the universal Lax pair formalism [7,9], in particular the root type and minimal type Lax pairs. We apply it to the models based on B r and D r root systems. For the rational potential, the B r , C r and BC r models are equivalent. Let us choose R = {±e j ∈ R r | e j · e k = δ jk } as the representation space of the Coxeter group consisting of orthogonal vectors and their negatives. They are the set of short roots of B r and the set of vector weights of D r in the parametrisation of roots given in (B.1) and (4.13), respectively.
The conserved quantities in the ℓ operator form are given by in which we abbreviate ℓ ± e j as ℓ ± j . In this case the commutation relation among ℓ operators (6.14) are greatly simplified thanks to the orthogonality of {e j }'s and the explicit forms of the roots: (8.10) in whichg = g L B r model, g D r model,š jk ≡š e j −e k ,s jk ≡š e j +e k . (8.11) By repeating them we obtain , m jk ≡š jk +s jk = m kj , j = k, (8.12) Here and later the identity and (8.14) to Summing over j and k and adding (8.15) and (8.16) together with the interchange of the dummy indices j ↔ k in the latter produces

Lax pair with spectral parameter
In the theory of classical Calogero-Moser models, Lax pair with spectral parameter (ξ) plays an important role, in particular, in elliptic potential models [34,8,6,7] for derivation of spectral curves, etc. In quantum theory, however, the meaning and use of the spectral parameter are yet to be established, partly because of the underdeveloped stage of the quantum models with elliptic potentials. Here we point out a small use of the quantum Lax pair with spectral parameter in the quantum model with trigonometric potential. Namely, it accounts for the useful trick by Polychronakos [15] for the proof of the involution of conserved quantities in trigonometric A r model. (Now we have a universal proof of involution for type I, II and III models, see the previous subsection.) From the theory of the generalised Lax pair [7] and its quantum version for degenerate potential models [9], we find that the L operator can contain one additional complex parameter ξ: in which the function x are given in Table I for the degenerate potentials. With the same M operator as before, the quantum equations of motion can be written in a matrix form: In other words, the ξ dependent part decouples. This allows us to define a one parameter family of conserved quantities ξ Q n = Ts(L ξ ) n , n = 1, 2, . . . , (8.21) which turns out to be a ξ dependent sum of Q n and the lower order conserved quantities Q m , m < n. A special limit ξ → −i∞ in the trigonometric models provides a convenient combination which allows easy proof of involution in the A r model. (In the rational model, the limit reduces to the Lax pair without spectral parameter. In the hyperbolic models this limit is ill-defined.) In the rest of this subsection we consider only the trigonometric potential models. Let us denote the limiting L ξ operator by L ∞ which reads L ∞ = L + a ρ∈∆ + g |ρ| |ρ ·Ĥ|ŝ ρ (8.22) and the corresponding ℓ operators are given by For the A r model in the vector representation R = {µ j ∈ R r , j = 1, . . . , r + 1} with the standard normalisation of roots ρ 2 = 2, the above expression simplifies to ℓ ∞ j = ℓ j + t j , t j = ag k =jš jk , (8.24) in which as before we abbreviate ℓ µ j as ℓ j andš jk ≡š µ j −µ k . These are the operators introduced in [15], ℓ j ↔ π j , ℓ ∞ j ↔π j . The Lax operator with the spectral parameter gives an 'explanation' for the rather ad hoc introduction ofπ j . It is straightforward to show [ℓ j , ℓ k ] = [t j , t k ], (8.25) which leads to This has the same structure as (8.12) in the previous subsection. By repeating the same argument we arrive at [15] [ ∞ Q n , ∞ Q m ] = 0, n, m = 1, . . . , r, ∞ Q n =

Summary and comments
We have discussed various issues related to quantum integrability of Calogero-Moser models based on all root systems. These are construction of quantum conserved quantities and a unified proof of their involution, the relationship between the Lax pair and the differentialreflection (Dunkl) operators formalisms, construction of excited states by creation operators, etc. They are mainly generalisations of the results known for the models based on A r root systems, which are shown to apply to the models based on any root systems. There are some interesting works discussing the integrability issues of the models based on other classical root systems and the exceptional ones including the non-crystallographic models [32], [35]- [38].
Here we list some comments on interesting issues which are not treated in this paper. The structure and properties of the eigenfunctions of the trigonometric potential models, which are generalisations of the Jack polynomials [25]- [28], will be discussed in future publications.
A comprehensive treatment of Liouville integrability of rational models with harmonic force is wanted. Our starting point, the factorised Hamiltonian (2.5) for degenerate potential models, is closely related with supersymmetry and shape invariance [39,38]. Further investigation in this direction is a future problem. It is a great challenge to formulate various aspects of quantum Calogero-Moser models with elliptic potentials; Lax pair, the differential-reflection operators [40,41], conserved quantities, supersymmetry and excited states wavefunctions.
where s α (β) = β − (α ∨ · β)α, α ∨ ≡ 2α/|α| 2 . (A. 2) The set of reflections {s α , α ∈ ∆} generates a group G ∆ , known as a Coxeter group, or finite reflection group. The orbit of β ∈ ∆ is the set of root vectors resulting from the action of the Coxeter group on it. The set of positive roots ∆ + may be defined in terms of a vector U ∈ R r , with α · U = 0, ∀α ∈ ∆, as those roots α ∈ ∆ such that α · U > 0. Given ∆ + , there is a unique set of r simple roots Π = {α j , j = 1, . . . , r} defined such that they span the root space and the coefficients {a j } in β = r j=1 a j α j for β ∈ ∆ + are all non-negative. The highest root α h , for which r j=1 a j is maximal, is then also determined uniquely. The subset of reflections {s α , α ∈ Π} in fact generates the Coxeter group G ∆ . The products of s α , with α ∈ Π, are subject solely to the relations (s α s β ) m(α,β) = 1, α, β ∈ Π. (A. 3) The interpretation is that s α s β is a rotation in some plane by 2π/m(α, β). The set of positive integers m(α, β) (with m(α, α) = 1, ∀α ∈ Π) uniquely specify the Coxeter group. The weight lattice Λ(∆) is defined as the Z-span of the fundamental weights {λ j }, j = 1, . . . , r, defined by α ∨ j · λ k = δ jk , α j ∈ Π. (A.4) The root systems for finite reflection groups may be divided into two types: crystallographic and non-crystallographic. Crystallographic root systems satisfy the additional condition α ∨ · β ∈ Z, ∀α, β ∈ ∆, (A. 5) which implies that the Z-span of Π is a lattice in R r and contains all roots in ∆. We call this the root lattice, which is denoted by L(∆). These root systems are associated with  12. H 4 : For all powers, the representation consisting of all the 120 roots is enough.