C Duval
email: duval@cpt.univ-mrs.fr

G Valent
email: valent@lpthe.jussieu.fr

A new integrable system on the sphere and conformally equivariant quantization

Keywords: CPT-P051-2010 Classical integrability, projectively equivalent metrics, Stäckel systems, conformally equivariant quantization, quantum integrability

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi-Moser system, we disclose a novel integrable system on the sphere S n , namely the dual Moser system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann-Uhlenbeck and dual Moser systems is insured by means of the conformally equivariant quantization procedure.

Introduction

In the wake of the celebrated results of Moser [START_REF] Moser | Various aspects of the integrable Hamiltonian systems[END_REF] concerning the classical integrability of the geodesic flow on the ellipsoid (first proved by Jacobi in the threedimensional case), and of the Neumann-Uhlenbeck problem on the n-dimensional spheres, we will present a (to our knowledge) new integrable system which relies on a preferred conformally flat metric on S n . This integrable system is actually dual (in the sense of projective equivalence [START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF][START_REF] Matveev | Quantum integrability of Beltrami-Laplace operator as geodesic equivalence[END_REF]) to the above-mentioned Jacobi-Moser system.

Among the techniques found in the literature, we will mention two different constructs that produce integrable systems together with their Poisson-commuting first integrals:

• Bihamiltonian systems initiated by Magri [START_REF] Magri | A simple model of the integrable Hamiltonian equation[END_REF] and Benenti [START_REF] Benenti | Inertia tensors and Stäckel systems in the Euclidean spaces[END_REF], and further elaborated by, e.g., Ibort, Magri, and Marmo [START_REF] Ibort | Bihamiltonian structures and Stäckel separability[END_REF], Falqui and Pedroni [START_REF] Falqui | Separation of Variables for Bi-Hamiltonian Systems[END_REF].

• Projectively equivalent systems discovered by Levi-Civita [START_REF] Levi-Civita | Sulle trasformazioni delle equazioni dinamiche[END_REF], and geometrically developed by Tabachnikov [START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF][START_REF] Tabachnikov | Ellipsoids, complete integrability and hyperbolic geometry[END_REF], Topalov and Matveev [START_REF] Topalov | Geodesic equivalence and integrability[END_REF].

The equivalence of these two theories has been proved by Bolsinov and Matveev [START_REF] Bolsinov | Geometrical interpretation of Benenti systems[END_REF].

We have chosen to work using Tabachnikov's approach. With the help of his general construction [START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF][START_REF] Ovsienko | Projective Differential Geometry Old And New: From The Schwarzian Derivative To The Cohomology Of Diffeomorphism Group[END_REF] providing, e.g., the Jacobi-Moser first integrals, and, adopting a "dual" approach, we derive the Poisson-commuting first integrals of the new geodesic flow, which we call the dual Moser system. This enables us to provide a global expression for these new first integrals.

Our dual Moser system turns out to be locally Stäckel (in ellipsoidal coordinates); it shows up as a "mirror image" of Jacobi-Moser relatively to Neumann-Uhlenbeck (see Table 1).

In contradistinction to the Jacobi-Moser system, conformal flatness of the new system is a fundamental input at the classical, and at the quantum level as well.

To deal with quantum integrability of these systems, we will resort to its commonly accepted definition, namely that the quantized first integrals should still be in involution. This, of course, leaves open the choice of an adapted quantization procedure.

It has been shown [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF] that Stäckel systems, using Carter's quantization prescription [START_REF] Carter | Killing tensor quantum numbers and conserved currents in curved space[END_REF], do remain integrable at the quantum level provided Robertson's condition holds [START_REF] Robertson | Bermerkung über separierbare Systeme in der Wellenmechanik[END_REF]. It is therefore worthwhile to study quantum integrability making use of a genuine quantization theory that takes into account the conformal geometry underlying the dual Moser system: the conformally equivariant quantization [START_REF] Duval | Conformally equivariant quantization: existence and uniqueness[END_REF][START_REF] Duval | Conformally equivariant quantum Hamiltonians[END_REF]. One striking discovery is that the dual Moser system passes the quantum test using the conformally equivariant quantization. We note that the same is true for the Neumann-Uhlenbeck system.

Prolegomena

Let us recall that two independent metrics on a Riemannian manifold are said to be projectively equivalent if they have the same unparametrized geodesics. A shown in, e.g., [START_REF] Topalov | Geodesic equivalence and integrability[END_REF][START_REF] Bolsinov | Geometrical interpretation of Benenti systems[END_REF] this entails that the associated geodesic flows are Liouville-integrable. We will resort to this pathbreaking result in the specific, historical, yet fundamental example of the geodesic flow of the ellipsoid.

The key point of our approach to Liouville/quantum integrability of the geodesic flow of the ellipsoid, E n , lies in the fact that E n = {Q ∈ R n+1 | n α=0 (Q α ) 2 /a α = 1} with metric g 1 = n α=0 (dQ α ) 2 | E n admits, as a matter of fact, another independent, and projectively equivalent Riemannian metric, g 2 , that we will introduce shortly.

Put Q α = q α √ a α for all α = 0, . . . , n, so that we have q • q = n α=0 (q α ) 2 = 1. The mapping Q → q : E n → S n is a diffeomorphism and the metric on S n , induced from the Euclidean ambient metric, reads now 1 where g is, at the moment, viewed as a (flat) metric on R n+1 . The equations of the geodesics of the ellipsoid are well-known, and retain the form qα + Γ α βγ qβ qγ = 0, (

for all α = 0, . . . , n, where the Christoffel symbols of (S n , g 1 ) are given by

Γ α βγ = q α a α δ βγ q 2 λ /a λ (1.3)
for all α, β, γ = 0, . . . , n. (See also Equation (2.12) yielding the associated geodesic spray.) We note that the constraint equation q2 + q • q = 0 (1.4) is indeed satisfied by (1.2). Following [START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF][START_REF] Topalov | Geodesic equivalence and integrability[END_REF], let us define, on S n , the conformally flat metric

g 2 = g| S n with g = 1 q 2 β /a β n α=0 dq 2 α (1.5)
where the conformally flat metric g is defined on R n+1 \{0}.

The equations of the geodesics for the latter metric are readily found by using the expression of the Christoffel symbols of g, namely

Γ α βγ = 1 q 2 λ /a λ δ βγ q α a α -δ α β q γ a γ -δ α γ q β a β (1.6)
for all α, β, γ = 0 . . . , n. One obtains the equations of the geodesic of (R n+1 \{0}, g), viz.,

qα + q α a α q2 β q 2 γ /a γ = 2 qα q β qβ /a β q 2 γ /a γ (1.7)
1 To avoid clutter, we will oftentimes write q α ≡ q α , as no confusion occurs in Euclidean space.

for all α = 0, . . . , n. The latter, suitably restrained to S n , is precisely Equation (1.2) with a different parametrization (again, the constraint (1.4) is duly preserved by Equation (1.7)); the metrics g 1 and g 2 are projectively equivalent. We will put this fact in broader perspective within Section 2.

Remark 1.1. The Γ α βγ and Γ α βγ , given by (1.3) and (1.6) respectively, may be viewed as the components of two projectively equivalent linear connections ∇ and ∇ on R n+1 \{0}. While ∇ is clearly the Levi-Civita connection of the metric g, the connection ∇ is, instead, a Newton-Cartan connection (see, e.g., [START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF]). This means that ∇ is a symmetric linear connection that parallel-transports a (spacelike) contravariant symmetric 2-tensor γ = n α,β=0 γ αβ ∂ q α ⊗ ∂ q β and a (timelike) 1-form θ = n α=0 θ α dq α spanning ker(γ). Here, the degenerate "metric" is given by

γ αβ = 1 a α δ αβ - q α q β a α a β 1 q 2 λ /a λ and θ α = q α . (1.8)

Main results

The main results of our article can be summarized as follows.

Theorem 1.2. The geodesic flow on (T * S n , α dp α ∧ dq α ) above the conformally flat manifold (S n , g 2 ) is Liouville-integrable, and admits the following set of Poissoncommuting first integrals

F α = q 2 α n β=0 a β p 2 β + β =α (a α p α q β -a β p β q α ) 2 a α -a β (1.9)
with α = 0, . . . , n. We will call the system (F 0 , . . . , F n ) the dual Moser system.

This theorem follows directly from Propositions 2.7, 2.8, and 2.10. Let us call D 1 2 , 1 2 (S n ) the space of differential operators on S n with arguments and values in the space of 1 2 -densities of the sphere S n ; we, likewise, denote by Pol(T * S n ) the space of fiberwise polynomial functions on T * M. It has been proved [START_REF] Duval | Conformally equivariant quantization: existence and uniqueness[END_REF] that there exists a unique invertible linear mapping Q1 This last result stems from Theorem 3.4, Propositions 3.6, and 3.12.

2 , 1 2 : Pol(T * S n ) → D 1 2 , 1 2 (S n ) that (i)

Plan of the article

The paper is organized as follows.

Section 2 gives us the opportunity to introduce three distinguished classically integrable systems on the sphere, namely, the Jacobi-Moser system, its dual counterpart, and the Neumann-Uhlenbeck system. The construction of the set of mutually Poisson-commuting first integrals is reviewed and specialized to the case of the dual Moser system. The resulting system is shown to be Stäckel, the ellipsoidal coordinates being the separating ones.

In Section 3 we address the quantum integrability issue of these systems, in terms of the conformally equivariant quantization. We prove that the Neumann-Uhlenbeck and the dual Moser systems are, indeed, quantum integrable using this quantization method, well adapted to the conformal flatness of configuration space.

Section 4, provides a conclusion to the present article, and gathers some perspectives for future work.

2 A novel integrable system on the sphere: the dual Moser system

Projectively equivalent metrics and conservation laws

Let us recall, almost verbatim, Tabachnikov's construction [START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF] of a maximal set of independent Poisson-commuting first integrals for a special Liouville-integrable system, namely a bi-Hamiltonian system associated with two projectively equivalent metrics, g 1 and g 2 , on a configuration manifold M. 2 See also [START_REF] Matveev | Quantum integrability of Beltrami-Laplace operator as geodesic equivalence[END_REF][START_REF] Bolsinov | Geometrical interpretation of Benenti systems[END_REF] for an alternative construction.

We start with two Riemannian manifolds (M, g 1 ) and (M, g 2 ) of dimension n. The tangent bundle T M is endowed with two distinguished 1-forms λ 1 and λ 2 , namely λ N = g * N θ, where θ is the canonical 1-form of T * M, and g N : T M → T * M is viewed as a bundle isomorphism. We then write, locally, λ N = g N ij u i dx j where g N = g N ij (x)dx i ⊗ dx j for N = 1, 2. Likewise, the Lagrangian functions to consider are the fiberwise quadratic polynomials L N = 1 2 g N ij (x)u i u j . Denote by ω N = dλ N the corresponding symplectic 2-forms of T M, and also by X N = X L N the associated geodesic sprays. We have

λ N (X N ) = 2L N and ω N (X N ) = -dL N (2.1)
for all N = 1, 2. The (maximal) integral curves of the vector fields X N on T M project onto configuration space as the geodesics of (M, g N ).

Introduce then the diffeomorphism φ : T M → T M defined by φ(x, u) = (x, ũ) where ũ = u L 1 (x, u)/L 2 (x, u). 3 This diffeomorphism is, indeed, designed to relate the two Lagrangians, viz.,

L 1 = φ * L 2 . (2.2)
Clearly, the two metrics g 1 and g 2 have the same unparametrized geodesics (we write g

1 ∼ g 2 ) iff φ * (X 1 ) ∧ X 2 = 0 (2.3)
i.e., iff the the push-forward φ * (X 1 ) and X 2 are functionally dependent.

The method, to obtain a generating function for the conserved quantities in involution, consists then in singling out, apart from ω 1 , a preferred X 1 -invariant 2-form constructed in terms of ω 2 .

Proposition 2.1. Suppose that g 1 ∼ g 2 , and define

ω ′ 2 = d(L -1 2 2 λ 2 ), then L X 1 (φ * ω ′ 2 ) = 0. (2.4) Proof. We have L X 1 (φ * ω ′ 2 ) = d((φ * ω ′ 2 )(X 1 )) = φ * d(ω ′ 2 (φ * X 1 )) = φ * d(h ω ′ 2 (X 2 )), for some function h (see (2.3)). The definition of ω ′ 2 then readily yields L X 1 (φ * ω ′ 2 ) = φ * d(h d(L -1 2 2 λ 2 )(X 2 )) = φ * d(h(-1 2 L -3 2 2 (dL 2 ∧ λ 2 )(X 2 ) + L -1 2 2 ω 2 (X 2 ))) = 0 in view of (2.1). So, the sought extra X 1 -invariant 2-form is φ * ω ′ 2 . (Note that L X 1 (φ * ω 2 ) = 0.
) It enters naturally into the definition of a "generating function" f t ∈ C ∞ (T M, R) of first-integrals given below.

Corollary 2.2. The function

f t = (t -1 ω 1 + φ * ω ′ 2 ) n ω n 1 (2.5) is X 1 -invariant whenever t = 0.

The example of two projectively equivalent geodesic sprays for the n-sphere

We recall the construction of the first-integrals in involution yielding the Liouvilleintegrability of the geodesic flow on T E n ∼ = T S n . Let us parametrize T R n+1 by the couples q, v ∈ R n+1 . The constraints defining the embedding T S n ֒→ T R n+1 are

q 2 := n α=0 q 2 α = 1 and v • q := n α=0 v α q α = 0. (2.6)
As already mentioned in Section 1, the (unparametrized) geodesics of the ellipsoid E n with semi-axes4 a 0 , a 1 , • • • , a n are precisely given by those of the sphere S n = {q ∈ R n+1 | n α=0 q 2 α = 1} endowed with either projectively equivalent metrics

g 1 = n α=0 a α dq 2 α S n & g 2 = 1 B n α=0 dq 2 α S n (2.7)
where

B = n α=0 q 2 α a α . (2.8)
The corresponding Lagrangians on T S n are respectively

L 1 = 1 2 A & L 2 = 1 2B n α=0 v 2 α (2.9)
where

A = n α=0 a α v 2 α .
(2.10)

The associated Cartan 1-forms then read in this case

λ 1 = n α=0 a α v α dq α & λ 2 = 1 B n α=0 v α dq α . (2.11)
Proposition 2.3. (i) The geodesic sprays for the metrics g N are given by the Hamiltonian vector fields X N = X L N , for N = 1, 2, namely

X 1 = n α=0 v α ∂ ∂q α - v 2 B n α=0 q α a α ∂ ∂v α (2.12)
and

X 2 = n α=0 v α ∂ ∂q α - 1 B n α=0 2v α n β=0 v β q β a β -v 2 q α a α ∂ ∂v α (2.13)
respectively.

(ii) Condition (2.3) holds true, implying g 1 ∼ g 2 .

Proof. Using (2.1) together with the constraints (2.6), we thus have to solve for X 1 , resp. X 2 , the equation ω N (X N ) + dL N + λ d(q 2 -1) + µ d(v • q) = 0 where λ and µ are Lagrange multipliers. The latter are, in fine, completely determined and readily yield (2.12), resp. (2.13). Now, the diffeomorphism φ : (q, v) → (q, ṽ) introduced in Section 2.1 is given by ṽ = v AB/v 2 ; routine calculation yields φ

* (∂ qα ) = ∂ qα + q α /(a α B) E with E = v α ∂ vα the Euler vector field; also φ * (∂ vα ) = AB/v 2 (∂ vα -v α (v -2 -a α /A) E).
This, along with the constraint v α q α = 0, helps us prove Equation (2.3).

The Jacobi-Moser system

Let us review here, and in some detail, the main result obtained by Tabachnikov [START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF] via the general procedure of Section 2.1, starting with the geodesic flow on (S n , g 1 ).

The diffeomorphism φ of T S n , namely φ(q, v)

= (q, ṽ = v L 1 /L 2 ), is such that ṽ = v AB/v 2 where v 2 = v 2 α ; whence φ * λ 2 = C √ A v α dq α , where C = 1 √ Bv 2 . (2.14) Now, since φ * ω ′ 2 = d(L -1 2 1 φ * λ 2 )
, easy computation then leads to

φ * ω ′ 2 = C √ 2 Ω 1 where Ω 1 = n α=0 dv α ∧ dq α + dC C ∧ n α=0 v α dq α . (2.15)
The function C defined by (2.14) is a first integral of the system, namely

X 1 C = 0. (2.16)
This function C is the Joachimsthal first-integral of the geodesic flow of (S n , g 1 ).

The next step consists in lifting the 1-parameter family of first-integrals (2.5) to T R n+1 by taking advantage of the constraints (2.6), and to put

f t = (t -1 ω 1 + Ω 1 ) n ∧ d(v.q) ∧ q • dq ω n 1 ∧ d(v.q) ∧ q • dq (2.17)
with a slight abuse of notation using the constancy of C (see (2.16)) in (2.15). Elementary calculation yields

f t = ω n * -(n/v 2 ) ω n-1 * ∧ v • dv ∧ v • dq ∧ q • dv ∧ q • dq ω n 1 ∧ (dv.q + v • dq) ∧ q • dq (2.18)
where

ω * = b α dv α ∧ dq α -(1/v 2 ) v • dv ∧ v • dq, together with b α = t -1 a α + 1
, for all α = 0, . . . , n. The following lemma [START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF] will be used to complete the calculation.

Lemma 2.4. Let ω = c α dv α ∧ dq α and ω 0 = dv α ∧ dq α , then ω n ∧ q • dv ∧ q • dq ω n+1 0 = n! n α=0 c α n α=0 q 2 α c α ω n-1 ∧ v • dv ∧ v • dq ∧ q • dv ∧ q • dq ω n+1 0 = (n -1)! n α=0 c α α<β (v α q β -v β q α ) 2 c α c β .
Using the two above formulae, we find f t = N/D where

N = n! b α q 2 α b α - n(n -1)! v 2 b α α<β (v α q β -v β q α ) 2 b α b β D = n! a α q 2
α /a α . This entails f t = g t /C (up to a constant overall factor), where C is the Joachimsthal first-integral, and

g t = v 2 n α=0 q 2 α b α - 1 2 α =β (v α q β -v β q α ) 2 b α b β .
Taking into account the expression b α = t -1 a α + 1, and the constraints (2.6), we end up with

g t = n α=0 a α v 2 α a α + t - n α=0 a α v 2 α a α + t n α=0 a α q 2 α a α + t + n α=0 a α v α q α a α + t 2 . (2.19)
At last, the first-integrals F α defined by

g t = n α=0 F α a α + t (2.20)
are easily found to be

F α = a α v 2 α + β =α a α a β (v α q β -v β q α ) 2 a α -a β . (2.21)
These are the Jacobi-Moser first-integrals. In terms of the momenta p α = a α v α (see (2.11)), they read

F α = p 2 α a α + β =α (p α a β q β -p β a α q α ) 2 a α a β (a α -a β ) . (2.22)
Remark 2.5. We indeed recover the Moser first integrals

F α = P 2 α + β =α (P α Q β -P β Q α ) 2 a α -a β (2.23)
by means of the canonical transformation (p, q) → (P, Q) where P α = p α / √ a α , and

Q α = q α √ a α .

The dual Moser system

Let us now adopt a "dual" standpoint by exchanging the rôle of the two projectively equivalent metrics on the n-sphere, i.e., by letting g 1 ↔ g 2 in the above derivation. In doing so, we will work out a complete set of commuting first-integrals of the geodesic flow on the conformally flat manifold (S n , g 2 ). This will turn out to provide a new integrable system on the n-sphere.

The general construction

Let us now apply the general procedure outlined in Section 2.1 starting with the geodesic flow on (S n , g 2 ), and replacing mutatis mutandis all reference to g 1 by that of g 2 .

We first need to work out the expression of the 2-form

φ * ω ′ 1 in Proposition 2.1. The diffeomorphism φ of T S n , viz., φ(q, v) = (q, ṽ = v L 2 /L 1 ), is ṽ = v v 2 /(AB) where, again, v 2 = v 2 α . We hence get φ * λ 1 = v 2 /(AB) a α v α dq α . Now, since φ * ω ′ 1 = d(L -1 2 2 φ * λ 1 ), thanks to φ * L 1 = L 2 , we are led to φ * ω ′ 1 = 2 A Ω 2 where Ω 2 = n α=0 a α dv α ∧ dq α - dA 2A ∧ λ 1 , (2.24)
with λ 1 = a α v α dq α (see (2.11)). We then find the new Joachimsthal first-integral, J, of the geodesic flow of (S n , g 2 ) with the help of the expression (2.13) of the geodesic spray X 2 . In fact, easy calculation yields X 2 A = (2A/B)X 2 (B), hence X 2 J = 0 where

J = A B 2 . (2.25)
Utilizing this constant of the motion, and Equation (2.4), we obtain, see (2.24),

L X 2 Ω 2 = 0 where Ω 2 = Ω 2 B .
(2.26)

Again, and to ease the calculation, we will lift the 1-parameter family of firstintegrals (2.5) to T R n+1 , using the constraints (2.6), and put this time

f t = (t ω 2 + Ω 2 ) n ∧ d(v • q) ∧ q • dq ω n 2 ∧ d(v • q) ∧ q • dq . (2.27)
We trivially get

f t = (t ω 2 + Ω 2 ) n ∧ q • dv ∧ q • dq ω n 2 ∧ q • dv ∧ q • dq (2.28)
where

ω 2 = Bω 2 , i.e., ω 2 = dv α ∧ dq α -d log B ∧ v α dq α and Ω 2 is as in (2.

24).

Let us mention the following somewhat technical lemma.

Lemma 2.6. Upon defining ω * = b α dv α ∧ dq α , where b α = a α + t, for every α = 0, 1, . . . , n, we have

(t ω 2 + Ω 2 ) k = ω k * -k ω k-1 * ∧ t d log B ∧ v • dq + 1 2 d log A ∧ λ 1 +k(k -1) ω k-2 * ∧ t d log B ∧ v • dq ∧ 1 2 d log A ∧ λ 1
for all k = 1, . . . , n, and

ω n-1 * = (n -1)! α<β γ =α,β b γ dv γ ∧ dq γ
A somewhat demanding computation yields

f t = ω n * -n ω n-1 * ∧ dA/(2A) ∧ λ 1 ∧ q • dv ∧ q • dq ω n 2 ∧ q • dv ∧ q • dq . (2.29)
Resorting to Lemmas 2.4 and 2.6 in order to evaluate f t , we obtain the partial result

ω n * ∧ q • dv ∧ q • dq = n! n β=0 b β n α=0 q 2 α b α ω n+1 0 (2.30)
where ω 0 = dv α ∧ dq α . Likewise, some more effort is needed to find

ω n-1 * ∧ 1 2 d log A∧λ 1 ∧q•dv∧q•dq = (n-1)! n γ=0 b γ α<β (a α v α q β -a β v β q α ) 2 b α b β ω n+1 0 A . (2.31)
We just have to plug Equations (2.30) and (2.31) into the expression (2.28) to find5

f t = n γ=0 b γ n α=0 q 2 α b α - 1 A α<β (a α v α q β -a β v β q α ) 2 b α b β . (2.32)
We will, again, deal with the rescaled first-integral

g t = J b α f t (2.33)
where J is an (2.25), as a generating function of the sought conservative system F 0 , F 1 , . . . , F n . Using the definition b α = a α +t, and Equation (2.20), we readily prove that the geodesic flow on T S n above (S n , g 2 ) admits the following first integrals, viz.,

F α = 1 B 2 Aq 2 α + β =α (a α v α q β -a β v β q α ) 2 a α -a β (2.34)
with α = 0, 1, . . . , n, where A and B are given by Equations (2.10) and (2.8), respectively.

With the help of the bundle isomorphism g -1 2 : T * S n → T S n provided by the metric g 2 , we can pull-back the previous first integrals to the cotangent bundle of S n . Whence the following result.

Proposition 2.7. The geodesic flow on the cotangent bundle of (S n , g 2 ) admits the following set of first integrals, viz.,

F α = q 2 α n β=0 a β p 2 β + β =α (a α p α q β -a β p β q α ) 2 a α -a β (2.35)
with α = 0, . . . , n.

In the next section we will prove that these first integrals are actually independent and are mutually Poisson commuting.

Liouville integrability of the unconstrained system

From now on, we choose to work in a purely Hamiltonian framework which will turn out to be well-suited to the quantization procedure that we will examine in the next section.

Let us recall that the Hamiltonian of the system is given by

H = 1 2 n α,β=0 g αβ 2 p α p β = 1 2 B n α=0 p 2 α (2.36)
where B = n α=0 q 2 α /a α (see (2.8)). We will show that the new set (2.35) of first-integrals of motion indeed turns the geodesic flow on the sphere (S n , g 2 ) into an integrable system dual to the Jacobi-Moser geodesic flow on the ellipsoid.

Proposition 2.8. The functions F 0 , . . . , F n of (T * R n+1 , n α=0 dp α ∧ dq α ) given by (2.35) are in involution, namely

{F α , F β } = 0 (2.37)
for all α, β = 0, . . . , n. Moreover the following holds true

{H, F α } = 2 B a α p 2 α -q 2 α β p 2 β γ p γ q γ . (2.38) Proof. Write F α = A α + B α with A α = q 2 α J, J = n β=0 a β p 2 β , (2.39)
where J is the Joachimsthal first integral (2.25), and

B α = β =α M 2 αβ a α -a β , M αβ = a α p α q β -a β p β q α .
(2.40)

One can check the following relationships

{A α , A β } = -4Jq α q β M αβ , {A α , B β } = 4Ja α q α q β M αβ a α -a β , {B α , B β } = 0,
for all α, β = 0, . . . , n, which readily imply Equation (2.37).

Let us furthermore observe that we have the following Poisson brackets, viz.,

{H, A α } = 2JBq α p α -4q 2 α H B n β=0 p β q β and {H, B α } = -2JBq α p α + 2Ba α p 2 α n β=0 p β q β ,
which proves Equation (2.38). The proof is complete.

Remark 2.9. Notice in contradistinction to the Jacobi-Moser case, that (i) the metric g given by (1.5) on the ambient space R n+1 \{0} is no longer flat, and (ii) the conservation relations {H, F α } = 0 for α = 0, . . . , n are only valid for the constrained system, see (2.38), where p • q = 0.

Let us also mention the interesting relations

n α=0 F α = n α=0 q 2 α n β=0 a β p 2 β (2.41) n α=0 F α a α = n α=0 p α q α 2 (2.42) n α=0 F α a 2 α = -2H + 2 n α=0 p α q α n β=0 p β q β a β , (2.43)
of which the last one leads to another proof of (2.38).

The Dirac brackets

Our goal is now to deduce from the knowledge of (2.35) independent quantities in involution I 1 , . . . , I n on (T * S n , n i=1 dξ i ∧ dx i ) from the symplectic embedding ι : T * S n ֒→ T * R n+1 defined by the constraints Proof. We get, using the Dirac brackets [START_REF] Abraham | Foundations of Mechanics[END_REF][START_REF] Marsden | Introduction to Mechanics and Symmetry[END_REF],

Z 1 (p, q) = n α=0 q 2 α -1 = 0, Z 2 (p, q) = n α=0 p α q α = 0. ( 2
{F α | T * S n , F β | T * S n } = {F α , F β }| T * S n - 1 {Z 1 , Z 2 } [{Z 1 , F α }{Z 2 , F β } -{Z 1 , F β }{Z 2 , F α }] | T * S n (2.45)
for second-class constraints. The denominator {Z 1 , Z 2 } = -2 n α=0 q 2 α does not vanish; one can also check that {Z 1 , F α } = -4 a α p α q α (1 + Z 1 ) and that {Z 2 , F α } = 0 for all α = 0, . . . , n. The fact that {F α , F β } = 0 completes the proof.

The constrained integrable system as a Stäckel system

In order to provide explicit expressions for the sought functions in involution I 1 , . . . , I n , we resort to Jacobi ellipsoidal coordinates x 1 , . . . , x n on S n . Those are defined by

Q λ (q, q) = n α=0 q 2 α a α -λ = - U x (λ) V (λ) (2.46)
where

U x (λ) = n i=1 (λ -x i ) and V (λ) = n α=0 (λ -a α ) (2.47)
and are such that a 0 < x 1 < a 1 < x 2 < . . . < x n < a n .

(2.48)

Notice that Equation (2.46) yields the local expressions

q 2 α (x) = n i=1 (a α -x i ) β =α (a α -a β ) (2.49)
for all α = 0, 1, . . . , n.

Let us mention the following identity, deduced from (2.49), viz.,

∂q α ∂x i = - 1 2 q α a α -x i (2.50)
for all i = 1, . . . , n and α = 0, 1, . . . , n It is easy to show that the induced metric g

2 = (1/B) n α=0 dq 2 α | S n , see (2.7), is indeed given by g 2 = n i,j=1 g ij (x)dx i dx j with g ij (x) = 1 4B n α=0 q 2 α (a α -x i )(a α -x j ) = g i (x) δ ij (2.51)
where, using a result taken from [START_REF] Moser | Various aspects of the integrable Hamiltonian systems[END_REF], we have

g i (x) = - 1 4B U ′ x (x i ) V (x i ) = - 1 4B j =i (x i -x j ) α (x i -a α ) . (2.52)
This metric is actually positive-definite because of the inequalities (2.48). In these ellipsoidal coordinates, we obtain

B = n α=0 q 2 α a α = 1 a 0 x 1 • • • x n a 1 • • • a n .
Upon defining the constrained "momenta" ξ i (for i = 1, . . . , n), via the induced canonical 1-form λ| T * S n = n i=1 ξ i dx i = ι * n α=0 p α dq α , we find

p α (ξ, x) = - q α (x) 2B n i=1 g i (x)ξ i a α -x i . (2.53)
We express, for convenience, the Hamiltonian (2.36) on (T * S n , n i=1 dξ i ∧ dx i ), which is then found to be

H = 1 2 n i=1 g i (x)ξ 2 i (2.54)
where g i (x) = 1/g i (x).

Let us now compute the expression of the conserved quantities (2.35) on T * S n .

Proposition 2.11. The dual Moser conserved quantities (F α | T * S n ) α=0,...,n retain the form

F α | T * S n = a α q 2 α (x) B n i=1 x i g i (x)ξ 2 i a α -x i .
(2.55)

Proof. On the one hand, in view of Equations (2.49) and (2.53), one gets, see (2.39),

A α | T * S n = q 2 α (x) B n i=1 x i g i (x)ξ 2 i , using the identities n α=0 q 2 α a α -x i = 0
for all i = 1, . . . , n.

On the other hand, a similar computation gives

M αβ | T * S n = a α -a β 2B q α (x)q β (x) n i=1 x i g i (x)ξ i (a α -x i )(a β -x i ) , so that B α | T * S n = q 2 α (x) B n i=1 (x i ) 2 g i (x)ξ 2 i a α -x i , proving that F α | T * S n (where F α = A α + B α ) is, indeed, as in (2.55).
Proposition 2.12. The following holds on T * S n , viz As a preparation to the proof that our system is, indeed Stäckel, let us introduce, for convenience, the symmetric functions, σ k (x) and σ i k (x) with x = (x 1 , . . . , x n ), that will be useful in the sequel, namely,

n α=0 F α | T * S n = 1 B n i=1 x i g i (x)ξ 2 i = J| T * S n (2.56) n α=0 F α | T * S n a α = 0 (2.57) n α=0 F α | T * S n a 2 α = -2H. ( 2
U x (λ) ≡ n j=1 (λ -x j ) = n k=0 (-1) k λ n-k σ k (x)
(2.59)

U x (λ) λ -x i ≡ j =i (λ -x j ) = n k=1 (-1) k-1 λ n-k σ i k-1 (x).
(2.60)

We will also use σ k (a) and σ α k (a) with a = (a 0 , a 1 , . . . a n ), which are defined similarly. Let us notice that the previous conserved quantities (2.55) can be written as

F α | T * S n = a α G aα (ξ, x) β =α (a α -a β )
where

G λ (ξ, x) = 1 B n i=1 x i g i (x) j =i (λ -x j ) ξ 2 i .
(2.61) Proposition 2.13. Let the functions I 1 , . . . , I n of T * S n be defined by

G λ (ξ, x) = n k=1 (-1) k-1 λ n-k I k (ξ, x).
(2.62)

Then I k (ξ, x) = n i=1 A i k (x)ξ 2 i with A i k (x) = 1 B x i g i (x)σ i k-1 (x).
(2.63)

Proof. By plugging the definition (2.60) of the symmetric functions σ i k (x) of order k = 0, 1, . . . , n -1 (in the variables (x 1 , . . . , x n ), with the exclusion of index i) into (2.61), one gets the desired result.

Theorem 2.14. The dual Moser system I 1 , . . . , I n on T * S n , given by (2.63), defines a Stäckel system, with Stäckel matrix B = A -1 of the form

B i k (x k ) = (-1) i (x k ) n-i-1 4V (x k ) (2.64)
for i, k = 1, . . . , n. This implies [START_REF] Perelomov | Integrable Systems of Classical Mechanics and Lie Algebras[END_REF] that the functions I 1 , . . . , I n are independent and in involution, which entails that the Stäckel coordinates x 1 , . . . , x n are separating for the Hamilton-Jacobi equation.

Proof. It is obvious from its expression (2.64) that B is a Stäckel matrix [START_REF] Perelomov | Integrable Systems of Classical Mechanics and Lie Algebras[END_REF]. We just need to prove that A is the inverse matrix of B. To this aim we first prove a useful identity. Let us consider the integral in the complex plane

1 2iπ |z|=R z n-i (z -λ) U x (λ) U x (z) dz.
When R → ∞ the previous integral vanishes because the integrand decreases as 1/R 2 for large R (let us recall that i ≥ 1). We then compute this integral using the theorem of residues and get the identity

n k=1 (x k ) n-i U ′ x (x k ) j =k (λ -x j ) = λ n-i . ( 2

.65)

Equipped with this identity let us now prove that n k=1 B i k A k j = δ i j . Multiplying this relation by (-1) j-1 λ n-j and summing over j from 1 to n, we get the equivalent relation

n k=1 B i k n j=1 (-1) j-1 λ n-j A k j = (-1) i-1 λ n-i ,
which becomes, using (2.63) and (2.60):

n k=1 B i k g k (x) j =k (λ -x j ) = (-1) i-1 λ n-i .
Using the explicit form of g k (x) given via (2.52) and of the matrix B, this last relation reduces to the identity (2.65), which completes the derivation of (2.64).

Remark 2.15. A few remarks are in order.

1. The first integral defined by (2.56) is precisely the Joachimsthal invariant (2.25) of the dual Moser system.

2. It should be emphasized that the bihamiltonian character of our system is obvious with our choice of ellipsoidal coordinates since, from their very definition, the x i are the eigenvalues of the Benenti (1, 1)-tensor field (L j i ) associated with a special conformal Killing tensor.

3. One can give some simple potentials for dual Moser. Denoting by J k the new first integrals, we have

J k = I k -v k , v k = µσ k (x) + ν(σ 1 (x) σ k (x) -σ k+1 (x)) (2.66)
with k = 1, . . . , n. Those will pairwise Poisson commute (see [START_REF] Perelomov | Integrable Systems of Classical Mechanics and Lie Algebras[END_REF], p. 101) if the potential terms can be written in the form

v k = n i=1 A i k (x) f i (x i ), implying f i (x i ) = n k=1 B k i v k .
A short computation, using the explicit form (2.64) of the matrix B and the relation (2.59), indeed gives

f i (x i ) = (x i ) n-1 4V (x i ) (µ + ν x i ).

Three Stäckel systems 2.5.1 The Neumann-Uhlenbeck system

In addition to the two previously studied integrable systems, it may be useful to consider the well-known Neumann-Uhlenbeck system [START_REF] Uhlenbeck | Minimal 2-spheres and tori in S k[END_REF][START_REF] Uhlenbeck | Equivariant harmonic maps into spheres[END_REF][START_REF] Moser | Various aspects of the integrable Hamiltonian systems[END_REF] on the cotangent bundle of the round sphere S n . It is initially defined on (T * R n+1 , n α=0 dp α ∧ dq α ) by the Hamiltonian

H = 1 2 n α=0 p 2 α + a α q 2 α (2.67)
with the parameters 0 < a 0 < a 1 < . . . < a n . This system is classically integrable, with the following commuting first integrals of the Hamiltonian flow in T * R n+1 :

F α (p, q) = q 2 α + β =α (p α q β -p β q α ) 2 a α -a β with α = 0, 1, . . . , n.
(2.68)

Under symplectic reduction, with the second class constraints (2.44), it becomes an integrable system on (T * S n , n i=1 dξ i ∧ dx i ). Writing g = n i=1 g i (x)(dx i ) 2 the induced Euclidean metric on S n with g i (x) = -1 4 U ′ x (x i )/V (x i ), the independent Poisson-commuting functions I k (k = 1, . . . , n) are

I k (ξ, x) = n i=1 g i (x)σ i k-1 (x)ξ 2 i -σ k (x) with H = 1 2 I 1 , (2.69)
where g i = 1/ g i .

A synthetic presentation

Let us observe that the previous calculation enables us to have a synthetic viewpoint unifying the Jacobi-Moser, Neumann-Uhlenbeck, and dual Moser systems. This highlights the novelty of the dual Moser system spelled out in this article.

In Table 1, we display in each row, and for each system, the metric, the first integrals in involution, the Hamiltonian, and the Stäckel matrix. (See, e.g., [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF] for a derivation of the formulae in the first two columns of this table .). Let us emphasize that in all three cases, the metric in the first row is indeed the Stäckel metric coming from I 1 , and which will be, later on, involved in the quantization procedures.

Jacobi-Moser

Neumann-Uhlenbeck dual Moser

g i = x i g i g i = -U ′ x (x i ) 4V (x i ) g i = 1 x i g i I k = i g i x i σ i k-1 ξ 2 i I k = i g i σ i k-1 ξ 2 i -σ k (x) I k = i x i g i σ i k-1 ξ 2 i H = 1 2 I 1 H = 1 2 I 1 H = 1 2σ n+1 (a) I n x k B i k (x k ) B i k (x k ) = (-1) i (x k ) n-i 4V (x k ) 1 x k B i k (x k )
Table 1: Three Stäckel systems

Quantum integrability

Start with a configuration manifold M of dimension n, and consider the space, S(M), of Hamiltonians on T * M that are fiberwise polynomial. A quantization prescription is a linear isomorphism Q between this space of symbols, S(M), and the space, D(M), of linear differential operators on M; this identification is, in addition, assumed to preserve the principal symbol.

It is well-known that there is, in general, no uniquely defined quantization. However, no matter how the quantization is chosen, we will adhere to the following, usual, definition of quantum integrability; see, e.g., [START_REF] Toth | Various quantum mechanical aspects of quadratic forms[END_REF][START_REF] Matveev | Quantum integrability of Beltrami-Laplace operator as geodesic equivalence[END_REF][START_REF] Benenti | Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. I. The completeness and Robertson conditions[END_REF][START_REF] Benenti | Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operators[END_REF]. Definition 3.1. A classically integrable system with independent, and mutually Poisson-commuting observables I 1 , . . . I n , is integrable at the quantum level iff

[Q(I k ), Q(I ℓ )] = 0 (3.1)
for all k, ℓ = 1, . . . , n

As a consequence, for a given integrable classical system, depending on the quantization procedure used, quantum integrability may be achieved or not. In what follows we will consider and use two quantization schemes for quadratic Hamiltonians: (i) the theory of conformally equivariant quantization, and (ii) Carter's minimal prescription.

Conformally equivariant quantization

Let us recall that there exists no quantization mapping that intertwines the action of Diff(M). To bypass this obstruction, equivariant quantization [START_REF] Lecomte | Projectively invariant symbol calculus[END_REF][START_REF] Duval | Conformally equivariant quantization: existence and uniqueness[END_REF] proposes to further endow M with a G-structure, and to look under which conditions the existence and uniqueness of a G-equivariant quantization can be guaranteed (the proper subgroup G ⊂ Diff(M) only is assumed to intertwine the quantization mapping Q).

We recall that the space F λ (M) of λ-densities on M, where λ is some complexvalued weight, is the space of sections of the complex line bundle |Λ n T * M| λ ⊗ C. If M is orientable, (M, vol), such a λ-density can be, locally, cast into the form φ = f |vol| λ with f ∈ C ∞ (M); this entails that φ transforms under the action of a ∈ Diff(M) according to f → a * f |(a * vol)/vol| λ , or infinitesimally as

L λ X (f ) = X(f ) + λ Div(X) f (3.2)
for all X ∈ Vect(M).

Remark 3.2. Note that the completion H(M) of the space of compactly supported half-densities, F c (M), is a Hilbert space canonically attached to M that will be used in the sequel. The scalar product of two half-densities reads

φ, ψ = M φ ψ
where the bar stands for complex conjugation.

We will denote by S δ (M) = S(M) ⊗ F δ (M) the graded space of symbols of weight δ. This space is turned into a Vect(M)-module using the definition (3.2) of the Lie derivative extended to the canonical lift of Vect(M) to T * M.

Likewise, we will introduce the filtered space D λ,µ (M) of differential operators sending F λ (M) to F µ (M). A differential operator of order k is, locally, written as

A = A i 1 ...i k k (x)∂ i 1 . . . ∂ i k + • • • + A i 1 (x)∂ i + A 0 (x) (3.3)
where

A i 1 ...i ℓ ℓ ∈ C ∞ (M) for ℓ = 0, 1, . . . , k.
It is clear that this space of weighted differential operators, D λ,µ (M), becomes a Vect(M)-module via the following definition of the Lie derivative, namely,

L λ,µ X (A) = L µ X • A -A • L λ X (3.4)
for all X ∈ Vect(M).

From now on, we will be dealing with the case of a conformal (Riemannian) structure, G = SO(n + 1, 1), with n > 2, dictated by the conformal flatness of our main example: the dual Moser system.

Theorem 3.3 ([10]

). Given a conformally flat Riemannian manifold (M, g), there exists (except for a discrete set of values of δ = µ -λ called resonances) a unique conformally-equivariant quantization, i.e., a linear isomorphism

Q λ,µ : S δ (M) → D λ,µ (M) (3.5)
that (i) preserves the principal symbol, and (ii) intertwines the actions of the Lie algebra o(n + 1, 1) ⊂ Vect(M).

In the particular and pivotal case of symbols of degree two, at the core of the present study, explicit formulae are given by the following theorem.

Theorem 3.4 ([9]

). (i) Let (M, g) be a conformally flat Riemannian manifold of dimension n ≥ 3. The conformally equivariant quantization mapping (3.5) restricted to symbols

P = P ij 2 (x)ξ i ξ j + P i 1 (x)ξ i + P 0 (x) of degree two is given, for non-resonant values of δ, by Q λ,µ (P ) = -P ij 2 • ∇ i • ∇ j +i β 1 ∇ i P ij 2 + β 2 g ij g kℓ ∇ i P kℓ 2 + P j 1 • ∇ j +β 3 ∇ i ∇ j (P ij 2 ) + β 4 g ij g kℓ ∇ i ∇ j (P kℓ 2 ) + β 5 R ij P ij 2 + β 6 R g ij P ij 2 +α∇ i (P i 1 ) + P 0
where ∇ denotes the levi-Civita connection,6 R ij (resp. R) the components of the Ricci tensor in the chosen chart (resp. the scalar curvature) of the metric g; the coefficients α, β 1 , . . . , β 6 depend on λ, µ, and n in an explicit fashion. 7(ii) The quantization mapping Q λ,µ depends only on the conformal class of g.

The above formula can be specialized to the case of half-density quantization of quadratic symbols P = P ij (x)ξ i ξ j ; one finds8 Q1 2 , 1 2

(P ) = P + β 3 ∇ i ∇ j (P ij ) + β 4 g ij g kℓ ∇ i ∇ j (P kℓ ) + β 5 R ij P ij + β 6 R g ij P ij (3.6) where P = -∇ i • P ij • ∇ j . (3.7)
Remark 3.5. The quantization prescription (3.7), called "minimal" in [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF], has been put forward by Carter [START_REF] Carter | Killing tensor quantum numbers and conserved currents in curved space[END_REF], who dealt with polynomial symbols of degree at most two. A great many studies of the quantum spectrum for various integrable models use naturally Carter's quantization [START_REF] Sklyanin | Non-linear equations in classical and quantum field theory[END_REF][START_REF] Toth | Various quantum mechanical aspects of quadratic forms[END_REF][START_REF] Matveev | Quantum integrability of Beltrami-Laplace operator as geodesic equivalence[END_REF][START_REF] Bellon | The quantum Neumann model: refined semiclassical results[END_REF]. Along with Equation (3.7), the formulae for the minimal quantization of lower degree monomials are respectively

P 0 = P 0 (3.8)
P 1 = i 2 P i 1 • ∇ i + ∇ i • P i 1 (3.9)
so that

P k = Q1 2 , 1 2
(P k ), ∀k = 0, 1. (3.10)
Accordingly, a generalization to cubic monomials has been proposed in [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF]:

P 3 = - i 2 ∇ i • P ijk 3 • ∇ j • ∇ k + ∇ i • ∇ j • P ijk 3 • ∇ k . (3.11)
All previously defined operators are formally self-adjoint on F c 1 2

(M); see Remark 3.2.

In the case where the quadratic observable P = P ij (x)ξ i ξ j stems from a Killing tensor,9 i.e., if ∇ (i P jk) = 0 for all i, j, k = 1 . . . , n, we can rewrite Equation (3.6) as

Q1 2 , 1 2 (P ) = P + f (P ) (3. 12
)
where P is as in (3.7), and the scalar term is given by

f (P ) = c 1 ∆ g Tr(P ) + c 2 R ij P ij + c 3 R • Tr(P ) (3.13)
where ∆ g is the Laplace operator of (M, g), and Tr(P ) = P ij g ij ; the coefficients in (3.13) are respectively

c 1 = n 2 8(n + 1)(n + 2) , c 2 = n 2 4(n + 1)(n -2) , c 3 = -n 2 2(n 2 -1)(n 2 -4)
. (3.14)

Quantum commutators

In order to implement Definition 3.1 of quantum integrability, we will need some preparation regarding the quantum commutators of Poisson-commuting symbols. In doing so, we will opt for the conformally equivariant quantization Q ≡ Q1

(T * M, ω = n i=1 dξ i ∧ dx i ).
The commutator of the two operators Q(P ) and Q(Q), given by (3.12), retains the form

[Q(P ), Q(Q)] = i Q(A P,Q + V P,Q ), (3.15)
where

A P,Q = - 2 3 ∇ j B jk P,Q ξ k (3.16)
with10

B jk P,Q = P ℓ[j ∇ ℓ ∇ m Q k]m + P ℓ[j R k] m,nℓ Q mn -(P ↔ Q) -∇ ℓ P m[j ∇ m Q k]ℓ -P ℓ[j R ℓm Q k]m (3.17) and V P,Q = 2 P jk ∂ j f (Q) -Q jk ∂ j f (P ) ξ k . (3.18)
Proof. Start with two quadratic observables P and Q. As shown in [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF], we have -i[ P , Q] = {P, Q}+ A P,Q , where the monomial A P,Q , and the skew-symmetric tensor B P,Q are as in (3.16), and (3.17), respectively. If it is then assumed that {P, Q} = 0, Equation (3.15) follows directly from the explicit expression (3.12) of the conformally equivariant quantization mapping, Q, and from Equation (3.10).

Now, for the Liouville-integrable systems considered below, all Poisson-commuting fiberwise polynomial symbols have the form P = P 2 +P 0 , where the indices 0 and 2 refer to the homogeneity degree. In view of Equation (3.15), and of results obtained in [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF], we find [Q

(P 2 +P 0 ), Q(Q 2 +Q 0 )] = i Q(A P 2 ,Q 2 +V P 2 ,Q 2 )
, which means that the zero degree terms P 0 and Q 0 produce no quantum corrections.

The structure of the quantum corrections (the right hand side of Equation (3.15)) is rather involved, because of the complexity of the tensor B P,Q ; see (3.17). Nevertheless, for Stäckel systems major simplifications occur. Indeed, the observables I k = I 2,k + I 0,k with I 2,k = i g i (x)σ i k-1 (x)ξ 2 i generate diagonal Killing tensors. Using the separating coordinates x i and considering H = 1 2 I 2,1 = 1 2 i g i (x)ξ 2 i for the Hamiltonian fixes up the diagonal metric to be g = i g i (x)(dx i ) 2 , with g i = 1/g i for all i = 1, . . . , n. Under these assumptions, Proposition 3.9 in [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF] gives

B kℓ I 2,i ,I 2,j = -2 I s[k 2,i R st I ℓ]t 2,j (3.19)
for all i, j, k, ℓ = 1, . . . n, which entails:

Proposition 3.7. A sufficient condition for a Stäckel system to be integrable at the quantum level is

R ij = 0, ∀i = j (3.20)
where i, j = 1, . . . , n, in the special separating coordinate system (x i ).

Proof. The Killing tensors I 2,i are diagonal, for i = 1, . . . , n, in the Stäckel coordinate system, and the result follows from (3.19).

Remark 3.8. 1. Condition (3.20) is the well-known Robertson condition [START_REF] Robertson | Bermerkung über separierbare Systeme in der Wellenmechanik[END_REF], which has to hold in the separating coordinates system. The relation (3.19) was also obtained in [START_REF] Benenti | Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operators[END_REF] by a direct computation of the commutator in separating coordinates; however the explicit form of the tensor B P,Q (3.17) was not given there.

2. In Corollary 3.10 of [START_REF] Duval | Quantum Integrability of Quadratic Killing Tensors[END_REF] the Robertson condition was misleadingly claimed to be also necessary. Benenti et al. [4] that the Robertson condition (3.20) is necessary and sufficient for the separability of the Schrödinger equation, comforting the above definition of quantum integrability.

It has been shown by

In the next subsections we will examine, successively, quantum integrability for the following Stäckel systems: the Neumann-Uhlenbeck, the dual Moser and the Jacobi-Moser systems. As previously explained, the potential, i.e., zero degree terms in the classical observables never induce quantum corrections; they will therefore be systematically omitted.

The quantum Neumann-Uhlenbeck system

Let us recall that, for the Neumann-Uhlenbeck system (see Table 1), the Stäckel metric g = i g i (x)(dx i ) 2 , is

g i (x) = - 1 4 U ′ x (x i ) V (x i ) = - 1 4 j =i (x i -x j ) α (x i -a α ) (3.21)
for i = 1, . . . , n. If we put g i = 1/ g i , the independent and Poisson-commuting observables are given by

I k = n i=1 g i (x)σ i k-1 (x) ξ 2 i for k = 1, 2, . . . , n, and the Stäckel Hamiltonian is H = 1 2 I 1 = 1 2 i g i (x) ξ 2 i .
Proposition 3.9. The conformally equivariant quantization does preserve quantum integrability of the Neumann-Uhlenbeck system.

Proof. From the fact that (S n , g) is the round sphere, we have

R ij = (n -1) g i δ ij & R = n(n -1). (3.22)
Straightforward computation then leads to Relations (3.14) readily imply the vanishing of c 4 . As a consequence, the f (I k ) are just constant, ensuring that V I k ,I ℓ = 0 (see (3.18)). Equation (3.15) and the fact that B I k ,I ℓ = 0 (since the Ricci tensor is diagonal in this coordinate system) entail that the conformally equivariant quantization (which coincides, up to a constant term, with Carter's) preserves integrability of the system at the quantum level.

f (I k ) = (n -k + 1)[c 4 σ k-1 (x) + 2(n -k + 2)c 1 σ k-1 (

The quantum dual Moser system

In the basic geometrical construction of the Poisson-commuting conserved quantities I k , we have been considering the conformally flat metric g 2 given by (2.51) and (2.52). Now, in the quantum approach to integrability, we choose to use, again, the Stäckel metric, g, associated with I 1 . One has

g = n i=1 g i (x)(dx i ) 2 , with g i (x) = 1 x i g i (x), (3.25)
where the Neumann-Uhlenbeck metric g is given by (3.21), while the first integrals for k = 1, . . . , n are

I k = n i=1 g i (x) σ i k-1 (x)ξ 2 i , g i (x) = 1 g i (x) . ( 3
R ij = (n -2)x i + n n k=1 x k -(n -1) n α=0 a α g i δ ij (3.27)
and scalar curvature

R = (n -1) (n + 2) n k=1 x k -n n α=0 a α . (3.28)
It is conformally flat for n = dim(M) ≥ 3.

Proof. The Ricci tensor can be computed with the help of classical formulae for a diagonal metric (see for instance [START_REF] Eisenhart | Riemannian geometry[END_REF], p. 119). The only possibly non-vanishing components of the Riemann tensor are R ik,kj , for i = j = k, and R ij,ji , for i = j.

Using the relations

∂ i (ln g j ) = 1 x i -x j (i = j), ∂ ij (ln g k ) = 0 (i = j = k)
one easily gets R ik,kj = 0, implying

R ij = - n k=1 g k R ik,kj = 0, ∀i = j.
The computation of the remaining components involves a sum which is conveniently computed using the theorem of residues, giving

R ik,ik = (x i + x k + n s=1 x s - n α=0 a α ) g i g k
from which one deduces easily the diagonal part of the Ricci tensor, given by (3.27), and the scalar curvature (3.28). Some extra computation shows that the conformal Weyl tensor vanishes in dimension n ≥ 4, and that the Cotton-York tensor vanishes as well for n = 3.

Remark 3.11. Although the metric g 2 given by (2.51) is clearly conformally flat, it is by no means trivial that the same is true for the Stäckel metric, g, given by (3.25) on S n .

We are now in position to prove the following proposition.

Proposition 3.12. The conformally equivariant quantization procedure (3.12) does preserve quantum integrability of the dual Moser system.

Proof. Using the definition (3.13) of the scalar term in the formula (3.12) for the conformally equivariant quantization of the I k , we find

f (I k ) = [(n -2)c 2 + k(c 6 -c 4 )]σ 1 (x) σ k (x) + [kc 5 -(n -2)c 2 ]σ k+1 (x) -kc 4 σ 1 (a) σ k (x) -2k(n -k + 1)c 1 σ k+1 (a)
where c 4 was already defined in (3.24) and shown to vanish in the proof of Proposition 3.9; we also have where we posit σ n+1 (x) = 0.

Let us now compute V I k ,I ℓ defined by (3.18) needed to check quantum integrability via the commutator (3.15).

In view of (3.26), one finds

V I k ,I ℓ = 2 n i=1 g i σ i k-1 ∂ i f (I ℓ ) -σ i ℓ-1 ∂ i f (I k ) ξ i .
Now, using the relations [START_REF] Benenti | Inertia tensors and Stäckel systems in the Euclidean spaces[END_REF] ∂ i σ k (x) = σ i k-1 (x), ∀i, k = 1, . . . , n, and σ k (x) = σ i k (x) + x i σ i k-1 (x), ∀k = 1, . . . , n -1, as well as σ n (x) = x i σ i n-1 (x), one gets ∂ i f (I k ) = 2(n + 2)c 1 [x i + σ 1 (x)]σ i k-1 (x) which obviously yields V I k ,I ℓ = 0, implying, at last [Q(I k ), Q(I ℓ )] = 0 (3.30) for all k, ℓ = 1, . . . , n.

Remark 3.13. Carter's (minimal) prescription (3.7) also leads to quantum integrability of the system because of the diagonal form (3.27) of the Ricci tensor in the separating coordinates. Now, in contradistinction with the Neumann-Uhlenbeck quantum system, the scalar terms f (I k ) given by (3.29) are no longer constant, yielding quite different quantum observables Q(I k ) and Îk . So, the fact that quantum integrability is not only preserved by Carter's quantum prescription, but also by conformally equivariant quantization is a new and noteworthy phenomenon.

The quantum Jacobi-Moser system

The Stäckel metric, associated to I 1 is now (see Table 1):

g = n i=1
g i (x)(dx i ) 2 , with g i (x) = x i g i (x), (3.31) where the Neumann-Uhlenbeck metric g i (x) is given by (3.21), and the first integrals by

I k = n i=1 g i (x)σ i k-1 (x)ξ 2 i , g i (x) = 1 g i (x) . ( 3

29

A simple computation gives V i I 1 ,I 2 = ∂V I 1 ,I 2 /∂ξ i = ∂ i f (I 2 ) -(σ 1 (x) -x i )∂ i f (I 1 ), hence the non-vanishing result

V i I 1 ,I 2 = (-c 3 )
σ n+1 (a)

x i σ 2 n (x) -2(σ i 1 (x)σ n-2 (x) + σ 1 (x)σ i n-2 (x))

+(n 2 -3n + 4)σ n-1 (x) + n(3n -5)σ i n-1 (x) , showing that the system looses its quantum integrability via conformally equivariant quantization.

Remark 3.16. Let us mention that quantum integrability of the Neumann-Uhlenbeck and Jacobi-Moser systems has first been established, in terms of Carter's quantum prescription (3.7), by Toth [START_REF] Toth | Various quantum mechanical aspects of quadratic forms[END_REF].

Conclusion and outlook

To sum up the main results of the article, let us mention that we have disclosed a new integrable system on S n , in duality with the well-known Jacobi-Moser system in terms of projective equivalence. As opposed to that of the generic ellipsoid, the "dual" metric is conformally flat. This remarkable fact enables us to have naturally recourse to conformally equivariant quantization. The latter turns out to preserve integrability at the quantum level. It is, to our knowledge, the first instance of conformally driven quantum integrability. This opens new perspectives related, e.g., to the determination of the conditions under which a classically integrable system, stemming from second-order Killing tensors on a conformally flat configuration manifold, remains quantum-integrable via the conformally equivariant quantization. Also, possible generalizations of the Jacobi-Moser system and its dual counterpart might conceivably be put to light in a similar manner.

Theorem 1 . 3 ., 1 2 .

 132 intertwines the action of the conformal group O(n + 1, 1) and (ii) preserves the principal symbol: we call it the conformally equivariant quantization mapping. Quantum integrability of the dual Moser system holds true in terms of the conformally equivariant quantization Q1 2

.44) Proposition 2 . 10 .

 210 The restrictions F α | T * S n = F α • ι of the functions (2.35) do Poisson-commute on T * S n .

  k = 1, . . . , n, where c 4 = -2(n + 1)c 1 + (n -1)c 2 + n(n -1)c 3 .(3.24)

. 26 ) 3 . 10 .

 26310 Lemma The metric (3.25) has Ricci tensor

c 5

 5 = 2(n + 2)c 1 -(n -2)c 2 , c 6 = -2c 1 + c 2 + 2(n -1)c 3 .Taking into account the relations (3.14) one gets c 5 = c 6 = 0, and we are left, for k = 1, . . . , n, withf (I k ) = 2c 1 (n + 2) [σ k (x)σ 1 (x) -σ k+1 (x)] -k(n -k + 1) σ k+1 (a)(3.29)

.32) Lemma 3 . 14 .- 1 )

 3141 The Ricci tensor and the scalar curvature of the metric (3.31) are given byR ij = σ n+1 (a)It is completely similar to the proof of Lemma 3.10.This allows us to prove: Proposition 3.15. Carter's prescription (3.7) preserves quantum integrability of the Jacobi-Moser system, while the conformally equivariant quantization does not.Proof. Since the Ricci tensor is diagonal, quantum integrability is established for the prescription (3.7). For the conformally equivariant quantization (3.12), we will just give a counter-example. One hasf (I 1 ) = 2(c 2 + nc 3 ) σ n+1 (a)σ n-2 (x) σ n-1 (x) -n σ 1 (x) σ n-2 (x) -2n(n -1) c 1 .

  Let P and Q be two, Poisson-commuting, quadratic symbols on

	2 , 1 2	.
	Proposition 3.6.

The formalism can be easily extended to the case of Finsler structures[START_REF] Tabachnikov | Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid[END_REF]; here, we will not need such a generality.

Note that φ is the identity on the zero section of T M .

We will, later on, deal with the choice 0 < a 0 < a 1 < • • • < a n .

We have ω n 2 ∧ q • dv ∧ q • dq = n! ω n+1 0 .

The covariant derivative of λ-densities φ = f |vol g | λ , locally defined in terms of the Riemannian density, |vol g |, reads ∇φ = df |vol g | λ .

See Equations (3.3),(3.4), and (4.4) in[START_REF] Duval | Conformally equivariant quantum Hamiltonians[END_REF].

The value δ = 0 is non-resonant[START_REF] Duval | Conformally equivariant quantization: existence and uniqueness[END_REF].

This is the case for the integrable systems of Stäckel type we are studying.

We use the following convention for the Riemann and Ricci tensors, namely,R ℓ i,jk = ∂ j Γ ℓ ik + Γ ℓ sj Γ s ik -(j ↔ k), and R ij = R s i,sj .

Acknowledgements: We express our deep gratitude to Serge Tabachnikov and Valentin Ovsienko for helpful discussions which have triggered this work. We are also indebted to John Harnad for useful correspondence.


Christophe Fonte
email: christophe.fonte@cran.uhp-nancy.fr

Cédric Delattre
email: cedric.delattre@iut-longwy.uhp-nancy.fr

Wronskian-based tests for stability of polynomial combinations

Keywords: hyperbolic polynomial, Hurwitz polynomial, Hermite-Biehler theorem, uncertain polynomial, Wronskian, Kharitonov polynomials

In this paper, a stability criterion based on counting the real roots of two specific polynomials is formulated. To establish this result, it is shown that a hyperbolicity condition and a strict positivity of a polynomial Wronskian are necessary and sufficient for the stability of any real polynomial. This result is extended to the stability study of some linear combinations of polynomials. Necessary and sufficient conditions of stability are obtained for polynomial segments and planes.

I. INTRODUCTION

A real polynomial is said to be stable (or Hurwitz) if all its roots lie in the open left half of the complex plane. This paper studies stability of polynomial combinations, particularly questions regarding stability of polynomial segments and polynomial planes.

For developing our approach, preliminary results are given in section 2. For that, we use an alternative to the Hermite-Biehler theorem without explicitly considering the zeroes interlacing property. This property is replaced by a double condition constituted on one hand by a polynomial positivity condition and on the other hand by a hyperbolicity condition (Recall that a polynomial is called hyperbolic if all its roots are real). Thereafter, this double requirement is reduced to a test for counting the real roots of polynomials by a discriminant method, see [START_REF] Gantmacher | The Theory of Matrices[END_REF] and [START_REF] Yang | A complete discrimination system for polynomials[END_REF]. Then a link between Hermite's criterion and this stability method is established.

Our framework permits to extend this issue on stability of one polynomial to stability of polynomial segments and planes.

In the first part of section 3, the convex polynomial segments are considered. We can note that the stability of convex segments with invariant degree has been examined by numerous authors, see [START_REF] Bialas | A necessary and sufficient condition for the stability of convex combinations of stable polynomials and matrices[END_REF], [START_REF] Blondel | A note on convex combinations of polynomials[END_REF], [START_REF] Bhattacharyya | Robust Control, The parametric Approach[END_REF], [START_REF] Bose | A system theoretic approach to stability of sets of polynomials[END_REF] and [START_REF] Barmish | New tools for robustness of linear systems[END_REF]. For example [START_REF] Bialas | A necessary and sufficient condition for the stability of convex combinations of stable polynomials and matrices[END_REF] was among the first to give a necessary and sufficient condition equivalent to eigenvalues negativity condition carrying on a product of matrices under the assumption that the endpoints are Hurwitz stable. Others approaches use methods based on the zero exclusion principle, see [START_REF] Barmish | New tools for robustness of linear systems[END_REF] or the stability boundary theorem, see [START_REF] Bose | A system theoretic approach to stability of sets of polynomials[END_REF] or even the strong stabilization, see [START_REF] Blondel | A note on convex combinations of polynomials[END_REF]. To solve this problem, we propose a new alternative that consists to apply our previous result on the stability of one polynomial to the polynomial segments. We show that a convex polynomial segment is stable under hyperbolicity assumptions if and only if a weak zeroes interlacing property and polynomial positivity conditions are satisfied.

In the second part of section 3, a generalization of these results are provided for different polynomial planes. Necessary and sufficient conditions are obtained for the Kharitonov's plane that complete these given in [START_REF] Chappellat | An alternative proof of Kharitonov's theorem[END_REF]. These results show that the stability of the Kharitonov's plane depends on a weak zeroes interlacing property and polynomial positivity conditions in connection with the four Kharitonov's polynomials. Then this approach is expanded to any polynomial plane. In this case, necessary and sufficient conditions are reduced to test some relations of polynomial positivity under assumptions of hyperbolicity and of zeroes interlacing property.

II. POLYNOMIAL STABILITY

In this section, a criterion of polynomial stability is established which is based on the counting of the real roots of two polynomials. As a preliminary result, a link is made between the polynomial stability and the sign of a polynomial Wronskian by using the roots interlacing property and the Hermite-Biehler theorem.

A. Hermite-Biehler stability

Definition 1: [START_REF] Gantmacher | The Theory of Matrices[END_REF], Zeroes Interlacing Property. Let P (u) and Q(u) be two real polynomials of respective degrees l and l -1 (or l). Let us assume the roots of these polynomials defined by the following sets

root(P (u)) = {u 1 , . . . , u l } root(Q(u)) = {v 1 , • • • , v l-1 } (or root(Q(u)) = {v 1 , • • • , v l }) Then P (u) and Q(u) interlace iff
• the leading coefficients of P (u) and Q(u) have the same sign,

• the roots of P (u) and Q(u) are real, simple and distinct, • the l roots of P (u) alternate with the l -1 (or l) roots of Q(u) as follows

u 1 < v 1 < u 2 < v 2 . . . < u l-1 < v l-1 < u l (or v 1 < u 1 < v 2 . . . < u l-1 < v l < u l )
May 31, 2011 DRAFT

Let f be in P m given by

f (s) = a m s m + a m-1 s m-1 • • • + a 2 s 2 + a 1 s + a 0 (1)
Let f be expanded as [START_REF] Barmish | New tools for robustness of linear systems[END_REF] where f e and f o denote the even part and the odd part of f respectively.

f (s) = f e (s 2 ) + sf o (s 2 ) (2)
The relationship between Zeroes Interlacing Property and Hurwitz stability is emphasized by the well known Hermite-Biehler Theorem that is recalled below.

Theorem 1: E.g. [START_REF] Gantmacher | The Theory of Matrices[END_REF], Hermite-Biehler's Theorem. The real polynomial f is stable iff f e (-s 2 ) and sf o (-s 2 ) satisfy the Zeroes Interlacing Property.

The next result arises by considering Theorem 1. This result proposes to use a polynomial Wronskian for testing polynomial stability.

Theorem 2: Let f be in P m . The polynomial f is stable iff the following two conditions hold a) the roots of f e (-s 2 ) are real, b) ∀s ∈ R,

W(f e , f o )(s) = f e (-s 2 ) sf o (-s 2 ) df e (-s 2 ) ds d (sf o (-s 2 )) ds > 0 (3)
Proof : See Appendix

B. Counting the number of distinct real roots of a polynomial.

In order to make this paper self-contained and to give a practical significance of algebraic conditions a) and b) given in Theorem 2, an explicit criterion to count the number of real roots of polynomials f e (-s 2 ) and W(f e , f o )(s) is produced in this subsection.

It is known that the discriminant of a polynomial is an expression which provides information about the nature of its roots. That is why a formulation based on the discrimination matrix of a real polynomial is hereafter given, see [START_REF] Gantmacher | The Theory of Matrices[END_REF], [START_REF] Yang | A complete discrimination system for polynomials[END_REF], [START_REF] Yang | Recent advances on determining the number of real roots of parametric polynomials[END_REF].

Consider a polynomial g(s) in P m g(s) = σ m s m + σ m-1 s m-1 • • • + σ 2 s 2 + σ 1 s + σ 0 Let R(g, g ′ ) be the 2m × 2m Sylvester matrix given below         σ m σ m-1 σ m-2 . . . σ 0 0 mσ m (m -1)σ m-1 . . . σ 1 0 0 σ m . . . σ 1 σ 0 . . . . . . σ m σ m-1 . . . σ 1 σ 0 0 mσ m (m -1)σ m-1 . . . σ 1         May 31, 2011 DRAFT
The even order minors of the matrix R(g, g ′ ) permit to determine the number of distinct (real and complex) roots of g. Denote by D k the determinant of the submatrix formed by the first k rows and the first k columns of R(g, g ′ ) for k = 1, 2 . . . , m.

sign(D i ) =    1 if D i > 0 0 if D i = 0 -1 if D i < 0 Definition 4: The revised sign list Π(D). Let [sign(D i ), sign(D i+1 ), ....sign(D i+j )], a section of the sign list Ψ(D) where sign(D i ) ̸ = 0, sign(D i+1 ) = sign(D i+2 ) = . . . = sign(D i+j-1 ) = 0, sign(D i+j ) ̸ = 0. The revised sign list is get by substituting in the sign list Ψ(D) the subsection [sign(D i+1 ), ....sign(D i+j-1 )] by [-sign(D i ), -sign(D i ), sign(D i ), sign(D i ), -sign(D i ), -sign(D i ), . . .]
Recall a result for counting the number of distinct real roots of a polynomial, see [START_REF] Yang | A complete discrimination system for polynomials[END_REF], [START_REF] Yang | Recent advances on determining the number of real roots of parametric polynomials[END_REF].

Lemma 1: [START_REF] Yang | A complete discrimination system for polynomials[END_REF], [START_REF] Yang | Recent advances on determining the number of real roots of parametric polynomials[END_REF] The number of distinct real roots of g is equal to µ -2γ where µ and γ are respectively the number of nonzero elements and the the number of sign changes in the revised sign list Π(D).

From this Lemma 1 and Theorem 2, a test is deduced for deciding of the polynomial stability.

Corollary 1: The real polynomial f in P m is stable iff all following conditions hold a) The revised sign list of the discriminant sequence of f e (-s 2 ) has no zero element and has no sign change, b) W(f e , f o )(s) has no real root, i.e. the revised sign list of the discriminant sequence of W(f e , f o )(s) has twice as many nonzero elements as sign changes, c) W(f e , f o )(s 0 ) > 0 for some real s 0 .

Proof : Obvious, see Theorem 2. Concerning conditions b) and c), we know that an univariate polynomial is strictly positive if this polynomial has no real roots and if it is strictly positive for some real value.

C. Connection between Hermite's criterion and Wronskian's condition.

A known result for the polynomial stability is the Hermite's criterion, see e.g. [START_REF] Olshevsky | Kharitonov's theorem and bezoutians[END_REF]. In this part, we examine the relationships between Hermite's criterion and the Wronskian's condition (3) of Theorem 2.

Theorem 3: (Hermite). A real polynomial f in P m is stable iff the m × m symmetric matrix B = [b ij ] is positive definite, where the entries b ij are defined by ∀(s 1 , s 2 ) ∈ R 2 , ∑ m i=1 ∑ m j=1 b ij s i-1 1 s j-1 2 = s 1 f o (-s 2 1 )f e (-s 2 2 ) -f e (-s 2 1 )s 2 f o (-s 2 2 ) s 1 -s 2 (4)
Rewrite right term of relation ( 4) as it follows f e (-s 2 1 )

s 1 f o (-s 2 1 ) -s 2 f o (-s 2 2 ) s 1 -s 2 -s 1 f o (-s 2 1
)

f e (-s 2 1 ) -f e (-s 2 2 ) s 1 -s 2 (5)
Considering s 1 = s and s 2 = s + ϵ, and computing limit of expression ( 5) as ϵ approaches 0 leads to W(f e , f o )(s). From Hermite's criterion, f is stable iff ( 4) is a positive form. Then we have

∀s ∈ R, W(f e , f o )(s) = ∑ m i=1 ∑ m j=1 b ij s i+j-2 = V T (s)BV (s) > 0 where V T (s) = [1 s s 2 . . . s m-1 ].
Note that condition b) in Theorem 2 is not equivalent to the definite-positivity of the matrix B given by expression (4). Indeed B is definite positive iff x T Bx > 0 for any x ∈ R m while the Wronskian in equation ( 3) is positive iff V T BV > 0 for any V Vandermonde vector of R m . We prove through the below example that condition a) of Theorem 2 is necessary.

Example. Consider the following unstable polynomial f (s) = (1s)(s + 1) 2 . We get

sf o (-s 2 ) = s 3 + s, f e (-s 2 ) = s 2 + 1 This yields to ∀s ∈ R, W(f e , f o )(s) = (s 2 + 1) 2 > 0 Moreover we have B = ( 1 1 1 1
)
Then B is not definite positive. This is in accordance with the fact that f e (-s 2 ) has no real root although W(f e , f o )(s) is positive.

May 31, 2011 DRAFT III. STABILITY OF SOME POLYNOMIAL COMBINATIONS Theorem 2 establishes a stability test for an unique polynomial. This result is extended to polynomial sets.

A. Stability of polynomial segments

Firstly, we consider the issue of polynomial stability of segments of the form αf 0 + (1α)f 1 with α ∈ [0, 1] without assuming the stability of the two endpoints of same degree f 0 and f 1 . For that, a preliminary result is derived from Theorem 2.1 given in [START_REF] Dedieu | Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials[END_REF], [START_REF] Dedieu | Corrigendum: Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials[END_REF].

Lemma 2: [START_REF] Dedieu | Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials[END_REF], [START_REF] Dedieu | Corrigendum: Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials[END_REF] Let f e 0 (-s 2 ) and f e 1 (-s 2 ), be two polynomials in H m with leading coefficients of same sign and with roots given by the sets below root(f e 0 (-s 2 )) = {s e 0 0 , . . . , s e 0 m } (6)

root(f e 1 (-s 2 )) = {s e 1 0 , • • • , s e 1 m } (7)
In these conditions, the two following statements are equivalent a) for each α ∈ [0, 1] the roots of f e α (-s 2 ) in ( 8) are real

f e α (-s 2 ) = αf e 0 (-s 2 ) + (1 -α)f e 1 (-s 2 ) (8)
b) for each k = 1, . . . , m -1, we have

max(s e 0 k , s e 1 k ) ≤ min(s e 0 k+1 , s e 1 k+1 ) (9)
Proof : In [START_REF] Dedieu | Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials[END_REF], [START_REF] Dedieu | Corrigendum: Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials[END_REF], the proof of Theorem 2.1 assumes that the two endpoints of (8) are two monic polynomials. This proof still holds if we consider as assumption that f e 0 (-s 2 ) and f e 1 (-s 2 ) have leading coefficients of same sign.

Note that (9) is a weaker condition than the Zeroes Interlacing Property, see Definition 1.

Theorem 4: Consider the polynomial segment f α given by ( 10)

∀ α ∈ [0, 1], f α (s) = αf 0 (s) + (1 -α)f 1 (s) ( 10
)
where f 0 and f 1 are two polynomials in P m . Denote by f e 0 and f e 1 respectively the even parts of f 0 and f 1 that satisfy the following properties

• the leading coefficients of f e 0 (-s 2 ) and f e 1 (-s 2 ) have the same sign, • the roots of f e 0 (-s 2 ) and f e 1 (-s 2 ) given by the sets ( 6) and ( 7) are real. Then the polynomial segment f α is stable iff the inequalities [START_REF]Positive Polynomials in Control[END_REF]

hold.        ∀s ∈ R, W(f e 0 , f o 0 )(s) > 0, W(f e 1 , f o 1 )(s) > 0, √ W(f e 0 , f o 0 )(s) W(f e 1 , f o 1 )(s) + 1 2 (W(f e 0 , f o 1 )(s) + W(f e 1 , f o 0 )(s)) > 0 ∀k = 1, . . . , m -1, max(s e 0 k , s e 1 k ) ≤ min(s e 0 k+1 , s e 1 k+1 ) (11)
May 31, 2011 DRAFT Proof : As previously defined for f in (2), denote by f e α and f o α , the even part and the odd part respectively of the polynomial segment f α . Then we have

f α (s) = f e α (s 2 ) + sf o α (s 2 )
where f e α (s 2 ) and sf o α (s 2 ) are given by the following expressions

{ f e α (s 2 ) = αf e 0 (s 2 ) + (1 -α)f e 1 (s 2 ) f o α (s 2 ) = αf o 0 (s 2 ) + (1 -α)f o 1 (s 2
) and where sf o 0 (s 2 ) and sf o 1 (s 2 ) are the odd parts of f 0 (s) and f 1 (s), respectively. Furthermore, we have

W(f e α , f o α )(s) = α 2 W(f e 0 , f o 0 )(s) + α(1 -α) [W(f e 0 , f o 1 )(s) + W(f e 1 , f o 0 )(s)] + (1 -α) 2 W(f e 1 , f o 1 )(s) (12)
Relation ( 12) is equivalent to the following one :

   W(f e α , f o α )(s) = [ (1 -α) √ W(f e 1 , f o 1 )(s) -α √ W(f e 0 , f o 0 )(s) ] 2 + 2α(1 -α) [ √ W(f e 0 , f o 0 )(s)W(f e 1 , f o 1 )(s) + 1 2 (W(f e 0 , f o 1 )(s) + W(f e 1 , f o 0 )(s)) ] (13)
If f α (s) is stable then we deduce from Theorem 2 1) Firstly, the roots of f e α (-s 2 ) are real for any α ∈ [0, 1]. Then, by considering Lemma 2, we get for k = 1, . . . , m -1 max(s e 0 k , s e 1 k ) ≤ min(s e 0 k+1 , s e 1 k+1 )

2) Secondly, for any s in R and for any α ∈ [0, 1], W(f e α , f o α )(s) > 0. Then, we deduce that W(f e 0 , f o 0 )(s) > 0 (case α = 0) and W(f e 1 , f o 1 )(s) > 0 (case α = 1). The next step is widely inspired from proof of Lemma 2.1 in [START_REF] Anderson | Criteria for copositive matrices using simplices and barycentric coordinates[END_REF]. Let α(s) defined by the following expression :

α(s) = √ W(f e 1 , f o 1 )(s) √ W(f e 1 , f o 1 )(s) + √ W(f e 0 , f o 0 )(s)
This yields to α(s) ∈ (0, 1). By taking α = α(s) for any s ∈ Rin (13), this yields to

∀s ∈ R, 2 α(s)(1 -α(s)) [ √ W(f e 0 , f o 0 )(s)W(f e 1 , f o 1 )(s) + 1 2 (W(f e 0 , f o 1 )(s) + W(f e 1 , f o 0 )(s)) ] > 0 That implies ∀s ∈ R, √ W(f e 0 , f o 0 )(s) W(f e 1 , f o 1 )(s) + 1 2 (W(f e 0 , f o 1 )(s) + W(f e 1 , f o 0 )(s)) > 0 Conversely,
)(s) > 0, W(f e 1 , f o 1 )(s) > 0, √ W(f e 0 , f o 0 )(s) W(f e 1 , f o 1 )(s) + 1 2 (W(f e 0 , f o 1 )(s) + W(f e 1 , f o 0 )(s)) > 0 then for any s ∈ R, W(f e α , f o α )(s) > 0. Hence by Theorem 2, f α (s) is stable for all α ∈ [0, 1].
From Theorem 4, let us examine now some particular cases.

A simplified form of Theorem 4 is presented in the following corollary when f 0 and f 1 are stable.

Corollary 2: Consider two stable polynomials f 0 and f 1 in P m with leading coefficients of same sign and f e 0 (-s 2 ) and f e 1 (-s 2 ) two polynomials in H m with their roots given by the sets ( 6) and [START_REF] Chappellat | An alternative proof of Kharitonov's theorem[END_REF]. The polynomial segment f α defined by ( 10) is stable iff the two relations below are verified.

   ∀s ∈ R, √ W(f e 0 , f o 0 )(s) W(f e 1 , f o 1 )(s) + 1 2 (W(f e 0 , f o 1 )(s) + W(f e 1 , f o 0 )(s)) > 0 ∀k = 1, . . . , m -1, max(s e 0 k , s e 1 k ) ≤ min(s e 0 k+1 , s e 1 k+1 )
Proof : Obvious, see Theorem 4.

Corollary 3: Consider two polynomials f 0 and f 1 in P m with leading coefficients of same sign and with the same even part f e . Then the polynomial segment f α defined by [START_REF] Gantmacher | The Theory of Matrices[END_REF] is stable iff f 0 and f 1 are stable.

Proof : It can be deduced from Theorem 2 than f α (s) is stable for all α ∈ [0, 1] if f 0 and f 1 are stable, thus 1) the polynomials f 0 and f 1 are stable. Then the roots of f e (-s 2 ) are real.

2) for any

s ∈ R, W(f e α , f o α )(s) > 0. Hence by developing W(f e α , f o α )(s), one get W(f e α , f o α )(s) = α W(f e , f o 0 )(s) + (1 -α) W(f e , f o 1 )(s) As for any s ∈ R, W(f e , f o 0 )(s) > 0, W(f e , f o 1 )(s) > 0, this implies W(f e α , f o α )(s) > 0.
Conversely, if we have for all s and for any α

∈ [0, 1], W(f e α , f o α )(s) > 0 , then for all s ∈ R, W(f e , f o 0 )(s) > 0, W(f e , f o 1 )(s) > 0.
Therefore f 0 and f 1 are stable.

Corollary 4: Consider two polynomials f 0 and f 1 in P m with leading coefficients of same sign and with the same even part f e . Then the polynomial segment f α defined by [START_REF] Gantmacher | The Theory of Matrices[END_REF] is stable iff the roots of f e (-s 2 ) are real and for any s ∈ R, W(f e , f o 0 )(s) > 0, W(f e , f o 1 )(s) > 0.

Proof : Obvious, see Corollary 3 and Theorem 2.

The result given in Corollary 4 holds if the two endpoints have the same odd parts.

May 31, 2011 DRAFT Corollary 6: Consider four stable polynomials of degree m, f 00 , f 01 , f 10 , f 11 such that

f 00 (s) = f e 0 (s 2 ) + sf o 0 (s 2 ), f 01 (s) = f e 0 (s 2 ) + sf o 1 (s 2 ), f 10 (s) = f e 1 (s 2 ) + sf o 0 (s 2 ) f 11 (s) = f e 1 (s 2 ) + sf o 1 (s 2 ) (17)
where f e 0 (-s 2 ) and f e 1 (-s 2 ) are two polynomials in H m with their roots given by the sets ( 6) and [START_REF] Chappellat | An alternative proof of Kharitonov's theorem[END_REF] and the leading coefficients of same sign. Then the convex plane of polynomials f α,µ defined by ( 14) is stable iff the following relation holds.

∀k = 1, . . . , m -1, max(s e 0 k , s e 1 k ) ≤ min(s e 0 k+1 , s e 1 k+1 )

Proof : The proof follows from Corollary 4 and Corollary 5.

The link between the stability of the four Kharitonov's polynomials and the stability of this Kharitonov's plane f α,µ is made by Corollary 5 and Corollary 6.

In the following theorem is produced a stability result for the polynomial combination αf 0 (s)+ µf 1 (s) with (α, µ) ∈ R 2 .

Theorem 5: Consider the polynomial combination f α,µ defined by ( 18)

∀(α, µ) ∈ R 2 , f α,µ (s) = αf 0 (s) + µf 1 (s) (18)
where f 0 and f 1 are two polynomials in P m . Denote by f e 0 , f e 1 and f o 0 , f o 1 respectively the even parts and the odd parts of f 0 and f 1 . Assume that the roots of f e 0 (-s 2 ) and f e 1 (-s 2 ) verify the Zeroes Interlacing Property. Then the polynomial plane f α,µ defined by ( 18) is stable iff the relationships (19) hold.

{ ∀s ∈ R, W(f e 0 , f o 0 )(s) > 0 ∀s ∈ R, W (f e 0 (-s 2 ), f e 1 (-s 2 ))(s) W (sf o 0 (-s 2 ), sf o 1 (-s 2 ))(s) > 0 (19)
In order to prove Theorem 5, we need the following Obreschkoff's theorem given in [START_REF] Obreschkoff | Verteilung und Berechnung der Nullstellen reeller Polynome[END_REF].

Theorem 6: [START_REF] Obreschkoff | Verteilung und Berechnung der Nullstellen reeller Polynome[END_REF] Let f e 0 (-s 2 ) and f e 1 (-s 2 ) two real polynomials of degree differing by 1 at most. The two following statements are equivalent i) the roots of f e 0 (-s 2 ) and f e 1 (-s 2 ) verify the Zeroes Interlacing Property. ii)

the polynomial combination αf e 0 (-s 2 ) + µf e 1 (-s 2 ) has only distinct real zeros for all reals α and µ. We are now in position to give a proof for Theorem 5.

Proof of Theorem 5 : By considering (18), define f e α,µ and f o α,µ as the following expressions f e α,µ (s 2 ) = αf e 0 (s 2 ) + µf e 1 (s 2 ) (20)

f o α,µ (s 2 ) = αf o 0 (s 2 ) + µf o 1 (s 2 ) (21)
From Theorem 2, f α,µ is stable iff the two conditions a) and b) given hereafter hold. a) Firstly, we can observe that f e α,µ (-s 2 ) defined by (20) has only distinct real roots for any (α, µ) ∈ R 2 since the roots of f e 0 (-s 2 ) and f e 1 (-s 2 ) verify the Zeroes Interlacing Property by assumption, see Theorem 6.

May 31, 2011 DRAFT b) Secondly, for any s ∈ R and for any (α, µ) ∈ R 2 , we have

W(f e α,µ , f o α,µ )(s) > 0. Namely { ∀(α, µ) ∈ R 2 , ∀s ∈ R, W(f e α,µ , f o α,µ )(s) = α 2 W(f e 0 , f o 0 )(s) + αµ W(f e 0 , f o 1 )(s) + µα W(f e 1 , f o 0 )(s) + µ 2 W(f e 1 , f o 1 )(s) > 0 (22)
Relation ( 22) is equivalent to the following

∀(α, µ) ∈ R 2 , ∀s ∈ R, [ α µ ]   W(f e 0 , f o 0 )(s) W(f e 0 , f o 1 )(s) W(f e 1 , f o 0 )(s) W(f e 1 , f o 1 )(s)   [ α µ ] > 0 (23)
Inequality ( 23) is satisfied iff the matrix W(s) hereafter is strictly positive definite for any real s.

W(s) =   W(f e 0 , f o 0 )(s) W(f e 0 , f o 1 )(s) W(f e 1 , f o 0 )(s) W(f e 1 , f o 1 )(s)   > 0 (24)
The matrix W(s) is strictly positive definite for any real s iff all the determinants of the principal submatrices W(s) are strictly positive for any real s, i.e. iff we have W(f e 0 , f o 0 )(s) > 0 and |W| > 0 for any real s. That is iff conditions (19) hold.

IV. CONCLUSION

In this paper, results have been produced for studying the stability of real polynomials. For beginning Theorem 2 establishes an algebraic criterion of stability that may be summarized as a test of positiveness of only one univariate polynomial. In addition, Corollary 1 provides a tractable test for deciding of the polynomial stability. Finally, these results have been extended to analyze the stability of polynomial combinations in the case of convex segments of polynomials and in the case of planes of polynomials.

As the stability criteria given in this paper are mainly based on a polynomial positivity, this work can be profitably linked with a wide panel of developments produced these last years surrounding positive polynomials and sums of squares of polynomials, see [START_REF]Positive Polynomials in Control[END_REF]. APPENDIX Proof of Theorem 2. "if" : According to Theorem 1, it is sufficient to prove that 1)

f e (-s 2 ) and sf o (-s 2 ) have simple and distinct roots, 2)

the leading coefficients of f e (s 2 ) and sf o (s 2 ) have the same sign, 3) the roots of sf o (-s 2 ) are all real and interlace with the roots of f e (-s 2 ). Let us show these three conditions. 1) If s i is a common zero of f e (-s 2 ) and sf o (-s 2 ) (or a common zero of f e (-s 2 ) and df e (-s 2 ) ds or a common zero of sf o (-s 2 ) and dsf o (-s 2 )

ds

) then W(f e , f o )(s i+1 ) = 0.

May 31, 2011 DRAFT

2) Let us consider that f is expressed by [START_REF] Anderson | Criteria for copositive matrices using simplices and barycentric coordinates[END_REF]. The leading coefficient of W(f e , f o )(s) is α m α m-1 , where α m and α m-1 are the leading coefficients of f e (s 2 ) and f o (s 2 ), respectively. Suppose that α m and α m-1 have a different sign, then α m α m-1 is negative and so is W(f e , f o )(s) for s large enough.

3) Let us show that the zeroes of f e (-s 2 ) and sf o (-s 2 ) interlace. Let s i and s i+1 be two negative and consecutive real zeroes of f e (-s 2 ). Then

W(f e , f o )(s i ) = -s i f o (-s 2 i ) df e (-s 2 ) ds | s=si > 0 W(f e , f o )(s i+1 ) = -s i+1 f o (-s 2 i+1 ) df e (-s 2 ) ds | s=si+1 > 0
Furthermore df e (-s 2 )

ds

| s=s i and df e (-s 2 )

ds | s=s i+1 have not the same sign . As the sign of W(f e , f o )(s) is constant, then the sign of s i f o (s i ) and s i+1 f o (s i+1 ) are opposite. Therefore there is an odd number of real simple zeroes of sf o (-s 2 ) between s i and s i+1 .

From this statement we consider two cases : i) m is even, i.e the number of roots of sf o (-s 2 ) is one less than f e (-s 2 ). Therefore the roots of sf o (-s 2 ) are all real and interlace with the roots of f e (-s 2 )

ii) m is odd, i.e. the number of roots of sf o (-s 2 ) is one more than f e (-s 2 ). Let us suppose that two roots of sf o (-s 2 ) are complex conjugate or that the roots of sf o (-s 2 ) and f e (-s 2 ) do not interlace. Then the greater root s k of f e (-s 2 ) is greater than the greater root of sf o (-s 2 ). This yields to

W(f e , f o )(s k ) = -s k f o (-s 2 k ) df e (-s 2 ) ds | s=s k > 0 (25)
Suppose that df e (-s 2 )

ds

| s=s k > 0 (the case df e (-s 2 ) ds | s=s k < 0 is similar). Then f e (-s 2 ) is positive for s > s k , and (25) implies that f o (-s 2 ) is negative for s > s k , therefore the leading coefficients of f e (-s 2 ) and f o (-s 2 ) have opposite signs. This is in contradiction with 2).

"only if" :

We know from Theorem 1 that any root s i of f e (-s 2 ) is real and simple, and that the roots of f e (-s 2 ) and sf o (-s 2 ) interlace. Following [START_REF] Gantmacher | The Theory of Matrices[END_REF], if m = 2k, then

sf o (-s 2 ) f e (-s 2 ) = k ∑ i=1 R i s -s i (26)
where s i are the roots of f e (-s 2 ) and with

R i = s i f o (-s 2 i )
d(f e (-s 2 )) ds

| s=s i = f o (-s 2 i ) -2 df e (u) du | u=-s 2 i < 0 (27) Indeed f o (u)
df e (u) du > 0 when f e (-s 2 ) and sf o (-s 2 ) interlace, see [START_REF] Gantmacher | The Theory of Matrices[END_REF].

We obtain f e (-s

Definition 2 :Definition 3 :

 23 The discriminant sequence D. The m-tuple D = [D 1 , D 2 . . . , D m ] is called the discriminant sequence of g. The sign list Ψ(D). Let D = [D 1 , D 2 . . . , D m ] be the discriminant sequence of g. We denote by Ψ(D) = [sign(D 1 ), sign(D 2 ) . . . , sign(D m )], the sign list of D where sign(.) is the sign function namely

2 )

 2 df o (-s 2 ) ds sf o (-s 2 ) df e (-s 2 ) ds (f e (-s 2 )) 2 = -k ∑ i=1 R i (ss i ) 2 > 0(28)May 31, 2011 DRAFTOn the other side, if m = 2k + 1 we have (from[START_REF] Gantmacher | The Theory of Matrices[END_REF]) :sf o (-s 2 ) f e (-s 2 ) = a 0 s + b 0 + k ∑ i=1 R i ss i (29)where a 0 and b 0 are two constants. According to Theorem 1 the leading coefficients of sf o (s 2 ) and f e (s 2 ) have the same sign and as well those of sf o (-s 2 ) and f e (-s 2 ). This impliesf e (-s 2 ) df o (-s 2 ) ds sf o (-s 2 ) df e (-s 2 ) ds (f e (-s 2 )) 2 = a 0 -k ∑ i=1 R i (ss i ) 2 > 0 (30)

  assume that conditions[START_REF]Positive Polynomials in Control[END_REF] holds.

	Then by considering Lemma 2, the roots of f e α (-s 2 ) are real thus by assumption the roots of 0 (-s 2 ) and f e f e 1 (-s 2 ) are real. Moreover, as relation (13) holds and if the following inequalities
	are satisfied
	∀s ∈ R, W(f e 0 , f o 0
	May 31, 2011	DRAFT

May 31, 2011 DRAFT

B. Stability of polynomial planes

The next corollary is an extension of Corollary 4 and Lemma 3.1 in [START_REF] Chappellat | An alternative proof of Kharitonov's theorem[END_REF], in regards to a convex plane f α,µ built from convex combinations of the four real polynomials of same degree f e 0 , f e 1 , f o 0 , f o 1 as given hereafter by relation [START_REF] Yang | Recent advances on determining the number of real roots of parametric polynomials[END_REF].

Corollary 5: Consider f o 0 , f o 1 two polynomials in P m and f e 0 (-s 2 ) and f e 1 (-s 2 ) two polynomials in H m with the leading coefficients of f e 0 , f e 1 of same sign and their roots defined by the sets ( 6) and [START_REF] Chappellat | An alternative proof of Kharitonov's theorem[END_REF]. Then the convex plane f α,µ defined by ( 14) is stable iff conditions [START_REF] Yang | A complete discrimination system for polynomials[END_REF] hold

Proof: If f α,µ is stable then 1) the roots of the polynomial combination (1α)f e 0 (-s 2 ) + αf e 1 (-s 2 ) are real, then from Lemma 2 we have ∀k = 1, . . . , m -1, max(s e 0 k , s e 1 k ) ≤ min(s e 0 k+1 , s e 1 k+1 ) 2) for any s ∈ R, for any α ∈ [0, 1], and for any µ

Conversely, if the roots of f e 0 (-s 2 ) and f e 1 (-s 2 ) are real and ∀k = 1, . . . , m -1, max(s e 0 k , s e 1 k ) ≤ min(s e 0 k+1 , s e 1 k+1 ) then the roots of the polynomial combination (1α)f e 0 (-s 2 ) + αf e 1 (-s 2 ) are real, see Lemma

1 )(s) > 0, then we have W(f e α , f o µ )(s) > 0 for all α ∈ [0, 1] and for all µ ∈ [0, 1]. Therefore we deduce from Theorem 2 that f α,µ is stable.

May 31, 2011 DRAFT