Quantum integrable systems and Geometry September 3-7, 2012, Olhao - - PDF document

Quantum integrable systems and Geometry September 3-7, 2012, Olhao Analytic prolongation of normal forms JP Fran¸ coise Universit´ e P.-M. Curie, Paris 6, France

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Birkhoff normal forms

• G. D. Birkhoff studied the local expression of a Hamiltonian system

near a critical point of Morse type up to symplectic changes of coordinates. Under some generic conditions, this local normal form exists as a formal series in any dimension. It is convergent in one degree of freedom and it is generically divergent in (m > 1) degrees of freedom [(Siegel, 54)(Moser, 76)]

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Hamiltonian systems integrable in Liouville sense A Hamiltonian system (H, ω) is said to be (completely) integrable if there exist m generically independent integrals H = (H1, ..., Hm) such that {Hi, Hj} = 0, H = H1. If the fibers H−1(c) are compact and connected, they are torii and the flows of all the Hi are linear on these torii. Action- angles coordinates allow to compute the frequencies of the Hamiltonian flow of H on these invariant torii.

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Analytic Hamiltonian systems which are Liouville integrable display a convergent Birkhoff normal form Theorem (J. Vey, 76) Assume H are analytic near 0 ∈ Rn, {Hi, Hj} = 0, H = H1 displays a Morse critical point, assume that the HessHi(0) generate a Cartan sub- algebra of Sp(2m, R), then the Birkhoff normal form of H is a convergent series. For instance, pi = x2

i + y2 i or pi = xiyi generate a Cartan sub-algebra

• f Sp(2m, R) (Precise definition : commutative and selfnormalizing).

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Birkhoff normal form of an integrable Hamiltonian system is in general a convergent series but a priori only defined in the neighborhood of the critical point. What can be said of its analytic prolongation if the Hamiltonian system is itself globally defined (for instance is a polynomial, rational function) ? In such case, if the HS displays different critical points, is it possible to compare the analytic prolongation of the Birkhoff form in one critical point to the Birkhoff form in the other critical point ? Begin with one degree of freedom !

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The pendulum has been studied recently (P.L.Garrido, G. Gallavotti, JPF, Journal Maths Physics 10). Reading Jacobi in the text and Gradshtein-Ryzhik tables of formula for elliptic functions. Jacobi found a coordinate system in which the motion is linear but he did not computed the symplectic form in these coordinates. Although his computation could be used to obtain a piece of information

• n the symplectic form (its relative cohomology class associated with H).

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Singularity theory of functions Consider an analytic function H : (x, y) → 1

2(x2 + y2) + .... and a

symplectic (volume) form ω = dx∧dy. Morse lemma allows to find anew (analytic) coordinate system (X, Y ) such that H = 1

2(X2 + Y 2) but with

no control of ω : ω = [1 + F(X, Y )]dX∧dY . Definition Two volume forms ω and ω′ have the same relative cohomo- logy class if ω − ω′ = dH∧dξ. Moser isotopy method for volume form applies and shows : Given a function H and two volume forms which are relatively cohomologous, there is an isotopy φ so that φ ∗ (H) = H and φ ∗ (ω) = ω′. Any polynomial form ω decomposes into ω = ψ(H)dx∧dy + dH∧dξ.

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After these changes of coordinates we find H =

1 2(x2 + y2), ω =

ψ(H)dx∧dy, there is and easy change into, H = φ(1 2(x2 + y2)), ω = dx∧dy and this is the Birkhoff normal form !

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Free rigid body motion and the geodesic motion on a revolution ellipsoid have been also studied (P.L. Garrido, G. Gallavotti, JPF, F18 in Ipparco Roma I, 2012.) Normal forms are derived via the analysis of relative cohomology.

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The Hamiltonian of the free rigid body in the coordinates B, β depending

• f the parameters I and of the ”parameter” A(in fact constant of motion) :

H′ = 1 2 B2 I3 + 1 2(cos2β I1 + sin2β I2 )((A2 − B2), (1) We consider instead : H = 2H′ − A2I−1

1

I−1

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− I−1

1

= B2 − r2sin2βA2 + r2sin2βB2, (2) with r2 = I−1

1

− I−1

2

I−1

3

− I−1

2

. (3)

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Consider the Hamiltonian : H = B2 + r2β2A2 + r2(sin2β − β2)A2 − r2sin2βB2. (4) Change (B, β) into (X = B, Y = rAβ), this multiplies the symplectic form by 1/rA and this yields to consider in the following the couple : H = X2 + Y 2 + ...., ω = dX∧dY (5)

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We have here an explicit version of the Morse lemma : (X, Y ) → (X′, Y ′) X′ = X

• 1 − r2sin2( Y

rA) = X + ...h.o.t.

Y ′ = rAsin( Y

rA) = Y + ...h.o.t.

(6) In these coordinates : H = X′2 + Y ′2, ω = dX∧dY =

1

• (1−Y ′2

A2 )(1− Y ′2 r2A2)

dX′∧dY ′. (7)

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We compute the cohomology class of the volume form ω by introducing the coefficients ak defined as follows : 1 √ 1 − u2 = Σkaku2k, (8) and the binomial coefficients Ck

• n. One can check that

ω = g(ξ)dX′∧dY ′ + dξ∧du, (9) where g(ξ) = Σh(Σk,l;k+l=hakalr2h)Ch

2h

4h ξh r2h. (10)

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It remains to perform a final change of coordinates of type : x = X”u(ξ), y = Y ”u(ξ), so that U(ξ) = u2(ξ) satisfies : U(ξ) + ξU ′(ξ) = g(ξ). Note that this last equation is easily solved in terms of formal series, if g(ξ) = Σngnξn, then U(ξ) = Σn

gn n+1.Inverting the series : ξ = HU(H) :

U(H) = Σh(Σk+l=hakalr2l) Ch

2h

4h(h + 1) Hh r2h into H = ξV (ξ) yields the Birkhoff normal form.

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The coefficients of the Birkhoff normal form are polynomials in a variable r depending on the inertia moments. We checked numerically that their roots are on the unit circle. We proved that this is true for the series issued from the chomology class and discuss the link with D. Ruelle’s articles on the extensions of the Lee-Yang theorem.

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Computation of canonical partition functions This second part closely relates to a question posed by G. Gallavotti and

• C. Marchioro in 1973.

Let us denote by ∆ a root system of rank r. It is a set of vectors of Rr which is invariant under reflections in the hyperplane perpendicular to each vector in ∆. A reflection sρ in terms of a root ρ is defined by sρ(x) = x − ρ(2ρ.x)/ρ2. Thus ∆ is characterized by sρ(η) ∈ ∆, ρ, η ∈ ∆

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The dynamical variables are the coordinates qi, i = 1, ..., r. and their canonically conjugate momenta pi, i = 1, ..., r. The Hamiltonian for the generalized Calogero system with an external quadratic potential (Bordner- Corrigan-Sasaki, 00) is : H = 1 2p2 + 1 2Σρ∈∆+ g|ρ| | ρ |2 (ρ.q)2 + 1 2ω2q2 in which the coupling constants g|ρ| are defined on orbits of the correspon- ding Coxeter group.

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Choose a representation E of dimension D of the Coxeter group, then define the D×D matrix :

• p. ˆ

H : (p ˆ H)αβ = (p.β)δαβ where α and β are vectors belonging to the representation. Introduce next the D×D matrices X, L and M : X = iΣρ∈∆+g|ρ|(ρ ˆ H)1 ρ.qˆ sρ L = p ˆ H + X M = i 2Σρ∈∆+g|ρ| (ρ)2 (ρ.q)2ˆ sρ

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where ˆ sραβ = δα,sρ(β), and a diagonal matrix : Q = q ˆ H, Qαβ = (q.α)δαβ The time evolution of the Hamiltonian displays ˙ L = [L, M] − ω2Q ˙ Q = [Q, M] + L Introduce next : L± = L ± iωQ with P = L+L−,

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we get a Lax Matrix : ˙ P = [P, M] Consider the symplectic form : Tr(dQ∧dL) = CDΣr

j=1dqj∧dpj

defined on the product of two copies of the representation. The constant CD depends actually on the representation. Let Λ be an eigenvalue of the matrix P and let T be the matrix of the projection onto the eigenspace corresponding to this eigenvalue. Classical result of linear perturbation theory yields : dΛ = Tr(TdP)

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The Hamiltonian flow generated by the function Λ and the symplectic form yields : ˙ Q = [Q, M] + iω[T, Q] + (LT + TL), ˙ L = [L, M] + iω[T, L] − ω2(QT + TQ), and thus : ˙ P = [P, M] + 2iω[T, P] = [P, M] This shows that the eigenvalues of the Lax matrix P are constants of motion for the Hamiltonian flow generated by any of its eigenvalues. In particular this proves that the Hamiltonian flows generated by the eigenvalues of the Lax matrix P Poisson commute. The Hamiltonian flows generated by any eigenvalues of the Lax matrix P have all orbits periodic of the same period π/ω.

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Introduce U the (time dependent) matrix solution of the Cauchy pro- blem : ˙ U = UM, U(0) = I The matrix UPU −1 is then a constant of the motion. Denote V a time- independent matrix which diagonalizes this matrix. Conjugate all the ma- trices UL±U −1, UTU −1 by the matrix V yields : ˙ L′± = ±2iωτL′± where L′± = V UL±U −1V −1 and τ is the constant diagonal matrix whose entries are equal to zero except the diagonal term equals to 1 in the position corresponding to the eigenvalue. These equations can be easily integrated and they yield the periodicity of the eigenvalues of the matrix Q =

1 2iω(L+ − L−) At this point we have obtained that there is a familly of

commuting flows which are all isochronous. (Caseiro, JPF, Sasaki, JMP00)

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• G. Gallavotti and C. Marchioro posed the question of finding a classical

proof (non quantum) for finding the value of the integral

• exp(−βH(p, q))dp∧dq.

A proof (JPF, 1988) was based on symplectic geometry : the use of these commuting isochronous flows and the Lojasiewicz inequality which shows that the involved integrals tend to zero at the boundary of the domain of

• integration. This proof was then extended to any root system by (Caseiro,

JPF, Sasaki, JMP00).

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In 2000, another proof was given by K. Aomoto-P. Forrester. It was based

• n the discovery of a Jacobian Identity Associated with Real Hyperplane
• Arrangements. The proof made use of the Griffiths residue theorem from

complex algebraic geometry. Let A be a finite arrangement of hyperplanes in Cr. Let N(A) be the union of hyperplanes of A in Cr and M(A) be its complement in Cr. It is also assumed that A is real, meaning that the defining function of every hyperplane H ∈ A fH(z) = uH,0 + Σr

ν=1uH,νzν

has real coefficients uH,0 and uH,ν. A connected component of M(A) ∩ Rr is called a chamber.

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Theorem, Aomoto-Forrester We fix a chamber ∆j. The mapping Tj : ∆j → Rr defined by : Tj : wν = xν − ΣH∈A λHUH,ν fH(x) from ∆j onto Rr is a diffeomorphism. Furthermore, the sum of the jacobian determinants of the T −1

j

is constant equal to one. The proof relies on complex analytic geometry techniques (Griffiths residue).

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Proposition Assume that f(x) is a summable function on Rr and that g(w) = f(T −1

j

(w)) does not depend on j. Then

• f(x)dx =
• g(w)dw.

This integral formula is deduced from the change of variables formula and the previous Jacobian identity.

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The special case r = 1 is already quite meaningful : +∞

−∞

h(x − Σp

j=1

λj x − aj )dx = +∞

−∞

h(w)dw, (λj > 0, aj ∈ R) was first obtained by G. Boole in the 19th century. Another special case is obtained when Tj : wν = xν − λ0Σ 1 xν − xµ , g(w) = φ(w2

1 + ... + w2 r),

then

• φ(Σx2

ν + λ0 2Σ

1 (xν − xµ)2 − r(r − 1)λ0)dx =

• φ(w2

1 + ... + w2 r)dw.

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In the case φ(u) = e−βu, this yields the initial Gallavotti-Marchioro formula.

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After the remarkable contributions of Aomoto-Forrester, a natural question was posed. Consider in full generality the (classical and/or quantum system) defined by : H = 1 2

• j

y2

j + 1

2

• j

[xj −

• H∈A

λHuH,i fH(x) ]2. Does this Hamiltonian system induces a symplectic action of the torus ?

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This is rather easily disproved in a joint work in progress by R. Caseiro, R. Sasaki and JPF with the 1-dimensional system H = 1 2y2 + 1 2(x − ΣN

j=1

λj x − aj )2 This Hamiltonian is defined on bands ]ai, ai+1[. Under the condition λj > 0, this HS displays a unique stationary point (bi, 0), i = 0, ..., N

• n each intervals ]ai, ai+1[ and these stationary points are minima and of

Birkhoff type. To each of these points one can associate a Birkhoff normal

• form. This Birkhoff normal form can be obtained as follows :

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H = 1 2y2 + 1 2w2, w = x −

N

• j=1

λj x − aj , ω = dx∧dy. Write w = f(x) = Q(x) P(x), Q(x) = ΠN

j=0(x − bj), P(x) = ΠN j=1(x − aj).

Let γ = H−1(c) be a circle of radius c centered at 0, positively oriented and parametrized by : γ = {y = csinθ, w = ccosθ, θ ∈ [0, 2π[}.

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Relative cohomology techniques developped around the isochore Morse lemma (JPF, 83) show that the (inverse of) Birkhoff series is essentially given by the following :

• γ0

xk(w)dy, where xk(w) denotes the unique series obtained by inverting w = f(x) near the point (bk, 0). This inverse series can be obtained in many different ways. Here we focus on the Lagrange inversion formula. It reads as follows : Denote

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gi(z) = ∂i−1 ∂zi−1{[z − bk f(z) ]i} , then xk(w) = bk +

+∞

• i=1

gi(bk)wi i!

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Note that the closed formula derived above for the coefficients of this series can be easily implemented. We did it with Mathematica. The final step is to include this formula in the integral :

• γ=H−1(c)

xk(w)dy. This yields ultimately :

• γ=H−1(c)

xk(w)dy =

• i

2π 1 i!gi(bk)ci+1(cosθ)i+1dθ

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=

• p

1 (2p − 1)!g2p−1(bk)c2p 2π cos2pθdθ =

• p

2p (2pp!)2g2p−1(bk)c2p.

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To conclude on the new results obtained : Relative cohomology class techniques allowed to find explicit analytic continuation of the Birkhoff normal form in the examples of the Pendulum and of the Free rigid body (Joint work with P. Garrido and G. Gallavotti) Lagrange inversion formula (which had been already used in several times the context of pertubation theory (Inoue)) provide an explicit expression for the Birkhoff series of the 1-dim Aomoto-Forrester system. It shows that in general there is no associated symplectic action of the torus but also provide a nice information for the semi-classical approach to the quantized system. (Joint work with R. Caseiro and R. Sasaki)

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