Duality between BC ( n )-type Sutherland and RuijsenaarsSchneider - - PowerPoint PPT Presentation

Introduction Reduction and Lax matrices Duality Open problems

Duality between BC(n)-type Sutherland and Ruijsenaars–Schneider models

B´ ela G´ abor Pusztai

Bolyai Institute, University of Szeged Aradi v´ ertan´ uk tere 1, H-6720 Szeged, Hungary

September 4, 2012

Introduction Reduction and Lax matrices Duality Open problems

References

Talk is based on the following papers: B.G. Pusztai, Action-angle duality between the Cn-type hyperbolic Sutherland and the rational Ruijsenaars–Schneider–van Diejen models,

• Nucl. Phys. B853 (2011) 139-173

B.G. Pusztai, The hyperbolic BCn Sutherland and the rational BCn Ruijsenaars–Schneider–van Diejen models: Lax matrices and duality,

• Nucl. Phys. B856 (2012) 528-551

Introduction Reduction and Lax matrices Duality Open problems

The hyperbolic BCn Sutherland model

Let c = {q = (q1, . . . , qn) ∈ Rn | q1 > . . . > qn > 0}. Phase space: PS = T ∗c ∼ = {(q, p) | q ∈ c, p ∈ Rn} Symplectic form: ωS = n

c=1 dqc ∧ dpc

Hamiltonian: HS = 1 2

n

• c=1

p2

c +

• 1≤a<b≤n
• g2

sinh2(qa − qb) + g2 sinh2(qa + qb)

• +

n

• c=1

g2

1

sinh2(qc) +

n

• c=1

g2

2

sinh2(2qc) (g, g1, g2 are real coupling parameters; g2 > 0, g2

1 + g2 2 > 0)

Introduction Reduction and Lax matrices Duality Open problems

The hyperbolic BCn Sutherland model

Let c = {q = (q1, . . . , qn) ∈ Rn | q1 > . . . > qn > 0}. Phase space: PS = T ∗c ∼ = {(q, p) | q ∈ c, p ∈ Rn} Symplectic form: ωS = n

c=1 dqc ∧ dpc

Hamiltonian: HS = 1 2

n

• c=1

p2

c +

• 1≤a<b≤n
• g2

sinh2(qa − qb) + g2 sinh2(qa + qb)

• +

n

• c=1

g2

1

sinh2(qc) +

n

• c=1

g2

2

sinh2(2qc) (g, g1, g2 are real coupling parameters; g2 > 0, g2

1 + g2 2 > 0)

Introduction Reduction and Lax matrices Duality Open problems

The hyperbolic BCn Sutherland model

Let c = {q = (q1, . . . , qn) ∈ Rn | q1 > . . . > qn > 0}. Phase space: PS = T ∗c ∼ = {(q, p) | q ∈ c, p ∈ Rn} Symplectic form: ωS = n

c=1 dqc ∧ dpc

Hamiltonian: HS = 1 2

n

• c=1

p2

c +

• 1≤a<b≤n
• g2

sinh2(qa − qb) + g2 sinh2(qa + qb)

• +

n

• c=1

g2

1

sinh2(qc) +

n

• c=1

g2

2

sinh2(2qc) (g, g1, g2 are real coupling parameters; g2 > 0, g2

1 + g2 2 > 0)

Introduction Reduction and Lax matrices Duality Open problems

The rational BCn RSvD model

Phase space: PR = T ∗c ∼ = {(λ, θ) | λ ∈ c, θ ∈ Rn} Symplectic form: ωR = n

c=1 dθc ∧ dλc

Hamiltonian: HR =

n

• c=1

cosh(2θc)

• 1 + ν2

λ2

c

1

2

1 + κ2 λ2

c

1

2

×

n

• a=1

(a=c)

• 1 +

4µ2 (λc − λa)2 1

2

1 + 4µ2 (λc + λa)2 1

2

+ νκ 4µ2

n

• c=1
• 1 + 4µ2

λ2

c

• − νκ

4µ2 (µ, ν and κ are real parameters satisfying µ = 0 = ν and νκ ≥ 0)

Introduction Reduction and Lax matrices Duality Open problems

The rational BCn RSvD model

Phase space: PR = T ∗c ∼ = {(λ, θ) | λ ∈ c, θ ∈ Rn} Symplectic form: ωR = n

c=1 dθc ∧ dλc

Hamiltonian: HR =

n

• c=1

cosh(2θc)

• 1 + ν2

λ2

c

1

2

1 + κ2 λ2

c

1

2

×

n

• a=1

(a=c)

• 1 +

4µ2 (λc − λa)2 1

2

1 + 4µ2 (λc + λa)2 1

2

+ νκ 4µ2

n

• c=1
• 1 + 4µ2

λ2

c

• − νκ

4µ2 (µ, ν and κ are real parameters satisfying µ = 0 = ν and νκ ≥ 0)

Introduction Reduction and Lax matrices Duality Open problems

The rational BCn RSvD model

Phase space: PR = T ∗c ∼ = {(λ, θ) | λ ∈ c, θ ∈ Rn} Symplectic form: ωR = n

c=1 dθc ∧ dλc

Hamiltonian: HR =

n

• c=1

cosh(2θc)

• 1 + ν2

λ2

c

1

2

1 + κ2 λ2

c

1

2

×

n

• a=1

(a=c)

• 1 +

4µ2 (λc − λa)2 1

2

1 + 4µ2 (λc + λa)2 1

2

+ νκ 4µ2

n

• c=1
• 1 + 4µ2

λ2

c

• − νκ

4µ2 (µ, ν and κ are real parameters satisfying µ = 0 = ν and νκ ≥ 0)

Introduction Reduction and Lax matrices Duality Open problems

Motivation

Why to study the Calogero–Moser–Sutherland (CMS) and the Ruijsenaars–Schneider–van Diejen (RSvD) type classical many-particle systems? Nice problems in classical mechanics Close relationship between the scattering theory of the CMS/RSvD models and the scattering characteristic of certain soliton equations (Ruijsenaars–Schneider 1986, Babelon–Bernard 1993, Kapustin–Skorik 1994) The CMS/RSvD models naturally generate certain integrable random matrix ensembles (Bogomolny–Giraud–Schmit 2009) (The links between the particle systems, the soliton equations, and the integrable random matrix ensembles are well understood only in the An case.)

Introduction Reduction and Lax matrices Duality Open problems

Motivation

Why to study the Calogero–Moser–Sutherland (CMS) and the Ruijsenaars–Schneider–van Diejen (RSvD) type classical many-particle systems? Nice problems in classical mechanics Close relationship between the scattering theory of the CMS/RSvD models and the scattering characteristic of certain soliton equations (Ruijsenaars–Schneider 1986, Babelon–Bernard 1993, Kapustin–Skorik 1994) The CMS/RSvD models naturally generate certain integrable random matrix ensembles (Bogomolny–Giraud–Schmit 2009) (The links between the particle systems, the soliton equations, and the integrable random matrix ensembles are well understood only in the An case.)

Introduction Reduction and Lax matrices Duality Open problems

Motivation

Why to study the Calogero–Moser–Sutherland (CMS) and the Ruijsenaars–Schneider–van Diejen (RSvD) type classical many-particle systems? Nice problems in classical mechanics Close relationship between the scattering theory of the CMS/RSvD models and the scattering characteristic of certain soliton equations (Ruijsenaars–Schneider 1986, Babelon–Bernard 1993, Kapustin–Skorik 1994) The CMS/RSvD models naturally generate certain integrable random matrix ensembles (Bogomolny–Giraud–Schmit 2009) (The links between the particle systems, the soliton equations, and the integrable random matrix ensembles are well understood only in the An case.)

Introduction Reduction and Lax matrices Duality Open problems

Motivation

Why to study the Calogero–Moser–Sutherland (CMS) and the Ruijsenaars–Schneider–van Diejen (RSvD) type classical many-particle systems? Nice problems in classical mechanics Close relationship between the scattering theory of the CMS/RSvD models and the scattering characteristic of certain soliton equations (Ruijsenaars–Schneider 1986, Babelon–Bernard 1993, Kapustin–Skorik 1994) The CMS/RSvD models naturally generate certain integrable random matrix ensembles (Bogomolny–Giraud–Schmit 2009) (The links between the particle systems, the soliton equations, and the integrable random matrix ensembles are well understood only in the An case.)

Introduction Reduction and Lax matrices Duality Open problems

Motivation

Why to study the Calogero–Moser–Sutherland (CMS) and the Ruijsenaars–Schneider–van Diejen (RSvD) type classical many-particle systems? Nice problems in classical mechanics Close relationship between the scattering theory of the CMS/RSvD models and the scattering characteristic of certain soliton equations (Ruijsenaars–Schneider 1986, Babelon–Bernard 1993, Kapustin–Skorik 1994) The CMS/RSvD models naturally generate certain integrable random matrix ensembles (Bogomolny–Giraud–Schmit 2009) (The links between the particle systems, the soliton equations, and the integrable random matrix ensembles are well understood only in the An case.)

Introduction Reduction and Lax matrices Duality Open problems

Summarizing the results

In a symplectic reduction framework we work out the following: Construction of a Lax matrix for the rational BCn RSvD model with three independent coupling parameters (µ, ν and κ) Construction of action-angle variables for both the hyperbolic BCn Sutherland and the rational BCn RSvD models Establishing the action-angle duality between these BCn-type many-particle systems

Introduction Reduction and Lax matrices Duality Open problems

Summarizing the results

In a symplectic reduction framework we work out the following: Construction of a Lax matrix for the rational BCn RSvD model with three independent coupling parameters (µ, ν and κ) Construction of action-angle variables for both the hyperbolic BCn Sutherland and the rational BCn RSvD models Establishing the action-angle duality between these BCn-type many-particle systems

Introduction Reduction and Lax matrices Duality Open problems

Summarizing the results

In a symplectic reduction framework we work out the following: Construction of a Lax matrix for the rational BCn RSvD model with three independent coupling parameters (µ, ν and κ) Construction of action-angle variables for both the hyperbolic BCn Sutherland and the rational BCn RSvD models Establishing the action-angle duality between these BCn-type many-particle systems

Introduction Reduction and Lax matrices Duality Open problems

Group theoretic background

We need the following objects: C = 0n 1n 1n 0n

• ∈ U(2n)

U(n, n) = {y ∈ GL(2n, C) | y∗Cy = C} K = U(n, n) ∩ U(2n) (maximal compact subgroup in U(n, n)) u(n, n) = {Y ∈ gl(2n, C) | Y ∗C + CY = 0} k = Lie(K) = u(n, n) ∩ u(2n) p = {Y ∈ u(n, n) | Y ∗ = Y } a = {diagonal elements of p} (maximal Abelian subspace in p) u(n, n) = k ⊕ p (Cartan decomposition)

Introduction Reduction and Lax matrices Duality Open problems

Group theoretic background

We need the following objects: C = 0n 1n 1n 0n

• ∈ U(2n)

U(n, n) = {y ∈ GL(2n, C) | y∗Cy = C} K = U(n, n) ∩ U(2n) (maximal compact subgroup in U(n, n)) u(n, n) = {Y ∈ gl(2n, C) | Y ∗C + CY = 0} k = Lie(K) = u(n, n) ∩ u(2n) p = {Y ∈ u(n, n) | Y ∗ = Y } a = {diagonal elements of p} (maximal Abelian subspace in p) u(n, n) = k ⊕ p (Cartan decomposition)

Introduction Reduction and Lax matrices Duality Open problems

Group theoretic background

We need the following objects: C = 0n 1n 1n 0n

• ∈ U(2n)

U(n, n) = {y ∈ GL(2n, C) | y∗Cy = C} K = U(n, n) ∩ U(2n) (maximal compact subgroup in U(n, n)) u(n, n) = {Y ∈ gl(2n, C) | Y ∗C + CY = 0} k = Lie(K) = u(n, n) ∩ u(2n) p = {Y ∈ u(n, n) | Y ∗ = Y } a = {diagonal elements of p} (maximal Abelian subspace in p) u(n, n) = k ⊕ p (Cartan decomposition)

Introduction Reduction and Lax matrices Duality Open problems

Symplectic reduction setup

Let ξ(E) = iµ(EE ∗ − 1N) + i(µ − ν)C ∈ k with column vector E = (1, . . . , 1, −1, . . . , −1)∗ ∈ R2n, and consider the (co-)adjoint orbit O = O(ξ(E)) through ξ(E) Smooth left action of K × K on the (extended) phase space Pext = T ∗U(n, n) × O ∼ = U(n, n) × u(n, n) × O: (kL, kR) . (y, Y , ρ) = (kLykR, kRYk−1

R , kLρk−1 L ),

where kL, kR ∈ K and y ∈ U(n, n), Y ∈ u(n, n), ρ ∈ O Momentum map Jext : U(n, n) × u(n, n) × O → k ⊕ k, (y, Y , ρ) →

• (yYy−1)k + ρ
• ⊕ (−Yk − κiC)

Perform Marsden–Weinstein reduction by fixing Jext to 0

Introduction Reduction and Lax matrices Duality Open problems

Symplectic reduction setup

Let ξ(E) = iµ(EE ∗ − 1N) + i(µ − ν)C ∈ k with column vector E = (1, . . . , 1, −1, . . . , −1)∗ ∈ R2n, and consider the (co-)adjoint orbit O = O(ξ(E)) through ξ(E) Smooth left action of K × K on the (extended) phase space Pext = T ∗U(n, n) × O ∼ = U(n, n) × u(n, n) × O: (kL, kR) . (y, Y , ρ) = (kLykR, kRYk−1

R , kLρk−1 L ),

where kL, kR ∈ K and y ∈ U(n, n), Y ∈ u(n, n), ρ ∈ O Momentum map Jext : U(n, n) × u(n, n) × O → k ⊕ k, (y, Y , ρ) →

• (yYy−1)k + ρ
• ⊕ (−Yk − κiC)

Perform Marsden–Weinstein reduction by fixing Jext to 0

Introduction Reduction and Lax matrices Duality Open problems

Symplectic reduction setup

Let ξ(E) = iµ(EE ∗ − 1N) + i(µ − ν)C ∈ k with column vector E = (1, . . . , 1, −1, . . . , −1)∗ ∈ R2n, and consider the (co-)adjoint orbit O = O(ξ(E)) through ξ(E) Smooth left action of K × K on the (extended) phase space Pext = T ∗U(n, n) × O ∼ = U(n, n) × u(n, n) × O: (kL, kR) . (y, Y , ρ) = (kLykR, kRYk−1

R , kLρk−1 L ),

where kL, kR ∈ K and y ∈ U(n, n), Y ∈ u(n, n), ρ ∈ O Momentum map Jext : U(n, n) × u(n, n) × O → k ⊕ k, (y, Y , ρ) →

• (yYy−1)k + ρ
• ⊕ (−Yk − κiC)

Perform Marsden–Weinstein reduction by fixing Jext to 0

Introduction Reduction and Lax matrices Duality Open problems

Symplectic reduction setup

Let ξ(E) = iµ(EE ∗ − 1N) + i(µ − ν)C ∈ k with column vector E = (1, . . . , 1, −1, . . . , −1)∗ ∈ R2n, and consider the (co-)adjoint orbit O = O(ξ(E)) through ξ(E) Smooth left action of K × K on the (extended) phase space Pext = T ∗U(n, n) × O ∼ = U(n, n) × u(n, n) × O: (kL, kR) . (y, Y , ρ) = (kLykR, kRYk−1

R , kLρk−1 L ),

where kL, kR ∈ K and y ∈ U(n, n), Y ∈ u(n, n), ρ ∈ O Momentum map Jext : U(n, n) × u(n, n) × O → k ⊕ k, (y, Y , ρ) →

• (yYy−1)k + ρ
• ⊕ (−Yk − κiC)

Perform Marsden–Weinstein reduction by fixing Jext to 0

Introduction Reduction and Lax matrices Duality Open problems

Identification of the level set (Sutherland picture)

First step: Solve the constraint Jext = 0 Parametrization of the level set (Jext)−1({0}) induced by the KAK decomposition of U(n, n): ΥS : PS × (K × K)/U(1)diag → (Jext)−1({0}), (q, p, [(ηL, ηR)]) → (ηLeQη−1

R , ηRL(q, p)η−1 R , ηLξ(E)η−1 L ),

where Q = diag(q1, . . . , qn, −q1, . . . , −qn) ∈ a, and L(q, p) = P − sinh(adQ|off-diag)−1ξ(E) + coth(adQ|off-diag)(κiC) − κiC ∈ g with P = diag(p1, . . . , pn, −p1, . . . , −pn) ∈ a. Note that L is the Lax matrix of the hyperbolic BCn Sutherland model.

Introduction Reduction and Lax matrices Duality Open problems

Identification of the level set (Sutherland picture)

First step: Solve the constraint Jext = 0 Parametrization of the level set (Jext)−1({0}) induced by the KAK decomposition of U(n, n): ΥS : PS × (K × K)/U(1)diag → (Jext)−1({0}), (q, p, [(ηL, ηR)]) → (ηLeQη−1

R , ηRL(q, p)η−1 R , ηLξ(E)η−1 L ),

where Q = diag(q1, . . . , qn, −q1, . . . , −qn) ∈ a, and L(q, p) = P − sinh(adQ|off-diag)−1ξ(E) + coth(adQ|off-diag)(κiC) − κiC ∈ g with P = diag(p1, . . . , pn, −p1, . . . , −pn) ∈ a. Note that L is the Lax matrix of the hyperbolic BCn Sutherland model.

Introduction Reduction and Lax matrices Duality Open problems

The reduced phase space (Sutherland picture)

Second step: Identify the reduced phase space Pext/ /0(K × K) The reduced phase space can be identified with the base manifold

• f the (trivial) principal (K × K)/U(1)diag-bundle

πS : MS =PS × (K × K)/U(1)diag ։ PS, (q, p, [(ηL, ηR)]) → (q, p) Theorem ( Feh´ er–BGP 2007) The (global) coordinates qc, pc form a Darboux system on the reduced manifold PS, i.e. for the reduced symplectic form we have ωS = n

c=1 dqc ∧ dpc.

Introduction Reduction and Lax matrices Duality Open problems

The reduced phase space (Sutherland picture)

Second step: Identify the reduced phase space Pext/ /0(K × K) The reduced phase space can be identified with the base manifold

• f the (trivial) principal (K × K)/U(1)diag-bundle

πS : MS =PS × (K × K)/U(1)diag ։ PS, (q, p, [(ηL, ηR)]) → (q, p) Theorem ( Feh´ er–BGP 2007) The (global) coordinates qc, pc form a Darboux system on the reduced manifold PS, i.e. for the reduced symplectic form we have ωS = n

c=1 dqc ∧ dpc.

Introduction Reduction and Lax matrices Duality Open problems

The reduced phase space (Sutherland picture)

Second step: Identify the reduced phase space Pext/ /0(K × K) The reduced phase space can be identified with the base manifold

• f the (trivial) principal (K × K)/U(1)diag-bundle

πS : MS =PS × (K × K)/U(1)diag ։ PS, (q, p, [(ηL, ηR)]) → (q, p) Theorem ( Feh´ er–BGP 2007) The (global) coordinates qc, pc form a Darboux system on the reduced manifold PS, i.e. for the reduced symplectic form we have ωS = n

c=1 dqc ∧ dpc.

Introduction Reduction and Lax matrices Duality Open problems

Lax matrix for the rational BCn RSvD models

For any λ = (λ1, . . . , λn) ∈ c, θ = (θ1, . . . , θn) ∈ Rn and a, b, c ∈ {1, . . . , n} we define: zc(λ) = −

• 1 + iν

λc

• n
• d=1

(d=c)

• 1 +

2iµ λc − λd 1 + 2iµ λc + λd

• ,

AC

a,b(λ, θ) = eθa+θb|za(λ)zb(λ)|

1 2

2iµ 2iµ + λa − λb , AC

n+a,n+b(λ, θ) = e−θa−θb za(λ)zb(λ)

|za(λ)zb(λ)|

1 2

2iµ 2iµ − λa + λb , AC

a,n+b(λ, θ) = An+b,a(λ, θ)

= eθa−θbzb(λ)|za(λ)zb(λ)−1|

1 2

2iµ 2iµ + λa + λb + i(µ − ν) iµ + λa δa,b.

Introduction Reduction and Lax matrices Duality Open problems

Lax matrix for the rational BCn RSvD models

We also need the functions α(x) =

• x +

√ x2 + κ2 √ 2x and β(x) = iκ 1 √ 2x 1

• x +

√ x2 + κ2 defined for x > 0, and the (2n) × (2n) Hermitian matrix h(λ) = diag(α(λ1), . . . , α(λn)) diag(β(λ1), . . . , β(λn)) −diag(β(λ1), . . . , β(λn)) diag(α(λ1), . . . , α(λn))

• .

The Lax matrix of the rational BCn RSvD model is the Hermitian (2n) × (2n) matrix ABC(λ, θ) = h(λ)−1AC(λ, θ)h(λ)−1 ∈ U(n, n).

Introduction Reduction and Lax matrices Duality Open problems

Lax matrix for the rational BCn RSvD models

We also need the functions α(x) =

• x +

√ x2 + κ2 √ 2x and β(x) = iκ 1 √ 2x 1

• x +

√ x2 + κ2 defined for x > 0, and the (2n) × (2n) Hermitian matrix h(λ) = diag(α(λ1), . . . , α(λn)) diag(β(λ1), . . . , β(λn)) −diag(β(λ1), . . . , β(λn)) diag(α(λ1), . . . , α(λn))

• .

The Lax matrix of the rational BCn RSvD model is the Hermitian (2n) × (2n) matrix ABC(λ, θ) = h(λ)−1AC(λ, θ)h(λ)−1 ∈ U(n, n).

Introduction Reduction and Lax matrices Duality Open problems

Identification of the level set (Ruijsenaars picture)

First step: Solve the constraint Jext = 0 Parametrization of the level set (Jext)−1({0}) induced by diagonalizing the Lie algebra part of Pext: ΥR : PR × (K × K)/U(1)diag → (Jext)−1({0}), (λ, θ, [(ηL, ηR)]) → (ηLAC(λ, θ)

1 2 h(λ)−1η−1

R , ηRh(λ)Λh(λ)−1η−1 R , ηLξ(V(λ, θ))η−1 L ),

where Λ = diag(λ1, . . . , λn, −λ1, . . . , −λn) ∈ a, V(λ, θ) = AC(λ, θ)− 1

2 F(λ, θ),

with column vector F(λ, θ) ∈ C2n given by Fc = eθc|zc|

1 2 ,

Fn+c = e−θczc|zc|− 1

2

(1 ≤ c ≤ n)

Introduction Reduction and Lax matrices Duality Open problems

Identification of the level set (Ruijsenaars picture)

First step: Solve the constraint Jext = 0 Parametrization of the level set (Jext)−1({0}) induced by diagonalizing the Lie algebra part of Pext: ΥR : PR × (K × K)/U(1)diag → (Jext)−1({0}), (λ, θ, [(ηL, ηR)]) → (ηLAC(λ, θ)

1 2 h(λ)−1η−1

R , ηRh(λ)Λh(λ)−1η−1 R , ηLξ(V(λ, θ))η−1 L ),

where Λ = diag(λ1, . . . , λn, −λ1, . . . , −λn) ∈ a, V(λ, θ) = AC(λ, θ)− 1

2 F(λ, θ),

with column vector F(λ, θ) ∈ C2n given by Fc = eθc|zc|

1 2 ,

Fn+c = e−θczc|zc|− 1

2

(1 ≤ c ≤ n)

Introduction Reduction and Lax matrices Duality Open problems

The reduced phase space (Ruijsenaars picture)

Second step: Identify the reduced phase space Pext/ /0(K × K) The reduced phase space can be identified with the base manifold

• f the (trivial) principal (K × K)/U(1)diag-bundle

πR : MR =PR × (K × K)/U(1)diag ։ PR, (λ, θ, [(ηL, ηR)]) → (λ, θ) Theorem (BGP 2011-2012) The (global) coordinates θc, λc form a Darboux system on the reduced manifold PR, i.e. for the reduced symplectic form we have ωR = n

c=1 dθc ∧ dλc.

Introduction Reduction and Lax matrices Duality Open problems

The reduced phase space (Ruijsenaars picture)

Second step: Identify the reduced phase space Pext/ /0(K × K) The reduced phase space can be identified with the base manifold

• f the (trivial) principal (K × K)/U(1)diag-bundle

πR : MR =PR × (K × K)/U(1)diag ։ PR, (λ, θ, [(ηL, ηR)]) → (λ, θ) Theorem (BGP 2011-2012) The (global) coordinates θc, λc form a Darboux system on the reduced manifold PR, i.e. for the reduced symplectic form we have ωR = n

c=1 dθc ∧ dλc.

Introduction Reduction and Lax matrices Duality Open problems

The reduced phase space (Ruijsenaars picture)

Second step: Identify the reduced phase space Pext/ /0(K × K) The reduced phase space can be identified with the base manifold

• f the (trivial) principal (K × K)/U(1)diag-bundle

πR : MR =PR × (K × K)/U(1)diag ։ PR, (λ, θ, [(ηL, ηR)]) → (λ, θ) Theorem (BGP 2011-2012) The (global) coordinates θc, λc form a Darboux system on the reduced manifold PR, i.e. for the reduced symplectic form we have ωR = n

c=1 dθc ∧ dλc.

Introduction Reduction and Lax matrices Duality Open problems

Dual reduction picture

Let L0 = (Jext)−1({0}). Since PS and PR are two models of the same reduced symplectic manifold Pext/ /0(K × K), there is a (unique) symplectomorphism S making the diagram Pext MS

ΥS ∼ =

• πS
• L0
• ι
• MR

ΥR ∼ =

• πR
• PS

S ∼ =

PR

• commutative. The relationship between the CMS and the RSvD

coupling parameters: g2 = µ2, g2

1 = 1

2νκ, g2

2 = 1

2(ν − κ)2

Introduction Reduction and Lax matrices Duality Open problems

Action-angle duality

Immediate consequence of the dual reduction picture: The (global) coordinates S∗λc, S∗θc on PS are action-angle variables for the Sutherland model The (global) coordinates (S−1)∗qc, (S−1)∗pc on PR are action-angle variables for the RSvD model Action-angle duality: The action variables of the hyperbolic BCn Sutherland model are the particle-positions of the rational BCn RSvD model, and vice versa.

• Remark. Just as for the A-type models (Ruijsenaars 1987), the

action-angle duality plays a crucial role in the precise formulation

• f the (time-dependent) scattering theory!

Introduction Reduction and Lax matrices Duality Open problems

Action-angle duality

Immediate consequence of the dual reduction picture: The (global) coordinates S∗λc, S∗θc on PS are action-angle variables for the Sutherland model The (global) coordinates (S−1)∗qc, (S−1)∗pc on PR are action-angle variables for the RSvD model Action-angle duality: The action variables of the hyperbolic BCn Sutherland model are the particle-positions of the rational BCn RSvD model, and vice versa.

• Remark. Just as for the A-type models (Ruijsenaars 1987), the

action-angle duality plays a crucial role in the precise formulation

• f the (time-dependent) scattering theory!

Introduction Reduction and Lax matrices Duality Open problems

Action-angle duality

Immediate consequence of the dual reduction picture: The (global) coordinates S∗λc, S∗θc on PS are action-angle variables for the Sutherland model The (global) coordinates (S−1)∗qc, (S−1)∗pc on PR are action-angle variables for the RSvD model Action-angle duality: The action variables of the hyperbolic BCn Sutherland model are the particle-positions of the rational BCn RSvD model, and vice versa.

• Remark. Just as for the A-type models (Ruijsenaars 1987), the

action-angle duality plays a crucial role in the precise formulation

• f the (time-dependent) scattering theory!

Introduction Reduction and Lax matrices Duality Open problems

Action-angle duality

Immediate consequence of the dual reduction picture: The (global) coordinates S∗λc, S∗θc on PS are action-angle variables for the Sutherland model The (global) coordinates (S−1)∗qc, (S−1)∗pc on PR are action-angle variables for the RSvD model Action-angle duality: The action variables of the hyperbolic BCn Sutherland model are the particle-positions of the rational BCn RSvD model, and vice versa.

• Remark. Just as for the A-type models (Ruijsenaars 1987), the

action-angle duality plays a crucial role in the precise formulation

• f the (time-dependent) scattering theory!

Introduction Reduction and Lax matrices Duality Open problems

Numerical experiment (n = 4)

Introduction Reduction and Lax matrices Duality Open problems

Scattering theoretic result on the Cn Sutherland model

Theorem (BGP 2011) For each trajectory R ∋ t → (q(t), p(t)) ∈ c × Rn of the hyperbolic Cn Sutherland model (g1 = 0) we have the temporal asymptotics qc(t) ∼ q±

c + tp± c and pc(t) ∼ p± c as t → ±∞ with

the properties p+

c = −p− c and p+ 1 > . . . > p+ n > 0, together with

q+

c = − q− c − c−1

• a=1

δ(p−

c − p− a , g) + n

• a=c+1

δ(p−

c − p− a , g)

+

n

• a=1

(a=c)

δ(p−

c + p− a , g) + δ(2p− c ,

√ 2g2), where δ(p, µ) = 2−1 ln(1 + 4µ2p−2).

Introduction Reduction and Lax matrices Duality Open problems

Open problems on scattering theory

Consequences. Due to the relationship p+

c = −p− c the Cn Sutherland can be

seen as a pure soliton system in the sense of Ruijsenaars. The scattering map of the Cn Sutherland model has a factorized form, i.e., the classical phase shifts are entirely determined by the 2-particle processes and by the 1-particle scatterings on the external field.

• PROBLEMS. Work out the (rigorous!) time-dependent scattering

theory of the classical and the quantum BCn Sutherland/RSvD models (scattering states, comparison dynamics, Møller wave maps, scattering map, ...). Relate it to the soliton scattering properties of integrable field theories.

Introduction Reduction and Lax matrices Duality Open problems

Open problems on scattering theory

Consequences. Due to the relationship p+

c = −p− c the Cn Sutherland can be

seen as a pure soliton system in the sense of Ruijsenaars. The scattering map of the Cn Sutherland model has a factorized form, i.e., the classical phase shifts are entirely determined by the 2-particle processes and by the 1-particle scatterings on the external field.

• PROBLEMS. Work out the (rigorous!) time-dependent scattering

theory of the classical and the quantum BCn Sutherland/RSvD models (scattering states, comparison dynamics, Møller wave maps, scattering map, ...). Relate it to the soliton scattering properties of integrable field theories.

Introduction Reduction and Lax matrices Duality Open problems

Open problems on scattering theory

Consequences. Due to the relationship p+

c = −p− c the Cn Sutherland can be

seen as a pure soliton system in the sense of Ruijsenaars. The scattering map of the Cn Sutherland model has a factorized form, i.e., the classical phase shifts are entirely determined by the 2-particle processes and by the 1-particle scatterings on the external field.

• PROBLEMS. Work out the (rigorous!) time-dependent scattering

theory of the classical and the quantum BCn Sutherland/RSvD models (scattering states, comparison dynamics, Møller wave maps, scattering map, ...). Relate it to the soliton scattering properties of integrable field theories.

Introduction Reduction and Lax matrices Duality Open problems

Open problems on the r-matrix structures

Since the eigenvalues of the Lax matrix L are in involution, we have {L ⊗ , L} = [r12, L ⊗ 1] − [r21, 1 ⊗ L] with some (dynamical) r-matrix r12. PROBLEMS. Find the explicit form of r. Many things are known. (Avan–Talon 1993, Sklyanin 1994, Braden–Suzuki 1994, Avan–Babelon–Talon 1994, Suris 1997, Forger–Winterhalder 2002, BGP arXiv:12051029, ...) However, almost nothing is known about the r-matrix structure of the elliptic BCn CMS model, the Inozemtsev’s model, the non-A-type RSvD models, ... Find Yang–Baxter-type algebraic structures underlying these models.

Introduction Reduction and Lax matrices Duality Open problems

Open problems on the r-matrix structures

Since the eigenvalues of the Lax matrix L are in involution, we have {L ⊗ , L} = [r12, L ⊗ 1] − [r21, 1 ⊗ L] with some (dynamical) r-matrix r12. PROBLEMS. Find the explicit form of r. Many things are known. (Avan–Talon 1993, Sklyanin 1994, Braden–Suzuki 1994, Avan–Babelon–Talon 1994, Suris 1997, Forger–Winterhalder 2002, BGP arXiv:12051029, ...) However, almost nothing is known about the r-matrix structure of the elliptic BCn CMS model, the Inozemtsev’s model, the non-A-type RSvD models, ... Find Yang–Baxter-type algebraic structures underlying these models.

Introduction Reduction and Lax matrices Duality Open problems

Open problems on the r-matrix structures

Since the eigenvalues of the Lax matrix L are in involution, we have {L ⊗ , L} = [r12, L ⊗ 1] − [r21, 1 ⊗ L] with some (dynamical) r-matrix r12. PROBLEMS. Find the explicit form of r. Many things are known. (Avan–Talon 1993, Sklyanin 1994, Braden–Suzuki 1994, Avan–Babelon–Talon 1994, Suris 1997, Forger–Winterhalder 2002, BGP arXiv:12051029, ...) However, almost nothing is known about the r-matrix structure of the elliptic BCn CMS model, the Inozemtsev’s model, the non-A-type RSvD models, ... Find Yang–Baxter-type algebraic structures underlying these models.

Introduction Reduction and Lax matrices Duality Open problems

Open problems on the hyperbolic/trigonometric and the elliptic RSvD models

The Lax matrices of the A-type RS models are well understood (Ruijsenaars 1987). Also, we have constructed a Lax matrix for the rational BCn RSvD model with three independent parameters (BGP 2011-2012). However, the Lax matrices for the generic non-A-type hyperbolic/trigonometric/elliptic RSvD models are still missing! Derive the non-A-type hyperbolic/trigonometric RSvD models from an appropriate symplectic reduction framework. (Surprisingly, even the symplectic reduction derivation of the hyperbolic An RS model is missing!)

Introduction Reduction and Lax matrices Duality Open problems

Open problems on the hyperbolic/trigonometric and the elliptic RSvD models

The Lax matrices of the A-type RS models are well understood (Ruijsenaars 1987). Also, we have constructed a Lax matrix for the rational BCn RSvD model with three independent parameters (BGP 2011-2012). However, the Lax matrices for the generic non-A-type hyperbolic/trigonometric/elliptic RSvD models are still missing! Derive the non-A-type hyperbolic/trigonometric RSvD models from an appropriate symplectic reduction framework. (Surprisingly, even the symplectic reduction derivation of the hyperbolic An RS model is missing!)

Introduction Reduction and Lax matrices Duality Open problems

Open problems on the quantum mechanical models

The (non-elliptic) quantum CMS models can derived from quantum Hamiltonian reduction (Feh´ er–BGP 2007, 2009). Apply/Generalize this technique for the RSvD models and provide a conceptual understanding of the duality between the CMS and the RSvD models at the quantum level, too. (Bispectrality?)