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 C n -type hyperbolic Sutherland and the rational Ruijsenaars–Schneider–van Diejen models, Nucl. Phys. B853 (2011) 139-173 B.G. Pusztai, The hyperbolic BC n Sutherland and the rational BC n 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 BC n Sutherland model Let c = { q = ( q 1 , . . . , q n ) ∈ R n | q 1 > . . . > q n > 0 } . Phase space: P S = T ∗ c ∼ = { ( q , p ) | q ∈ c , p ∈ R n } Symplectic form: ω S = � n c =1 d q c ∧ d p c Hamiltonian: n g 2 g 2 � � H S = 1 � � p 2 c + sinh 2 ( q a − q b ) + sinh 2 ( q a + q b ) 2 c =1 1 ≤ a < b ≤ n n n g 2 g 2 � � 1 2 + sinh 2 ( q c ) + sinh 2 (2 q c ) c =1 c =1 ( g , g 1 , g 2 are real coupling parameters; g 2 > 0, g 2 1 + g 2 2 > 0)
Introduction Reduction and Lax matrices Duality Open problems The hyperbolic BC n Sutherland model Let c = { q = ( q 1 , . . . , q n ) ∈ R n | q 1 > . . . > q n > 0 } . Phase space: P S = T ∗ c ∼ = { ( q , p ) | q ∈ c , p ∈ R n } Symplectic form: ω S = � n c =1 d q c ∧ d p c Hamiltonian: n g 2 g 2 � � H S = 1 � � p 2 c + sinh 2 ( q a − q b ) + sinh 2 ( q a + q b ) 2 c =1 1 ≤ a < b ≤ n n n g 2 g 2 � � 1 2 + sinh 2 ( q c ) + sinh 2 (2 q c ) c =1 c =1 ( g , g 1 , g 2 are real coupling parameters; g 2 > 0, g 2 1 + g 2 2 > 0)
Introduction Reduction and Lax matrices Duality Open problems The hyperbolic BC n Sutherland model Let c = { q = ( q 1 , . . . , q n ) ∈ R n | q 1 > . . . > q n > 0 } . Phase space: P S = T ∗ c ∼ = { ( q , p ) | q ∈ c , p ∈ R n } Symplectic form: ω S = � n c =1 d q c ∧ d p c Hamiltonian: n g 2 g 2 � � H S = 1 � � p 2 c + sinh 2 ( q a − q b ) + sinh 2 ( q a + q b ) 2 c =1 1 ≤ a < b ≤ n n n g 2 g 2 � � 1 2 + sinh 2 ( q c ) + sinh 2 (2 q c ) c =1 c =1 ( g , g 1 , g 2 are real coupling parameters; g 2 > 0, g 2 1 + g 2 2 > 0)
Introduction Reduction and Lax matrices Duality Open problems The rational BC n RSvD model Phase space: P R = T ∗ c ∼ = { ( λ, θ ) | λ ∈ c , θ ∈ R n } Symplectic form: ω R = � n c =1 d θ c ∧ d λ c Hamiltonian: � 1 � 1 n 1 + ν 2 1 + κ 2 � 2 � 2 H R = � cosh(2 θ c ) λ 2 λ 2 c c c =1 � 1 � 1 n 4 µ 2 4 µ 2 � 2 � 2 � × 1 + 1 + ( λ c − λ a ) 2 ( λ c + λ a ) 2 a =1 ( a � = c ) n 1 + 4 µ 2 + νκ � � − νκ � 4 µ 2 λ 2 4 µ 2 c c =1 ( µ , ν and κ are real parameters satisfying µ � = 0 � = ν and νκ ≥ 0)
Introduction Reduction and Lax matrices Duality Open problems The rational BC n RSvD model Phase space: P R = T ∗ c ∼ = { ( λ, θ ) | λ ∈ c , θ ∈ R n } Symplectic form: ω R = � n c =1 d θ c ∧ d λ c Hamiltonian: � 1 � 1 n 1 + ν 2 1 + κ 2 � 2 � 2 H R = � cosh(2 θ c ) λ 2 λ 2 c c c =1 � 1 � 1 n 4 µ 2 4 µ 2 � 2 � 2 � × 1 + 1 + ( λ c − λ a ) 2 ( λ c + λ a ) 2 a =1 ( a � = c ) n 1 + 4 µ 2 + νκ � � − νκ � 4 µ 2 λ 2 4 µ 2 c c =1 ( µ , ν and κ are real parameters satisfying µ � = 0 � = ν and νκ ≥ 0)
Introduction Reduction and Lax matrices Duality Open problems The rational BC n RSvD model Phase space: P R = T ∗ c ∼ = { ( λ, θ ) | λ ∈ c , θ ∈ R n } Symplectic form: ω R = � n c =1 d θ c ∧ d λ c Hamiltonian: � 1 � 1 n 1 + ν 2 1 + κ 2 � 2 � 2 H R = � cosh(2 θ c ) λ 2 λ 2 c c c =1 � 1 � 1 n 4 µ 2 4 µ 2 � 2 � 2 � × 1 + 1 + ( λ c − λ a ) 2 ( λ c + λ a ) 2 a =1 ( a � = c ) n 1 + 4 µ 2 + νκ � � − νκ � 4 µ 2 λ 2 4 µ 2 c c =1 ( 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 A n 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 A n 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 A n 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 A n 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 A n 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 BC n RSvD model with three independent coupling parameters ( µ , ν and κ ) Construction of action-angle variables for both the hyperbolic BC n Sutherland and the rational BC n RSvD models Establishing the action-angle duality between these BC n -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 BC n RSvD model with three independent coupling parameters ( µ , ν and κ ) Construction of action-angle variables for both the hyperbolic BC n Sutherland and the rational BC n RSvD models Establishing the action-angle duality between these BC n -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 BC n RSvD model with three independent coupling parameters ( µ , ν and κ ) Construction of action-angle variables for both the hyperbolic BC n Sutherland and the rational BC n RSvD models Establishing the action-angle duality between these BC n -type many-particle systems
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