Quantum Calogero-Moser models: integrability for all root systems

Khastgir, S. Pratik; Pocklington, Andrew; Sasaki, Ryu

doi:10.1088/0305-4470/33/49/303

Cited by 62 publications

(110 citation statements)

References 55 publications

Supporting

Mentioning

110

Contrasting

Order By: Relevance

“…In this section we briefly recapitulate the essence of Calogero-Moser models based on any root system ∆ (applicable to the exceptional and non-crystallographic root system) and the associated universal Lax pair formalism along with appropriate notation [16,17,18,19] and background [14,15] for the main body of this paper. Those who are familiar with the universal Lax pair formulation may skip this section and return when necessity arises.…”

Section: Universal Lax Operator For Calogero-moser Model With Degenermentioning

confidence: 99%

“…The same remark applies to the conserved quantities of the spin exchange models to be discussed in the following section. For the quantum CM models without spin, the equivalence of the Lax pair formalism and Dunkl operator formalism was proven in [19].…”

Section: It Is Easy To Verify As In the Calogero-moser Models That mentioning

confidence: 99%

“…However, the CM models can be formulated n classical and quantum mechanics for any root system [14,15,16,17,18,19], and one can guess that introduction of spin exchange can be done at least for some root systems too. There were several attempts [8,9,10,12] in this direction, but they were far from being universal in a way for introducing spin into the CM models.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Universal Lax pairs for spin Calogero–Moser models and spin exchange models

Inozemtsev¹,

Sasaki²

2001

J. Phys. A: Math. Gen.

Self Cite

View full text Add to dashboard Cite

For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a "site", we associate a vector space V µ whose element is called a "spin". Its dynamical variables are the canonical coordinates {q j , p j } of a particle in R r , (r = rank of ∆), and spin exchange operators {P ρ } (ρ ∈ ∆) which exchange the spins at the sites µ and s ρ (µ). Here s ρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical CalogeroMoser model. For ∆ = A r and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.

show abstract

Section: Universal Lax Operator For Calogero-moser Model With Degenermentioning

confidence: 99%

Section: It Is Easy To Verify As In the Calogero-moser Models That mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Universal Lax pairs for spin Calogero–Moser models and spin exchange models

Inozemtsev¹,

Sasaki²

2001

J. Phys. A: Math. Gen.

Self Cite

View full text Add to dashboard Cite

show abstract

“…A Calogero-Moser model is a Hamiltonian system associated with a root system ∆ of rank r. Quantum versions of these models are also integrable, at least for degenerate potential functions [3] for any choice of ∆. The dynamical variables are the coordinates {q j } and their canonically conjugate momenta {p j }, with the Poisson brackets…”

mentioning

confidence: 99%

Liouville integrability of classical Calogero–Moser models

Khastgir

Sasaki

2001

Physics Letters A

View full text Add to dashboard Cite

Liouville integrability of classical Calogero-Moser models is proved for models based on any root systems, including the non-crystallographic ones. It applies to all types of elliptic potentials, i.e. untwisted and twisted together with their degenerations (hyperbolic, trigonometric and rational), except for the rational potential models confined by a harmonic force.In this note we demonstrate the Liouville integrability of classical Calogero-Moser models

show abstract

“…One can diagonalize the matrix D − κC by conjugation with the unitary matrix 69) where the real functions α(x), β(x) are defined on the interval [|κ|, ∞) ⊂ R by the formulae 70) at least if κ = 0. If κ = 0, then we set α(x) = 1 and β(x) = 0.…”

Section: The Ruijsenaars Gaugementioning

confidence: 99%

Integrable many-body systems of Calogero-Ruijsenaars type

Görbe

View full text Add to dashboard Cite

This thesis is submitted in accordance with the regulations for the Doctor of Philosophy degree at the University of Szeged. The results presented in the thesis are the author's original work (see Publications) with the exceptions of Section 1.1, which reviews some pre-existing material, and Subsections 4. 2.2, 4.3.1, 4.3.3, 4.3.4, 4.3.5 that contain results obtained by B.G. Pusztai. These are included to make the exposition self-contained.The research was carried out within the Ph.D. programme "Geometric and fieldtheoretic aspects of integrable systems" at the

show abstract

Quantum Calogero-Moser models: integrability for all root systems

Cited by 62 publications

References 55 publications

Universal Lax pairs for spin Calogero–Moser models and spin exchange models

Universal Lax pairs for spin Calogero–Moser models and spin exchange models

Liouville integrability of classical Calogero–Moser models

Integrable many-body systems of Calogero-Ruijsenaars type

Contact Info

Product

Resources

About