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R. Sasaki

Publications and source records attributed to R. Sasaki.

At least 37 records · Page 2Linked to original sources
arXiv2004

Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials

The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pmβ' p} (β' is a deformation parameter) instead of an ordinary polynomial in p in the hierarchies of C-M systems. We determine the polynomials describing the equilibrium positions of the rational and trigonometric Ruijsenaars-Schneider systems based on classical root systems. These are deformation of the classical orthogonal polynomials, the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium positions of the corresponding Calogero and Sutherland systems. The orthogonality of the original polynomials is inherited by the deformed ones which satisfy three-term recurrence and certain functional equations. The latter reduce to the celebrated second order differential equations satisfied by the classical orthogonal polynomials.

hep-th↗
arXiv2003

Classical vs Quantum Mechanics: role of elementary excitations

Simple theorems relating a quantum mechanical system to the corresponding classical one at equilibrium and connecting the quantum eigenvalues to the frequencies of normal modes oscillations are presented. Corresponding to each quantum eigenfunction, a ` classical eigenfunction' is associated. Those belonging to `elementary excitations' play an important role.

quant-ph↗
arXiv2003

Quantum & Classical Eigenfunctions in Calogero & Sutherland Systems

An interesting observation was reported by Corrigan-Sasaki that all the frequencies of small oscillations around equilibrium are " quantised" for Calogero and Sutherland (C-S) systems, typical integrable multi-particle dynamics. We present an analytic proof by applying recent results of Loris-Sasaki. Explicit forms of `classical' and quantum eigenfunctions are presented for C-S systems based on any root systems.

hep-th↗
arXiv2003

Affine Toda-Sutherland Systems

A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorisable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed.

hep-th↗
arXiv2003

Exact solution of $Z_n$ Belavin model with open boundary condition

$Z_n$ Belavin model with open boundary condition is studied. The double-row transfer matrices of the model are diagonalized by algebraic Bethe ansatz method in terms of the intertwiner and the face-vertex correspondence relation. The eigenvalues and the corresponding Bethe ansatz equations are obtained.

hep-th↗
arXiv2003

Quantum vs Classical Integrability in Ruijsenaars-Schneider Systems

The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider systems, which are one parameter deformation of Calogero-Moser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (Corrigan-Sasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". In this paper we report that similar features and results hold for the Ruijsenaars-Schneider type of integrable systems based on the classical root systems.

hep-th↗
arXiv2002

Polynomials Associated with Equilibrium Positions in Calogero-Moser Systems

In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". The equilibrium positions of Calogero and Sutherland systems for the classical root systems (A_r, B_r, C_r and D_r) correspond to the zeros of Hermite, Laguerre, Jacobi and Chebyshev polynomials. Here we define and derive the corresponding polynomials for the exceptional (E_6, E_7, E_8, F_4 and G_2) and non-crystallographic (I_2(m), H_3 and H_4) root systems. They do not have orthogonality but share many other properties with the above mentioned classical polynomials.

hep-th↗
arXiv2002

Quantum vs Classical Integrability in Calogero-Moser Systems

Calogero-Moser systems are classical and quantum integrable multi-particle dynamics defined for any root system $Δ$. The {\em quantum} Calogero systems having $1/q^2$ potential and a confining $q^2$ potential and the Sutherland systems with $1/\sin^2q$ potentials have "integer" energy spectra characterised by the root system $Δ$. Various quantities of the corresponding {\em classical} systems, {\em e.g.} minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices, etc. at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also "integers", or they appear to be "quantised". To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero-Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general.

hep-th↗
arXiv2001

Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry

Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q^6 (rational models) or sin^2(2q) (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-exactly solvable) multi-particle dynamical systems. They posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A new method for identifying and solving quasi-exactly solvable systems, the method of pre-superpotential, is presented.

hep-th↗
arXiv2001

On the Integrability of Classical Ruijsenaars-Schneider Model of $BC_{2}$ Type

The problem of finding most general form of the classical integrable relativistic models of many-body interaction of the $BC_{n}$ type is considered. In the simplest nontrivial case of $n=2$,the extra integral of motion is presented in explicit form within the ansatz similar to the nonrelativistic Calogero-Moser models. The resulting Hamiltonian has been found by solving the set of two functional equations.

hep-th↗
arXiv2001

Scalar Symmetries of the Hubbard Models with Variable Range Hopping

Examples of scalar conserved currents are presented for trigonometric, hyperbolic and elliptic versions of the Hubbard model with non-nearest neighbour variable range hopping. They support for the first time the hypothesis about the integrability of the elliptic version. The two- electron wave functions are constructed in an explicit form.

hep-th↗
arXiv2001

Hierarchies of Spin Models related to Calogero-Moser Models

The universal formulation of spin exchange models related to Calogero-Moser models implies the existence of integrable hierarchies, which have not been explored. We show the general structures and features of the spin exchange model hierarchies by taking as examples the well-known Heisenberg spin chain with the nearest neighbour interactions. The energy spectra of the second member of the hierarchy belonging to the models based on the $A_r$ root systems $(r=3,4,5)$ are explicitly and {\em exactly} evaluated. They show many many interesting features and in particular, much higher degree of degeneracy than the original Heisenberg model, as expected from the integrability.

hep-th↗
arXiv2001

Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models

For any root system $Δ$ and an irreducible representation ${\cal R}$ of the reflection (Weyl) group $G_Δ$ generated by $Δ$, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $μ$ of ${\cal R}$, to be called a "site", we associate a vector space ${\bf V}_μ$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{q_j,p_j\}$ of a particle in ${\bf R}^r$, ($r=$ rank of $Δ$), and spin exchange operators $\{\hat{\cal P}_ρ\}$ ($ρ\inΔ$) which exchange the spins at the sites $μ$ and $s_ρ(μ)$. Here $s_ρ$ is the reflection generated by $ρ$. For each $Δ$ and ${\cal R}$ a {\em 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 Calogero-Moser model. For $Δ=A_r$ and ${\cal R}=$ 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 {\em degenerate} potentials.

hep-th↗
arXiv2001

Quadratic Algebra associated with Rational Calogero-Moser Models

Classical Calogero-Moser models with rational potential are known to be superintegrable. That is, on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom, they possess an additional set of r-1 algebraically and functionally independent globally defined conserved quantities. At the quantum level, Kuznetsov uncovered the existence of a quadratic algebra structure as an underlying key for superintegrability for the models based on A type root systems. Here we demonstrate in a universal way the quadratic algebra structure for quantum rational Calogero-Moser models based on any root systems.

hep-th↗
arXiv2000

Liouville Integrability of Classical Calogero-Moser Models

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.

hep-th↗
arXiv2000

Quantum Calogero-Moser Models: Integrability for all Root Systems

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{μ,ν\in{\cal R}}(L^n)_{μν}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.

hep-th↗
arXiv1999

Calogero-Moser Models V: Supersymmetry and Quantum Lax Pair

It is shown that the Calogero-Moser models based on all root systems of the finite reflection groups (both the crystallographic and non-crystallographic cases) with the rational (with/without a harmonic confining potential), trigonometric and hyperbolic potentials can be simply supersymmetrised in terms of superpotentials. There is a universal formula for the supersymmetric ground state wavefunction. Since the bosonic part of each supersymmetric model is the usual quantum Calogero-Moser model, this gives a universal formula for its ground state wavefunction and energy, which is determined purely algebraically. Quantum Lax pair operators and conserved quantities for all the above Calogero-Moser models are established.

hep-th↗