Hideaki Ujino

Integrable semi-discretization of the coupled nonlinear Schrödinger equations

A system of semi-discrete coupled nonlinear Schrodinger equations is studied. To show the complete integrability of the model with multiple components, we extend the discrete version of the inverse scattering method for the single-component discrete nonlinear Schrodinger equation proposed by Ablowitz and Ladik. By means of the extension, the initial-value problem of the model is solved. Further, the integrals of motion and the soliton solutions are constructed within the framework of the extension of the inverse scattering method.

Integrable semi-discretization of the coupled modified KdV equations

The discrete version of the inverse scattering method proposed by Ablowitz and Ladik is extended to solve multi-component systems. The extension enables one to solve the initial value problem, which proves directly the complete integrability of a semi-discrete version of the coupled modified Korteweg–de Vries (KdV) equations and their hierarchy. It also provides a procedure to obtain conservation laws and multi-soliton solutions of the hierarchy.

Integrability of the Quantum Calogero-Moser Model

The integrability of the quantum Calogero-Moser model is proved through the quantum inverse scattering method. By use of the Lax operator L , a new formula for the conserved operators is found. The formula suggests that Tr L n , which are conserved in the classical case, are not necessarily conserved operators. The Henon-type conserved operators are also presented.

Rodrigues Formula for Hi-Jack Symmetric Polynomials Associated with the Quantum Calogero Model

The Hi-Jack symmetric polynomials, which are associated with the simultaneous eigenstates for the first and second conserved operators of the quantum Calogero model, are studied. Using the algebraic properties of the Dunkl operators for the model, we derive the Rodrigues formula for the Hi-Jack symmetric polynomials. Some properties of the Hi-Jack polynomials and the relationships with the Jack symmetric polynomials and with the basis given by the QISM approach are presented. The Hi-Jack symmetric polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.

The quantum calogero-moser model : algebraic structures

For the quantum Calogero-Moser model, we construct a set of conserved operators and another set of operators, named boost operators, from its Lax operator. We prove that each conserved operator satisfies both the Lax equation and a remarkable relation named additional relation. From these knowledge, we show that the conserved operators are involutive. Moreover, the conserved operators and the boost operators constitute the U(1)-current algebra. All the proofs are simplified a great deal due to the Lax equations and additional relations.

Algebraic Construction of the Eigenstates for the Second Conserved Operator of the Quantum Calogero Model

An algebraic construction of the eigenstates for the quantum Calogero model is investigated. Extending the method of Lapointe and Vinet, we construct the eigenstates for the second conserved operator of the quantum Calogero model. All the eigenstates can be factorized into symmetric polynomials which we call “Hi-Jack symmetric polynomials” and the ground state wave function. The conjectured formula for the eigenvalue of the second conserved operator is confirmed. The Hi-Jack polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.

Orthogonality of the Hi-Jack Polynomials Associated with the Calogero Model

Orthogonality of the Hi-Jack polynomials, which were introduced as the simultaneous eigenfunctions of the first and second conserved operators of the Calogero model, is proved by showing that they are non-degenerate simultaneous eigenfunctions for all the commuting conserved operators of the Calogero model. The fact that our definition of the Hi-Jack polynomials uniquely specifies the Hi-Jack polynomials, which is assured by the triangularity with respect to the dominance order among the Young tableaux, plays an essential role in our proof.Using a similarity transformation that maps the Calogero model into

The Quantum Calogero Model and the W-Algebra

N

Orthogonal Symmetric Polynomials Associated with the Quantum Calogero Model

decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions. The simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that is a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformation and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent sets of mutually commuting conserved operators, as is the case with the hydrogen atom. We also confirm the same story for the

The SU(ν) Quantum Calogero Model and W_∞ Algebra with SU(ν) Symmetry

B_{N}