Abstract. A system of semi-discrete coupled nonlinear Schrödinger 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 Schrödinger 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. †
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.KEYWORDS: quantum Calogero model, Hi-Jack symmetric polynomials, Rodrigues formula, quantum inverse scattering method (QISM), Dunkl operator §1. IntroductionExact solutions for the Schrödinger equations have provided important significance in physics and mathematical physics. Most of us have studied the Laguerre polynomials and the spherical harmonics in the theory of the hydrogen atom, and the Hermite polynomials and their Rodrigues formula in the theory of the quantum harmonic oscillator. The former is also a good example that shows the role of conserved operators in quantum mechanics. The hydrogen atom has three, independent and mutually commuting conserved operators, namely, the Hamiltonian, the total angular momentum and its z-axis component. The simultaneous eigenstates for the three conserved operators give the orthogonal basis of the hydrogen atom. A classical system with a set of independent and mutually Poisson commuting (involutive) conserved quantities whose number of elements is the same as the degrees of freedom of the system can be integrated by quadrature. This is guaranteed by the Liouville theorem. Such a system is called the completely integrable system. Quantum systems with enough number of such conserved operators are analogously called quantum integrable systems. The hydrogen atom is a simple example of the quantum integrable system. Among the various quantum integrable systems, onedimensional quantum many-body systems with inversesquare long-range interactions are now attracting much interests of theoretical physicists. Of the various integrable inverse-square-interaction models, the quantum Calogero model 1) has the longest history. Its Hamilto- *
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.
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