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Journal of the Physical Society of Japan

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

Hideaki Ujino, Miki Wadati

Published Mar 15, 1996• 33 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.65.653

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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.
The U(1) Calogero Sutherland Model with anti-periodic boundary condition is studied. The Hamiltonian is reduced to a convenient form by similarity transformation. The matrix representation of the Hamiltonian acting on a partially ordered state space is obtained in an upper triangular form. Consequently the diagonal elements become the energy eigenvalues. The eigenstates are constructed using Young diagram and represented in terms of Jack symmetric polynomials. The eigenstates so obtained are orthonormalized.
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.
The main characteristics of the quantum oscillator coherent states including the two-particle Calogero interaction are investigated. We show that these Calogero coherent states are the eigenstates of the second-order differential annihilation operator which is deduced via R-deformed Heisenberg algebra or Wigner-Heisenberg algebraic technique and correspond exactly to the pure uncharged-bosonic states. They possess the important properties of non-orthogonality and completeness. The minimum uncertainty relation for the Calogero interaction coherent states is investigated. New sets of even and odd Wigner oscillator coherent states are pointed out.
The present paper is an extended version of another paper [Phys. Rev. B 59, R2490 (1999)], where we have established the equivalence of the Calogero-Sutherland model to decoupled oscillators. Here, we first employ the same approach for finding the eigenstates of a large class of Hamiltonians, dealing with correlated systems. A number of few- and many-body interacting models are studied and the relationship between their respective Hilbert spaces, with that of oscillators, is found. This connection is then used to obtain the spectrum generating algebras for these systems and make an algebraic statement about correlated systems. The procedure to generate solvable interacting models is outlined. We then point out the inadequacies of the present technique and make use of a method for solving linear differential equations to diagonalize the Sutherland model and establish a precise connection between this correlated system's wave functions, with those of the free particles on a circle. In the process, we obtain an expression for the Jack polynomials. In two dimensions, we analyze the Hamiltonian having Laughlin wave function as the ground state and point out the natural emergence of the underlying linear ${W}_{1+\ensuremath{\infty}}$ symmetry in this approach.