[Paper Review] Equilibrium Positions and Eigenfunctions of Shape Invariant (`Discrete') Quantum Mechanics

Satoru Odake, Ryu Sasaki|arXiv (Cornell University)|May 9, 2005

Nonlinear Waves and Solitons15 references6 citations

TL;DR

This paper establishes a deep connection between classical multi-particle equilibrium positions and quantum single-particle eigenfunctions in shape-invariant quantum systems, showing that both are described by the same orthogonal polynomials—Hermite, Laguerre, Jacobi, continuous Hahn, Wilson, and Askey-Wilson—through the properties of factorization and shape invariance in rational and trigonometric potentials.

ABSTRACT

Certain aspects of the integrability/solvability of the Calogero-Sutherland-Moser systems and the Ruijsenaars-Schneider-van Diejen systems with rational and trigonometric potentials are reviewed. The equilibrium positions of classical multi-particle systems and the eigenfunctions of single-particle quantum mechanics are described by the same orthogonal polynomials: the Hermite, Laguerre, Jacobi, continuous Hahn, Wilson and Askey-Wilson polynomials. The Hamiltonians of these single-particle quantum mechanical systems have two remarkable properties, factorization and shape invariance.

Motivation & Objective

• To demonstrate that the equilibrium positions of classical Calogero-Sutherland-Moser and Ruijsenaars-Schneider-van Diejen systems correspond to the zeros of specific orthogonal polynomials.
• To show that the eigenfunctions of the corresponding single-particle quantum systems are described by the same orthogonal polynomials.
• To establish that the Hamiltonians of these quantum systems exhibit factorization and shape invariance, enabling exact solvability.
• To explore the possibility of extending these results to elliptic potentials, where eigenfunctions remain unknown.
• To highlight the role of shape invariance in constructing eigenfunctions and spectra from ground state data.

Proposed method

• Utilizes numerical analysis, functional equations, and three-term recurrence relations to determine equilibrium positions of RSvD systems.
• Applies the concept of shape invariance to construct eigenfunctions and energy spectra via iterative application of ladder operators.
• Employs similarity transformations using ground state wavefunctions to map the Hamiltonian to a form where eigenfunctions are identified as orthogonal polynomials.
• Derives the energy spectrum using the shape invariance condition, expressing it in terms of parameters and quantum numbers.
• Uses the Crum-type construction adapted to discrete quantum mechanics to generate isospectral Hamiltonians and their eigenfunctions.
• Relies on the Askey-scheme of hypergeometric orthogonal polynomials to classify and relate the polynomials arising in different systems.

Experimental results

Research questions

• RQ1 Do the equilibrium positions of classical multi-particle systems with rational and trigonometric potentials coincide with the zeros of classical orthogonal polynomials?
• RQ2 Are the eigenfunctions of the corresponding single-particle quantum systems described by the same orthogonal polynomials as the classical equilibrium positions?
• RQ3 To what extent does shape invariance in the quantum Hamiltonian determine the entire spectrum and eigenfunctions?
• RQ4 Can the correspondence between classical equilibria and quantum eigenfunctions be extended to elliptic potentials?
• RQ5 Is there a discrete analogue of Crum’s theorem that does not require shape invariance for constructing isospectral systems?

Key findings

• The equilibrium positions of the classical rational and trigonometric Calogero-Sutherland and Ruijsenaars-Schneider-van Diejen systems are given by the zeros of Hermite, Laguerre, Jacobi, continuous Hahn, Wilson, and Askey-Wilson polynomials.
• The eigenfunctions of the single-particle quantum systems with the same potentials are identified as the same orthogonal polynomials, confirming a direct correspondence.
• The Hamiltonians of these quantum systems are factorized and exhibit shape invariance, allowing exact determination of the spectrum and eigenfunctions from the ground state and first excited state energy.
• The energy spectrum for the Askey-Wilson case is explicitly given by $ E_n = rac{1}{2}(q^{-n}-1)(1 - a_1a_2a_3a_4q^{n-1}) $, consistent with known results.
• The eigenfunctions are constructed via iterative application of adjoint lowering operators, yielding $ ilde{ ho}_n(z) riangleq P_n( ext{Re} hinspace z; a_1,a_2,a_3,a_4|q) $, which are Askey-Wilson polynomials.
• In the $ c o ty $ limit, the energy spectrum reduces to the standard form $ E_n = rac{ ilde{ ho}^2 ilde{ ho}^2}{mL^2} 2n(n + g_1 + g_2 + g'_1 + g'_2) $, matching known results for rational and trigonometric cases.