Unified derivation of exact solutions for a class of quasi-exactly solvable models

Abstract

We present a unified treatment of exact solutions for a class of four quantum mechanical models, namely the singular anharmonic potential, the generalized quantum isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified oscillator. We show that all four cases are reducible to the same basic ordinary differential equation, which is quasi-exactly solvable. A systematic and closed form solution to the basic equation is obtained via the Bethe ansatz method. Using the result, general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations. A hidden $sl(2)$ algebraic structure is also discovered in these models.

Latex file, 14 pages, many modifications made and misprints corrected

References in corpus (3)

• Exact polynomial solutions of second order differential equations and their applications
• The generalized quantum isotonic oscillator
• Generalized quantum isotonic nonlinear oscillator in d dimensions

Cited by in corpus (10)

• Novel quasi-exactly solvable models with anharmonic singular potentials
• New quasi-exactly solvable class of generalized isotonic oscillators
• Quasi-exactly solvable relativistic soft-core Coulomb models
• On the classical and quantum dynamics of a class of nonpolynomial oscillators
• Quasi-exact Treatment of the Relativistic Generalized Isotonic Oscillator
• Exact solutions to relativistic singular fractional power potentials
• Ladder operators and squeezed coherent states of a 3-dimensional generalized isotonic nonlinear oscillator
• Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems
• Perturbation-based Non-perturbative Method
• Dirac Equation with Space Contributions Embedded in a Quantum-Corrected Gravitational Field