Prepotential approach to exact and quasi-exact solvabilities

Abstract

Exact and quasi-exact solvabilities of the one-dimensional Schrödinger equation are discussed from a unified viewpoint based on the prepotential together with Bethe ansatz equations. This is a constructive approach which gives the potential as well as the eigenfunctions and eigenvalues simultaneously. The novel feature of the present work is the realization that both exact and quasi-exact solvabilities can be solely classified by two integers, the degrees of two polynomials which determine the change of variable and the zero-th order prepotential. Most of the well-known exactly and quasi-exactly solvable models, and many new quasi-exactly solvable ones, can be generated by appropriately choosing the two polynomials. This approach can be easily extended to the constructions of exactly and quasi-exactly solvable Dirac, Pauli, and Fokker-Planck equations.

11 pages, no figures. New paragraphs added in the Introduction and Summary sections. New references added. Version to appear in Ann. Phys

References in corpus (7)

• Introduction to PT-Symmetric Quantum Theory
• Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics
• Quasi-exact solvability of Dirac equation with Lorentz scalar potential
• Explicit solutions of the classical Calogero & Sutherland systems for any root system
• Dirac oscillators and quasi-exactly solvable operators
• Prepotential approach to exact and quasi-exact solvabilities of Hermitian and non-Hermitian Hamiltonians
• Quasi-exact Solvability of Planar Dirac Electron in Coulomb and Magnetic Fields

Cited by in corpus (10)

• Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials
• Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism
• Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces
• Simple unified derivation and solution of Coulomb, Eckart and Rosen-Morse potentials in prepotential approach
• Deformed shape invariance symmetry and potentials in curved space with two known eigenstates
• Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators
• Prepotential approach to exact and quasi-exact solvabilities of Hermitian and non-Hermitian Hamiltonians
• Shape invariance in prepotential approach to exactly solvable models
• Prepotential approach to quasinormal modes
• Quasi-exactly solvable Schrödinger equations, symmetric polynomials, and functional Bethe ansatz method