Symmetrized quartic polynomial oscillators and their partial exact solvability

Abstract

Sextic polynomial oscillator is probably the best known quantum system which is partially exactly {\it alias} quasi-exactly solvable (QES), i.e., which possesses closed-form, elementary-function bound states $\psi (x)$ at certain couplings and energies. In contrast, the apparently simpler and phenomenologically more important quartic polynomial oscillator is {\em not\,} QES. A resolution of the paradox is proposed: The one-dimensional Schrödinger equation is shown QES after the analyticity-violating symmetrization $V(x)=A|x|+B{x}^2+C|x{|}^3+x^4$ of the quartic polynomial potential.

14 pp., 2 figures

References in corpus (1)

• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential

Cited by in corpus (8)

• The generalized Klein-Gordon oscillator in a Cosmic Space-Time with a Space-Like Dislocation and the Aharonov-Bohm Effect
• Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity
• Relativistic Solutions of Generalized-Dunkl Harmonic and Anharmonic Oscillators
• Morse potential, symmetric Morse potential and bracketed bound-state energies
• Symmetrized exponential oscillator
• Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators
• Displaced harmonic oscillator $V\sim \min [(x+d{)}^2,(x-d{)}^2]$ as a benchmark double-well quantum model
• Avoided level crossings in quasi-exact approach