Displaced harmonic oscillator $V\sim \min \,[(x+d)^2,(x-d)^2]$ as a benchmark double-well quantum model

TL;DR

This paper introduces a benchmark double-well quantum model based on a displaced harmonic oscillator potential, revealing exact polynomial bound states at specific displacements and discussing quasi-exact solvability for certain states.

Contribution

The work identifies and constructs polynomial bound states at specific displacements in a displaced harmonic oscillator model, expanding understanding of quasi-exact solvability in double-well systems.

Findings

Existence of exact polynomial bound states at certain displacements

Construction of N-plet quasi-exactly solvable states in closed form

Non-QES states are solvable through matching hypergeometric functions

Abstract

For the displaced harmonic double-well oscillator the existence of exact polynomial bound states at certain displacements $d\,$ is revealed. The $N-$plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, Schr\"{o}dinger equation can still be considered ``non-polynomially exactly solvable'' (NES) because the exact left and right parts of the wave function (proportional to confluent hypergeometric function) just have to be matched in the origin.