Exact solutions of the harmonic oscillator plus non-polynomial interaction

The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a x² + b x²/(1 + c x²) (a > 0, c > 0) are given by the confluent Heun functions Hc (α, β, γ, δ, η; z). The minimum value of the potential well is calculated as V min ( x ) = − ( a + | b | − 2 a | b | ) / c at x = ± [ ( | b | / a − 1 ) / c ] 1 / 2 (|b| > a) for the double-well case (b < 0). We illustrate the wave functions through varying the potential parameters a, b, c and show that they are pulled back to the origin when the potential parameter b increases for given values of a and c. However, we find that the wave peaks are concave to the origin as the parameter |b| is increased.