Exact solutions of the harmonic oscillator plus non-polynomial interaction: Exact solutions for HO plus non polynom

The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a x 2 + b x 2 /(1 + c x 2) (a > 0, c > 0) are given by the confluent Heun functions H c (a, ß, ?, d, ?;z). The minimum value of the potential well is calculated as Vmin(x)=-(a+|b|-2a |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.