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 x 2 + b x 2 /(1 + c x 2 ) ( a > 0, c > 0) are given by the confluent Heun functions H c ( 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.