Exact solutions of the harmonic oscillator plus non-polynomial interaction

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