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The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a x2 + b x2/(1 + c x2) (a 0, c 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 less then . We illustrate the wave functions through varying the potential parameters a, b, c and show that they are pulled back to the origin whe
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