Exact solutions of a nonpolynomial oscillator related to isotonic oscillator

We find that the analytical solutions to a quantum system with a nonpolynomial oscillator potential related to isotonic oscillator are given by the confluent Heun functions H_c(α, β, γ, δ, η; z). The properties of the wave functions, which are strongly relevant for the potential parameters a and g, are illustrated. It is shown that the wave functions are shrunk to the origin for a given a when the potential parameter g increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter | g| increases. Moreover, the wave peaks of the even wave functions become sharper when the potential parameter a< 1 decreases, but they become flat when the potential parameter a> 1 increases. When the minimum value V_min= - g/ a² tends to zero, this nonpolynomial oscillator reduces to a harmonic oscillator.