Exact solutions of a quartic potential

We find that the analytical solutions to quantum system with a quartic potential V (x) = ax2 + bx4 (arbitrary a and b > 0 are real numbers) are given by the triconfluent Heun functions HT(α,β,γ; z). The properties of the wave functions, which are strongly relevant for the potential parameters a and b, are illustrated. It is shown that the wave functions are shrunk to the origin for a given b when the potential parameter a increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter |a| increases or parameter b decreases for a given negative potential parameter a. The minimum value of the double well case (a < 0) is given by Vmin = -a2/(4b) at x = ±|a|/2b.