Exact solutions of an asymmetric double well potential

The analytical solutions of an asymmetric double well potential V(x)=ax2-bx3+cx4 are found to be a triconfluent Heun function H_T(α, β, γ; z). It should be emphasized that these potential parameters are taken arbitrarily without any restriction on them. The wave functions which depend on the potential parameters are shrunk toward to the origin for given b and c when the parameter a increases, while they are moved far from the origin and toward to the left when the parameter b increases for given a and c. Also, when the parameter c increases for given a and b they have the similar property to the case when the parameter a increases.