From quartic anharmonic oscillator to double well potential

It is already known that the quantum quartic single-well anharmonic oscillator $V_{ao}(x)=x^2+g^2 x^4$ and double-well $V_{dw}(x)= x^2(1 - gx)^2$ are essentially one-parametric, their eigenstates depend on a combination $(g^2 \hbar)$. Hence, these problems reduced to study potentials $V_{ao}=u^2+u^4$ $V_{dw}=u^2(1-u)^2$, respectively. shown by taking uniformly-accurate approximation for eigenfunction $\Psi_{ao}(u)$, obtained recently, see JPA 54 (2021) 295204 [1] Arxiv 2102.04623 [2], then forming function $\Psi_{dw}(u)=\Psi_{ao}(u) \pm \Psi_{ao}(u-1)$ allows get highly accurate both eigenfunctions of potential its eigenvalues.