Anharmonic oscillator: a solution

It is shown that for the one-dimensional quantum anharmonic oscillator with potential $V(x)= x^2+g^2 x^4$ Perturbation Theory (PT) in powers of $g^2$ (weak coupling regime) and semiclassical expansion $\hbar$ energies coincide. related to fact dynamics $x$-space $(gx)$-space corresponds same energy spectrum effective constant $\hbar g^2$. Two equations, which govern those two spaces, Riccati-Bloch (RB) Generalized Bloch (GB) respectively, are derived. The PT logarithmic derivative wave function leads (with polynomial $x$ coefficients) RB equation true GB equation, a loop density matrix path integral formalism. A 2-parametric interpolation these expansions uniform approximation wavefunction unprecedented accuracy $\sim 10^{-6}$ locally 10^{-9}-10^{-10}$ any $g^2 \geq 0$. generalization radial quartic briefly discussed.