Anharmonic oscillator: a solution

It is shown that for the one-dimensional quantum anharmonic oscillator with potential V(x)= x²+g² x⁴ the Perturbation Theory (PT) in powers of g² (weak coupling regime) and the semiclassical expansion in powers of hbar for energies coincide. It is related to the fact that the dynamics in x-space and in (gx)-space corresponds to the same energy spectrum with effective coupling constant hbar g². Two equations, which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, are derived. The PT in g² for the logarithmic derivative of wave function leads to PT (with polynomial in x coefficients) for the RB equation and to the true semiclassical expansion in powers of hbar for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A 2-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in x-space with unprecedented accuracy sim 10^-6 locally and unprecedented accuracy sim 10^-9-10^-10 in energy for any g² geq 0. A generalization to the radial quartic oscillator is briefly discussed.