Anharmonic oscillator and double-well potential: Approximating eigenfunctions

A simple uniform approximation of the logarithmic derivative of the ground state eigenfunction for both the quantum-mechanical an harmonic oscillator and the double-well potential given by V = m(2)x(2) + gx(4) at arbitrary g >= 0 for m(2) > 0 and m(2) < 0, respectively, is presented. It is shown that if this approximation is taken as unperturbed problem it leads to an extremely fast convergent perturbation theory.