Extended Rayleigh-Schrodinger perturbation theory for the quartic anharmonic oscillator

This paper follows the ideas of the author's earlier works that there is a possible metastable process passing from unperturbed to perturbed as a perturbation potential evolves. Furthermore, we extend the balance condition which was fully described in earlier works, from the total wave to each corresponding sth-order component wave Psi ((s))(n), s = 0, 1,.... Therefore, to our great benefit, an infinite set of coupled balance conditions can be gained. We add this new idea to perturbation theory, and hence present the so-called extended Rayleigh-Schrodinger perturbation theory. As is well known, earlier applications in PFSKB (principles of the first and second kind of balance) theory seldom went beyond the first to, at most, second order in the eigenvalue because of computational difficulties in evaluating the minimization of the total energy of infinite summations which appear in the equations for the higher-order terms. Fortunately, these difficulties can be overcome by individually balancing the conditions of the corresponding sth-order component waves Psi ((s))(n). However, it seems reasonable to hope that a better understanding of extended Rayleigh-Schrodinger perturbation theory may be gained. In many cases of interest, the quartic anharmonic oscillator is chosen as an example for the demonstration of extended Rayleigh-Schrodinger perturbation theory because of its divergent Rayleigh-Schrodinger perturbation expansions and wide applications.