A Recursion Technique for Deriving Renormalized Perturbation Expansions for One-dimensional Anharmonic Oscillator

In the past few decades intensive investigations have been carried out on the one-dimensional anharmonic oscillator because of both its role in the modeling of quantum field theory and its usefulness in atomic, and molecular physics . The conventional way to study the energy eigenvalues and eigenfunctions of this bound-state problem is the practical application of perturbation expansions in a coupling constant. But the nonconvergence of obtained expansions 5,6 compels us to resort to modern procedures of summation of divergent series. Ones of the most common among them are various versions of the renormalization technique . The inclusion of one or more free parameters corrects for the above mentioned deficiency, by controlling and accelerating the convergence of the expansions not only for eigenvalues but for eigenfunctions as well . However, to provide a reasonable accuracy this method needs to know high orders of the perturbation series. Unfortunately we can easily obtain the high-order corrections only in the case of ground states when the simple recursion relations of the logarithmic perturbation theory 12−18 or the Bender-Wu method 5 for performing the Rayleigh-Schrödinger perturbation expansion are applied. The description of radially excited states involves including the nodes of wave function in consideration that makes these approaches extremely cumbersome and, practically, inapplicable. Very recently, a new semiclassical technique for deriving results of logarithmic perturbation theory within the framework of the one-dimensional Schrödinger equation has been proposed . Based upon an h̄-expansion this technique leads to recursion formulae having the same simple form both for the ground and excited states. The object of this paper is to extend the above mentioned formalism and to adapt it to the treatment of any renormalization scheme in terms of handy recursion relations within the framework of the united approach. To this end the paper is presented as follows. Section 2 contains a general discussion and the necessary assumptions in the semiclassical treatment of logarithmic perturbation theory. In Sect.3 quantization conditions obtained are used for deriving the recursion formulae for perturbation expansions. Sect.4 focuses on a derivation of the recursion formulae for renormalized perturbation expansions. Sect.5 gives the example of explicit application of proposed technique. Finally, Sect.6 contains concluding remarks.