OPEN PERTURBATION AND THE RICCATI-EQUATION - ALGEBRAIC DETERMINATION OF THE QUARTIC ANHARMONIC-OSCILLATOR ENERGIES AND EIGENFUNCTIONS

An algebraic procedure is proposed for the analytical solution of Schr odinger equations that can be viewed as a factorizable equation with a n additional potential V(x). Once V(x) has been expanded in a series o f suitable x-basis functions u=u(x), which are specific to each factor ization type, the solution of the Riccati equation associated with the given equation is performed by means of an open perturbation techniqu e, i.e., at each order of the perturbation, an additional balance u-de pendent term is introduced so that the resulting equation becomes solv able. Since the unperturbed potential involves the whole given potenti al and since the balance term is expected to be small, improved result s are expected at low orders of the perturbation, even at the zeroth o rder. The procedure, well adapted to the use of computer algebra, is a pplied to the solution of the gx(4)-anharmonic oscillator equation: by means of very simple algebraic manipulations, the trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant (g = 0.002 to g = 20000). (C) 1997 Ame rican Institute of Physics.