Quasilinearization approach to the resonance calculations: the quartic oscillator

Abstract

We pioneered the application of the quasilinearization method (QLM) to resonance calculations. The quartic anharmonic oscillator (kx2/2)+λx4 with a negative coupling constant λ was chosen as the simplest example of the resonant potential. The QLM has been suggested recently for solving the bound state Schrödinger equation after conversion into Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. Comparison of our approximate analytic expressions for the resonance energies and wavefunctions obtained in the first QLM iteration with the exact numerical solutions demonstrate their high accuracy in the wide range of the negative coupling constant. The results enable accurate analytic estimates of the effects of the coupling constant variation on the positions and widths of the resonances.