AccurateNth-Order Iterative Method for Nonlinear Oscillators

The raising and lowering properties, the canonical commutation relation, and the equation of motion for the displacement operator for the one-dimensional nonlinear oscillator are used to define $N\mathrm{th}$-order iterative approximations for transition frequencies. In contrast to other approximation methods the orders of approximation are suggested by the commutation relation. It is numerically shown that the iterative approximation yields better results than other low-order methods; the fourth-order approximation yields eigenfrequencies accurate to one part in ${10}^5$ for the fourth-degree oscillator. It is concluded that in low-order approximations it is more important to take into account the commutation relations and thus adjacent-level frequency variations than higher-order matrix elements as is done in Rayleigh-Schrödinger perturbation theory.