Renormalized Perturbation Theory for the Weakly Nonlinear Oscillator

Abstract

The problem of finding solutions for the weakly nonlinear quantum oscillator is investigated in the Heisenberg representation in one dimension. The perturbation method developed makes allowance for the nonisochronous nature of nonlinear oscillations and avoids at any level of approximation the secularity—terms increasing without bound as t → ∞—intrinsic to the usual type of iteration scheme. The treatment of the general quantum equation (d²/dt² + ω²)x(t) = εf(x), for the Heisenberg position operator x(t), is first motivated by the classical analog. The iteration equations for the quantum case are derived, and the case f(x) = x³ is studied fully to order ε, and partially to order ε².