The quantum 2D-harmonic oscillator in 1:1 resonance with time-dependent perturbation : averaging applied to slowly varying quantum systems

Abstract

In this paper we will study the 2D-harmonic oscillator in 1:1 resonance. We add a general third- and fourth order perturbation which is slowly time-dependent, and are interested in the resulting interaction between the unperturbed orbits (states). By treating this system quantummechanically we get a linear system (Schrödinger’s equation) on the infinite dimensional space . We show that this system can be reduced to a large finite dimensional system by bounding the perturbation suitably at infinity. By applying perturbation theory and averaging we are able to split this large system into smaller subsystems. We keep track of all error terms and show explicitly under which conditions they can be neglected. By applying adiabatic perturbation theory we finally transform the positive time-axis into a finite time-interval. We conclude by indicating how the theory can be extended to other resonances and to the 3D harmonic oscillator.