Multiple-Scale Analysis of Quantum Systems
  Conventional weak-coupling Rayleigh-Schr\"odinger perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale analysis, a powerful and sophisticated perturbative method that quantitatively analyzes characteristic physical behaviors occurring on various length or time scales, avoids such problems by implicitly performing an infinite resummation of the conventional perturbation series. Multiple-scale perturbation theory provides a good description of the classical anharmonic oscillator. Here, it is extended to study (1) the Heisenberg operator equations of motion and (2) the Schr\"odinger equation for the quantum anharmonic oscillator. In the former case, it leads to a system of coupled operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory. In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function. 
Carl M. Bender and Luis M. A. Bettencourt