Method of Averaging and the Quantum Anharmonic Oscillator

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Abstract
The Krylov-Bogoliubov method of averaging is applied to the time-dependent quantum anharmonic oscillator. A regular perturbation expansion contains secular terms. The averaging approximation does not, and as a result has a validity over larger time intervals. A new variant of the usual averaging transformation is used and rigorous error bounds are derived. Rigorous averaging methods have been applied extensively to ordinary differential equations but our work appears to be the first generalization to partial differential equations.