EBK quantization of nonseparable systems: A Fourier transform method

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Abstract
We present an accurate and efficient method for EBK quantization of bound states in multidimensional systems, based upon a Fourier representation of invariant tori in the quasiperiodic regime. The key feature of the method is the direct determination of good action variables from Fourier transforms of trajectories, using a formula due to Percival. It is possible to calculate actions using this formula provided frequencies and Fourier components can be determined to high accuracy. We describe a special windowing technique for this purpose, which overcomes the stability problems encountered in the recent related work of Eaker, Schatz, De Leon and Heller. Quantized energy levels can be found either by direct search for quantizing trajectories or by linear extrapolation from nearby arbitrary tori. Results are presented for two-, three-, and four-dimensional vibrational Hamiltonians.