A PDE-Based Approach to Classical Phase-Space Deformations

This paper presents a PDE-based approach to finding an optimal canonical basis with which to represent a nearly integrable Hamiltonian. The idea behind the method is to continuously deform the initial canonical basis in such a way that the dependence of the Hamiltonian on the canonical position of the final basis is minimized. The final basis incorporates as much of the classical dynamics as possible into an integrable Hamiltonian, leaving a much smaller non-integrable component than in the initial representation. With this approach it is also possible to construct the semiclassical wavefunctions corresponding to the final canonical basis. This optimized basis is potentially useful in quantum calculations, both as a way to minimize the required size of basis sets, and as a way to provide physical insight by isolating those effects resulting from integrable dynamics.