Semiclassical eigenvalues for nonseparable systems: Nonperturbative solution of the Hamilton–Jacobi equation in action-angle variables

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Abstract
It is shown how the Hamilton–Jacobi equation for a multidimensional nonseparable system can be efficiently solved directly in action‐angle variables. This allows one to construct the total (classical) Hamiltonian as a function of the ’’good’’ action‐angle variables which are the complete set of constants of the motion of the system; requiring the action variables to be integers then provides the semiclassical eigenvalues. Numerical results are presented for a two‐dimensional potential well, and one sees that the semiclassical eigenvalues are in good agreement with the exact quantum mechanical values even for the case of large nonseparable coupling.

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