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It is shown that it is possible to construct the quantum wave functions fornon-separable but integrable two-dimensional Hamiltonian systems, by solvingsuitable Dirichlet boundary values problems inside and outside the regionsspanned by particular families of classical trajectories, in one-to-onecorrespondence with the quantum state. The method is applied both to theSchrodinger equation, and to the quantum Hamilton-Jacobi equation. The boundaryvalues are obtained by integrating the one-dim equations on the caustics arcsenveloping the classical trajectories. This approach gives the same results asthe usual methods, and furthermore clarifies the links between quantum andclassical mechanics.

Construction of quantum wavefunctions for non-separable but integrable  two-dimensional Hamiltonian systems by means of the boundary values on the  classical caustics