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It is known that on a compact manifold, there are quasimodes, approximate eigenfunctions of the Laplacian eigenfunction equation $-\Delta \psi = \lambda^2 \psi$Ralston has constructed quasimodes concentrated around periodic orbits for a class of operators which include the Laplacians with precise bounds depending on the order of stability.  Now given Benci’s existence theorem for periodic orbits for Hamiltonian flows we have a way of understanding the Schroedinger equation to be giving us a way of getting approximate solutions to the periodic orbits associated to the corresponding classical system. Given a Hamiltonian system (on a compact manifold), Benci’s theorem tells us of existence of periodic orbits and then Ralston’s quasimode result tells us that there exist approximate solution of the Schroedinger equation concentrating around these orbits.  Now the actual universe is a compact hypersurface of a scaled 4-sphere so these results produce a correspondence between classical and ‘quantum’ equations.