Gaussian representation of extended quantum states

Excited energy eigenstates and their superpositions typically exhibit a fine oscillatory structure near caustics. Semiclassical theory accesses these, but depends on detailed geometrical knowledge of the caustics. Here we show that a finite placement of coherent states on the classical region efficiently fits such extended states, reproducing structures that are much finer than the Gaussian width of the basis states. An extended state, evolved such that it becomes fully distinguishable from the original state, can also be faithfully reproduced by this finite basis. The ideal fitting follows from the projection of the extended state on the finite Hilbert space spanned by the Gaussians, rather than any discretization of the continuous (overcomplete) coherent state representation.