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The correspondence limit of the atomic elliptic state in three dimensions isdiscussed in terms of Nelson's stochastic mechanics. In previous work we haveshown that this approach leads to a limiting Nelson diffusion and here wediscuss in detail the invariant measure for this process and show that it isconcentrated on the Kepler ellipse in the plane z=0. We then show that thelimiting Nelson diffusion generator has a spectral gap; thereby proving that inthe infinite time limit the density for the limiting Nelson diffusion willconverge to its invariant measure. We also include a summary of the Cheeger andPoincare inequalities both of which are used in our proof of the existence ofthe spectral gap.

On the divine clockwork: the spectral gap for the correspondence limit of the Nelson diffusion generator for the atomic elliptic state