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

TL;DR

This paper investigates the spectral gap of a Nelson diffusion process derived from the atomic elliptic state, demonstrating convergence to an invariant measure concentrated on the Kepler ellipse using inequalities like Cheeger and Poincare.

Contribution

It provides a detailed analysis of the invariant measure and proves the existence of a spectral gap for the limiting Nelson diffusion generator in the atomic elliptic state.

Findings

Invariant measure is concentrated on the Kepler ellipse.

The Nelson diffusion generator has a spectral gap.

Density converges to the invariant measure over time.

Abstract

The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson's stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z=0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincare inequalities both of which are used in our proof of the existence of the spectral gap.