Comparison Theorems for the Spectral Gap of Diffusion Processes and Schrodinger Operators on an Interval

The spectral gaps and thus the exponential rates of convergenceto equilibrium are compared for ergodic one-dimensional diffusionson an interval. One of the results may be thought of as thediffusion analogue of a recent result for the spectral gap ofone-dimensional Schrödinger operators. The similaritiesand differences between spectral gap results for diffusionsand for Schrödinger operators are also discussed.