We consider the first Dirichlet eigenvalue of diffusion operators on the half line. A criterion for the equivalence of the first Dirichlet eigenvalue with respect to the maximum domain and that to the minimum domain is presented. We also describle the relationships between the first Dirichlet eigenvalue of transient diffusion operators and the standard Muckenhoupt's conditions for the dual weighted Hardy inequality. Pinsky's result  and Chen's variational formulas  are reviewed, and both provide the original motivation for this research.
"First Eigenvalue of One-dimensional Diffusion Processes." Electron. Commun. Probab. 14 232 - 244, 2009. https://doi.org/10.1214/ECP.v14-1464