ELECTRONIC COMMUNICATIONS in PROBABILITY FIRST EIGENVALUE OF ONE-DIMENSIONAL DIFFUSION PRO-

rence, diffusion operators 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 [17] and Chen’s variational formulas [8] are reviewed, and both provide the original motivation for this research. 1 Introduction and Main Results In this paper, we deal with explicit bounds of the first Dirichlet eigenvalue for diffusion operators on the half line R+: = [0,∞). The work is a continuation or a supplement of [17, 5, 6], and is also inspired by analogous research for birth-death processes in [7, 19]. Let a(x) be positive ev-erywhere on (0,∞). For any measurable function b(x) on (0,∞), define C(x)