The first eigenvalue of one-dimensional elliptic operators with killing

Abstract

In this paper, we investigate the first eigenvalue for one-dimensional elliptic operators with killing. Two-sided approximation procedures and basic estimates of the first eigenvalue are given in both the half line and the whole line. The proofs are based on the $h$-transform, Chen's dual variational formulas, and the split technique. In particular, a few examples are presented to illustrate the power of our results.