Sharp bounds for the first eigenvalue of symmetric Markov processes and their applications (Q1928125)

Let \(X\) be a reversible Markov process on a locally compact, separable metric space, with symmetric measure \(\mu \). Let \(L\) denote its infinitesimal operator on \(L^2(\mu )\). Introducing a transform of this operator, \(L_ {\rho } f := \frac{L(\rho f)}{\rho } - \frac{L(\rho )}{\rho } f\) (for a strictly positive function \(\rho \)), the author gives lower and upper bounds for the first eigenvalue of \(L\). This result covers the lower bound found by \textit{M. Chen} [Acta Math. Sin., Engl. Ser. 16, No. 3, 361--368 (2000; Zbl 0963.60078)]. Further, the author uses his result to obtain approximation procedures for the first eigenvalue of birth-death processes with killing and qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.