General formula for lower bound of the first eigenvalue on Riemannian manifolds (Q1368212)

Let \(M\) be a compact connected Riemannian manifold with possibly empty boundary \(\partial M\). Consider the operator \(L:=\Delta+\nabla V\) for some \(V\in C^2(M)\). Let \(\lambda_1\) be the first (non-trivial) eigenvalue of \(L\) with Neumann boundary conditions. The authors give a lower bound for \(\lambda_1\) using probabilistic methods (the coupling method). The estimate improves an estimate of \textit{A. Lichnérowicz} [`Géométrie des groupes de transformations' (Travaux et recherches mathématiques 3, Dunod, Paris) (1958; Zbl 0096.16001)] and of \textit{J.-Q. Zhong} and \textit{H.-C. Yang} [Sci. Sin., Ser. A 27, 1265-1273 (1984; Zbl 0561.53046)]. The results are extended to non-compact manifolds and the spectral gap is estimated.