Helffer, Bernard; Klein, Markus; Nier, Francis Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. (English) Zbl 1079.58025 Mat. Contemp. 26, 41-85 (2004). Let \(\Omega\) denote either a connected compact Riemannian manifold or Euclidean space \(\mathbb{R}^n\). Let \(f\) be a Morse function and let \(\Delta_{f,h}^0:=-h^2\Delta+| \nabla f| ^2-h\Delta f\) be the semiclassical Witten Laplacian. Let \(m_0\) be the number of local minima of \(f\). If \(\Omega\) is compact, then there are exactly \(m_0\) eigenvalues in some interval \([0,e^{-\alpha/h}]\) for \(h>0\) small enough. The authors derive accurate asymptotic formula for the \(m_0\) first eigenvalues of \(\Delta_{f,h}^0\). Reviewer: Peter B. Gilkey (Eugene) Cited in 2 ReviewsCited in 44 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J10 Differential complexes 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 58J37 Perturbations of PDEs on manifolds; asymptotics 35P15 Estimates of eigenvalues in context of PDEs Keywords:Witten Laplacian; Dirichlet form; WKB analysis PDFBibTeX XMLCite \textit{B. Helffer} et al., Mat. Contemp. 26, 41--85 (2004; Zbl 1079.58025)