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Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. (English) Zbl 1079.58025

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 \(\Delta_{f,h}^0\).

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
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