## 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

### Keywords:

Witten Laplacian; Dirichlet form; WKB analysis