Low lying eigenvalues of witten Laplacians and metastability (after Helffer-Klein-Nier and Helffer-Nier)

Abstract

The aim of this lecture is to present the recent results obtained in collaboration with M. Klein and F. Nier on the low lying eigenvalues of the Laplacian attached to the Dirichlet form: C0∞(Ω) ∋ v → h2 ∫Ω ∇v(x) 2 e-2f(x)/h dx, where f is a C∞ Morse function on ̄Ω and h > 0. We give in particular an optimal asymptotics as h → 0 of the lowest strictly positive eigenvalue, which will hold under generic assumptions. We discuss also some aspects of the proof. © Springer 2006.