Nilufer Koldan, Mikhail Shubin Northeastern University Tapas Seminar June 14, 2006 Nilufer Koldan, Mikhail Shubin Semiclassical asymptotics and Witten's deformation on manifolds with boundary Witten's deformation of the de Rham complex of differential forms on a compact closed manifold was introduced by E.Witten in his famous paper, published in 1982. It is a new complex which depends on a given Morse function $f$ and contains a small parameter $h$ (``Planck's constant"). For the corresponding Laplacians the eigenfunctions in the semiclassical limit (as $h$ tends to 0) concentrate near critical points of $f$ , so that the asymptotics of the eigenvalues contain information about critical points and relate them to topology of the manifold, thereby giving a new proof of the Morse inequalities. In 2005, B.Helffer and F.Nier found asymptotics of the lowest eigenvalue for each of the Laplacians on a compact manifold with boundary. A new feature which appears here is an influence of the behavior of $f$ near the critical points of its restriction to the boundary. We extend the Helffer-Nier result by providing semiclassical asymptotics for every eigenvalue with a fixed number (in increasing order). The talk is based on a joint work by Igor Prokhorenkov and the speakers (in preparation). ------------------------------------------------------------------------------------------------------