Semiclassical Asymptotics on Manifolds with Boundary

We find the semiclassical asymptotics for every eigenvalue of the Witten Laplacian up to any fixed index (in increasing order) for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead of the operators. We also utilize some important ideas and technical elements from Helffer and Nier (2006), who proved more accurate asymptotic expansions but only for the exponentially small eigenvalues.