Scales, blow-up and quasimode constructions

Abstract

In this expository article we show how the concepts of manifolds with corners, blow-ups and resolutions can be used effectively for the construction of quasimodes, i.e. approximate eigenfunctions of the Laplacian on certain families of spaces, mostly exemplified by domains $\Omega_h\subset\mathbb{R}^2$, that degenerate as $h\to0$. These include standard adiabatic limit families and also families that exhibit several types of scaling behavior. An introduction to manifolds with corners and resolutions, and how they relate to the idea of (multiple) scales and matching, is included.