Scales, blow-up and quasimode constructions

TL;DR

This paper demonstrates how manifolds with corners, blow-ups, and resolutions can be used to construct quasimodes, approximate eigenfunctions of the Laplacian, on degenerating families of domains with multiple scaling behaviors.

Contribution

It introduces a geometric framework using manifolds with corners and blow-ups for constructing quasimodes on degenerating spaces, connecting scale analysis with resolution techniques.

Findings

Effective construction of quasimodes on degenerating domains.

Application of geometric resolution techniques to spectral problems.

Illustration of multiple scale matching in eigenfunction approximation.

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.