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. of approximate eigenfunctions of the Laplacian on certain families of spaces, mostly exemplified by domains Ω h ⊂ R 2 , that degenerate as h → 0. 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 ideas of (multiple) scales and matching, is included.

Joint asymptotic expansions for Bessel functions

“…We also need to consider a variant on blow-up known as quasihomogeneous blow-up. This was originally introduced in [Mel96]; see [Gri17,GrHu09] for an introduction. In short, instead of introducing a new point for each distinct ray approaching P , we introduce a new point for each distinct parabola (or cubic, or quartic, et cetera) approaching p. We again illustrate this procedure with X = [0, 1] × [0, 1] and P = {(0, 0)}.…”

“…In the analytic setting these concepts were originally introduced by Richard Melrose [Mel96] and have been used, for example, to study index theory on singular manifolds [Mel93]. Expository treatments may be found in [Gri01,Gri17,Maz91].…”

Low energy limit for the resolvent of some fibered boundary operators

“…Here and throughout the paper, we will in particular assume that the reader has some familiarity with manifolds with corners as presented in [31]. What we will need can be found for instance in [16,Chapter 2] or [22, § 2].…”

“…Thus the composition on the right of (6.7) is the one induced by ff 0 T seen as a triple space for ff 0 , which is precisely composition as k,φ N tf Y -suspended operators. Finally, in terms of the vector bundle k,φ N sc Y of (3.10), the face ff does not quite correspond to the double space of k,φ N sc Y -suspended operators with respect to the fiber bundle 16), it is an adiabatic version of this suspended calculus, namely it is semi-classical in the suspension parameters with k playing the role of the semi-classical parameter. However, since suspended operators are already 'classical' in the suspension parameter, insisting on having rapid decay at ff ∩φbf 0 and ff ∩φbf, the boundary hypersurface ff can be seen as a double space for (k −1 ) k,φ N sc Ysuspended operators.…”

Polyhomogénéité des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci

“…Nous donnerons un aperçu général du concept de variétés à coins afin de développer les quelques notions de la géométrie des variétés avec une structure de Lie à l'infini qui nous seront nécessaires. Pour plus détails, on réfère aussi le lecteur à (Grieser, 2017).…”