Conformal Perturbations and Local Smoothing

2020/06/30 by Christianson, Hans, Muckerman, Dylan
#35P25 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2007.00078

Abstract

The purpose of this paper is to study the effect of conformal perturbations on the local smoothing effect for the Schrödinger equation on surfaces of revolution. The paper \citeChWu-lsm studied the Schrödinger equation on surfaces of revolution with one trapped orbit. The dynamics near this trapping were unstable, but degenerately so. Beginning from the metric g from this paper, we consider the perturbed metric g_s = e^sfg, where f is a smooth, compactly supported function. If s is small enough and finitely many derivatives of f satisfy appropriate symbolic estimates, then we show that a local smoothing estimate still holds.

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