Conformal Perturbations and Local Smoothing

TL;DR

This paper investigates how conformal metric perturbations affect local smoothing estimates for the Schrödinger equation on surfaces of revolution, demonstrating stability under certain small perturbations.

Contribution

It introduces conditions under which local smoothing persists despite conformal perturbations of the metric on surfaces of revolution.

Findings

Local smoothing estimates are stable under small conformal perturbations.

Perturbed metrics with specific symbolic estimates preserve smoothing effects.

The analysis extends previous results to a broader class of perturbed geometries.

Abstract

The purpose of this paper is to study the effect of conformal perturbations on the local smoothing effect for the Schr\"odinger equation on surfaces of revolution. The paper \cite{ChWu-lsm} studied the Schr\"odinger 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 $g_s = e^{sf}g$, 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.