Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Concentration of eigenfunctions on singular Riemannian manifolds

Charlotte Dietze, Larry Read

Abstract

We consider a compact Riemannian manifold with boundary and a metric that is singular at the boundary. The associated Laplace-Beltrami operator is of the form of a Grushin operator plus a singular potential. In a supercritical parameter regime, we identify the rate of concentration and profile of the high-frequency eigenfunctions that accumulate at the boundary. We give an application to acoustic modes on gas planets.

Concentration of eigenfunctions on singular Riemannian manifolds

Abstract

We consider a compact Riemannian manifold with boundary and a metric that is singular at the boundary. The associated Laplace-Beltrami operator is of the form of a Grushin operator plus a singular potential. In a supercritical parameter regime, we identify the rate of concentration and profile of the high-frequency eigenfunctions that accumulate at the boundary. We give an application to acoustic modes on gas planets.

Paper Structure

This paper contains 4 sections, 4 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Trace asymptotics
  3. Proof of Theorem \ref{['thm:main_grushin']}
  4. Application to the study of acoustic modes in gas giants

Key Result

Theorem 1

Let $\rho_\lambda$ be given by eqn:spectralfnc, $\beta>2/n$, then there exists a function $\mathop{\mathrm{B}}\nolimits\in L^1(0,\infty)$ depending on $\beta$ with ${\left\|\mathop{\mathrm{B}}\nolimits\right\|}_1=1$ that is defined in eq:Bdef below such that holds for any bounded $V\in C([0,\infty)\times M)$.

Figures (1)

  • Figure 1: Illustrative $X$, grey area represents $[0,1)\times M$

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['thm:main_grushin']}
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:main']}
  • Corollary 4
  • proof