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Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space

Julie Clutterbuck, Frieder Jäckel, Xuan Hien Nguyen

TL;DR

The paper proves that in hyperbolic space, constant potentials do not minimize the fundamental gap for convex domains. It achieves this by a variational argument: perturbing a constant potential along a convex direction and showing the gap decreases if a certain integral against the squared eigenfunctions is negative. The core mechanism reduces the problem to an Airy-type perturbation of a separable ODE via a carefully chosen domain in $\mathbb{H}^2$, and then extends the analysis to higher dimensions with analogous one-dimensional reductions and weighted perturbation arguments. This work highlights a nuanced behavior of spectral gaps in negatively curved spaces and delineates where Euclidean intuition breaks down.

Abstract

We show that for every $n \geq 2$ and $D > 0$ there exist a convex domain $Ω\subseteq \mathbb H^n$ with diameter $D$ and a convex potential $V$ on $Ω$ such that the fundamental gap of the operator $-Δ+V$ is strictly smaller than the fundamental gap of $-Δ$. In comparison to previous work, this result requires more refined control of the eigenfunctions.

Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space

TL;DR

The paper proves that in hyperbolic space, constant potentials do not minimize the fundamental gap for convex domains. It achieves this by a variational argument: perturbing a constant potential along a convex direction and showing the gap decreases if a certain integral against the squared eigenfunctions is negative. The core mechanism reduces the problem to an Airy-type perturbation of a separable ODE via a carefully chosen domain in , and then extends the analysis to higher dimensions with analogous one-dimensional reductions and weighted perturbation arguments. This work highlights a nuanced behavior of spectral gaps in negatively curved spaces and delineates where Euclidean intuition breaks down.

Abstract

We show that for every and there exist a convex domain with diameter and a convex potential on such that the fundamental gap of the operator is strictly smaller than the fundamental gap of . In comparison to previous work, this result requires more refined control of the eigenfunctions.

Paper Structure

This paper contains 14 sections, 7 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Strategy
  4. Structure of the article.
  5. Set up
  6. Choice of domain and potential
  7. The Airy equation
  8. Approximation by the Airy equation
  9. Preliminary perturbation estimates
  10. The eigenvalue problem as a perturbed Airy equation
  11. Proof of Theorem \ref{['Main Theorem']} when $n=2$
  12. Higher dimensions
  13. Domain and change of variables
  14. Perturbation arguments

Key Result

Theorem 1

For every $n \geq 2$ and every $D > 0$ there exist a convex domain $\Omega \subseteq \mathbb{H}^n$ and a convex function $V$ on $\Omega$ such that ${\rm diam}(\Omega)=D$ and $\Gamma(\Omega;V) < \Gamma(\Omega)$.

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 2.1: Strategy
  • Lemma 2.2: Model Integral
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Remark 3.4
  • ...and 6 more