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

TL;DR

This paper demonstrates that on convex domains in hyperbolic space, constant potentials do not necessarily minimize the fundamental gap of the Schrödinger operator, contrasting with expectations from Euclidean cases.

Contribution

It constructs convex domains with convex potentials in hyperbolic space where the fundamental gap is smaller than that of the Laplacian, showing a new phenomenon in non-Euclidean geometry.

Findings

Existence of convex domains with smaller gaps under convex potentials

Refined control of eigenfunctions in hyperbolic space

Contrasts with Euclidean gap minimization results

Abstract

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