The geometric size of the fundamental gap

TL;DR

This paper strengthens the fundamental gap inequality for convex sets by quantifying how geometric flatness influences the eigenvalue gap, using a novel variational approach and convex partitioning techniques.

Contribution

It provides a quantitative enhancement of the fundamental gap inequality, linking geometric flatness to eigenvalue differences, and extends Payne-Weinberger inequalities to Neumann eigenvalues.

Findings

Quantitative bounds on the eigenvalue gap based on flatness.

A new sharp one-dimensional Schrödinger eigenvalue result.

A proof of the Hang-Wang conjecture for Neumann eigenvalues.

Abstract

The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in $\mathbb{R}^N$. The question concerning the rigidity of the inequality, raised by Yau in 1990, was left open. Going beyond rigidity, our main result strengthens Andrews-Clutterbuck inequality, by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The proof relies on a localized, variational interpretation of the fundamental gap, allowing a dimension reduction via the use of convex partitions \`a la Payne-Weinberger: the result stems by combining a new sharp result for one dimensional Schr\"odinger eigenvalues with measure potentials, with a thorough analysis of the geometry of the partition into convex cells. As a by-product of our approach, we obtain a quantitative form of Payne-Weinberger inequality for the first nontrivial Neumann eigenvalue of a convex set in $\mathbb{R}^N$, thus proving, in a stronger version, a conjecture from 2007 by Hang-Wang.