Sharp Fundamental Gap Estimate on Convex Domains in Gaussian Spaces
Jin Sun, Kui Wang
TL;DR
This work proves a sharp lower bound for the fundamental gap in Gaussian spaces, showing that for strictly convex domains with diameter D the gap λ_2−λ_1 is bounded below by the 1D model gap, and the normalized 1D gap DF^2 is monotone in D. The authors achieve this via a gauge transformation to a Schrödinger operator with convex potential, enabling Ni-type elliptic maximum-principle methods to compare the n-dimensional problem with its 1D analogue. They also establish sharpened log-concavity properties for the Gaussian heat kernel on convex domains, and extend the framework to Schrödinger operators with convex potentials, connecting the Gaussian setting to known Euclidean and spherical results. The results provide a Gaussian-space analogue of Andrews–Clutterbuck’s fundamental gap theory and deepen understanding of heat-kernel log-concavity in probabilistic and geometric contexts.
Abstract
We prove a sharp lower bound for the fundamental gap on convex domains in Gaussian spaces, the difference between the first two eigenvalues of the Ornstein-Uhlenbeck operator with Dirichlet boundary conditions. Our main result establishes that the gap is bounded below by the gap of the corresponding one-dimensional model, confirming the Gaussian analogue of the fundamental gap conjecture. Furthermore, we demonstrate that the normalized gap of the one-dimensional model is monotonically increasing with the diameter and prove the sharpness of our estimate. Beyond the fundamental gap, we also establish improved log-concavity properties for the Dirichlet heat kernel on convex domains in Gaussian spaces. Our work on Gaussian spaces complements the existing results of Andrews and Clutterbuck and Ni for Euclidean domains, as well as the work of Seto, Wang, and Wei for spherical domains.