Sharp Fundamental Gap Estimate on Convex Domains of Gaussian Spaces

Abstract: We prove a sharp lower bound for the fundamental gap of convex domains in Gaussian space, 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 a corresponding one-dimensional Schr\"odinger operator, 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. This work extends the seminal results of Andrews and Clutterbuck for Euclidean domains and Seto, Wang and Wei for spherical domains to the fundamentally important setting of Gaussian spaces.