A Probabilistic Proof of the Fundamental Gap Conjecture Via the Coupling by Reflection

Abstract

Let \({\Omega }\subset \mathbb{R} ^{n}\) be a strictly convex domain with smooth boundary and diameter D. The fundamental gap conjecture claims that if \(V:\bar {\Omega }\to \mathbb{R} \) is convex, then the spectral gap of the Schrodinger operator −Δ+V with Dirichlet boundary condition is greater than \(\frac {3\pi ^{2}}{D^{2}}\). Using analytic methods, Andrews and Clutterbuck recently proved in (J. Amer. Math. Soc. 24(3), 889–916 2011) a more general spectral gap comparison theorem which implies this conjecture. In the first part of the current work, we shall give a probabilistic proof of their result via the coupling by reflection of the diffusion processes. Moreover, we also present in the second part a simpler probabilistic proof of the original conjecture.