Proof of the fundamental gap conjecture

2010/06/09 by Ben Andrews, Andrews, Ben, Julie Clutterbuck +1 · 8 citations
Computer Science · Mathematics · #35J10 (Primary) #35K05 #35P15 #58J35 (Secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1006.1686

openalex publication_date 2010/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential. More generally, for an arbitrary smooth potential in higher dimensions, our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction, in terms of the diameter of the domain and a modulus of convexity for the potential.

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• Gradient bounds for viscosity solutions to certain elliptic equations
• Generic failure of uniform separation in planar Dirichlet spectra
• Self-concordant Schrödinger operators: spectral gaps and optimization without condition numbers
• Sharp Fundamental Gap Estimate on Convex Domains in Gaussian Spaces
• Modulus of continuity for solutions of non-local heat equations
• An improved version of a spectral inequality by Payne

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