Proof of the fundamental gap conjecture

By Ben Andrews, Julie Clutterbuck. 2010

176 citations recorded by the source corpus.

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spec- tral gap) of a Schrodinger 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/def91926c883dc96e05678c34a8c30e91b6e23ad

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Geometrical Structure of Laplacian Eigenfunctions (2012, methodology): “The estimate was later rigorously proved by Andrews and Clutterbuck for any bounded convex domain Ω in R, even in the presence of a semiconvex potential [10] (for more background on the spectral gap, see notes by Ashbaugh [23]).”
Non-collapsing in fully non-linear curvature flows (2011, methodology): “For parabolic equations estimates on the modulus of continuity have been developed in [7,8] and were applied in [9,24] to eigenfunctions and heat kernels.”