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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Proof of the fundamental gap conjecture


Authors: Ben Andrews and Julie Clutterbuck
Journal: J. Amer. Math. Soc. 24 (2011), 899-916
MSC (2010): Primary 35P15, 35J10; Secondary 35K05, 58J35
Published electronically: March 16, 2011
MathSciNet review: 2784332
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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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Additional Information

Ben Andrews
Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia and Mathematical Sciences Center, Tsinghua University, Beijing, 100084, Peoples Republic of China and Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, 100190, Peoples Republic of China
Email: Ben.Andrews@anu.edu.au

Julie Clutterbuck
Affiliation: Centre for Mathematics and Its Applications, Australian National University, ACT 0200, Australia
Email: Julie.Clutterbuck@anu.edu.au

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00699-1
PII: S 0894-0347(2011)00699-1
Keywords: Parabolic equation, eigenvalue estimate, spectral gap
Received by editor(s): June 8, 2010
Received by editor(s) in revised form: July 9, 2010, and January 11, 2011
Published electronically: March 16, 2011
Additional Notes: This research was supported by Discovery Grant DP0985802 of the Australian Research Council
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.