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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Proof of the fundamental gap conjecture
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by Ben Andrews and Julie Clutterbuck PDF
J. Amer. Math. Soc. 24 (2011), 899-916 Request permission

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
  • MR Author ID: 317229
  • ORCID: 0000-0002-6507-0347
  • Email: Ben.Andrews@anu.edu.au
  • Julie Clutterbuck
  • Affiliation: Centre for Mathematics and Its Applications, Australian National University, ACT 0200, Australia
  • MR Author ID: 656875
  • ORCID: 0000-0002-3186-4050
  • Email: Julie.Clutterbuck@anu.edu.au
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 24 (2011), 899-916
  • MSC (2010): Primary 35P15, 35J10; Secondary 35K05, 58J35
  • DOI: https://doi.org/10.1090/S0894-0347-2011-00699-1
  • MathSciNet review: 2784332