Proof of the fundamental gap conjecture

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral 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.
Original languageEnglish
Pages (from-to)899 - 916
Number of pages18
JournalJournal of the American Mathematical Society
Volume24
Issue number3
DOIs
Publication statusPublished - 2011
Externally publishedYes

Cite this

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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 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.",
author = "Ben Andrews and Clutterbuck, {Julie Faye}",
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Proof of the fundamental gap conjecture. / Andrews, Ben; Clutterbuck, Julie Faye.

In: Journal of the American Mathematical Society, Vol. 24, No. 3, 2011, p. 899 - 916.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

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AU - Andrews, Ben

AU - Clutterbuck, Julie Faye

PY - 2011

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N2 - We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral 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.

AB - We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral 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.

UR - http://www.ams.org/journals/jams/2011-24-03/S0894-0347-2011-00699-1/S0894-0347-2011-00699-1.pdf

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DO - 10.1090/S0894-0347-2011-00699-1

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SP - 899

EP - 916

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

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