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Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group

Frank, Rupert L. and Laptev, Ari (2010) Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group. International Mathematics Research Notices, 2010 (15). pp. 2889-2902. ISSN 1073-7928.

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We prove that for any domain in the Heisenberg group the (k+1)'th Neumann eigenvalue of the sub-Laplacian is strictly less than the k'th Dirichlet eigenvalue. As a byproduct we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.

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Frank, Rupert L.0000-0001-7973-4688
Additional Information:© The Author 2010. Published by Oxford University Press. Received June 8, 2009; Accepted December 2, 2009 Communicated by Prof. Peter Sarnak The authors acknowledge interesting discussions with A. Hansson concerning the topics of this paper. The first author wishes to thank E. Lieb and R. Seiringer for helpful remarks. Support through DFG grant FR 2664/1-1 and U.S. NSF grant PHY 06 52854 (R.F.) is gratefully acknowledged. Recently, Bernard Helffer suggested a way of proving similar estimates for a large class of sub-elliptic operators. This will be the subject of a forthcoming publication.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)FR 2664/1-1
Issue or Number:15
Record Number:CaltechAUTHORS:20170526-092032067
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Official Citation:Rupert L. Frank, Ari Laptev; Inequalities between Dirichlet and Neumann Eigenvalues on the Heisenberg Group. Int Math Res Notices 2010; 2010 (15): 2889-2902. doi: 10.1093/imrn/rnp230
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77801
Deposited By: Ruth Sustaita
Deposited On:26 May 2017 18:58
Last Modified:09 Mar 2020 13:18

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