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Spectral estimates on
the sphere
Jean Dolbeault, Maria J. Esteban and Ari Laptev |
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Vol. 7 (2014), No. 2, 435–460
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Abstract |
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In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the -dimensional unit sphere. These estimates depend on norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semiclassical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space. |
Keywords
spectral problems, partial differential operators on
manifolds, quantum theory, estimation of eigenvalues,
Sobolev inequality, interpolation,
Gagliardo–Nirenberg–Sobolev inequalities, logarithmic
Sobolev inequality, Schrödinger operator, ground state, one
bound state Keller–Lieb–Thirring inequality
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Mathematical Subject Classification 2010
Primary: 35P15, 58J50, 81Q10, 81Q35
Secondary: 47A75, 26D10, 46E35, 58E35, 81Q20
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Milestones
Received: 7 January 2013
Accepted: 13 June 2013
Published: 30 May 2014
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Authors |
| Jean Dolbeault | |
| Ceremade CNRS UMR 7534 Université Paris-Dauphine Place de Lattre de Tassigny 75775 Paris 16 France |
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| Maria J. Esteban | |
| Ceremade CNRS UMR 7534 Université Paris-Dauphine Place de Lattre de Tassigny 75775 Paris 16 France |
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| Ari Laptev | |
| Department of Mathematics Imperial College London Huxley Building, 180 Queen’s Gate London SW7 2AZ United Kingdom |
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