Vol. 7, No. 2, 2014

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Spectral estimates on the sphere

Jean Dolbeault, Maria J. Esteban and Ari Laptev

Vol. 7 (2014), No. 2, 435–460

In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the d-dimensional unit sphere. These estimates depend on Lp 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.

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
Mathematical Subject Classification 2010
Primary: 35P15, 58J50, 81Q10, 81Q35
Secondary: 47A75, 26D10, 46E35, 58E35, 81Q20
Received: 7 January 2013
Accepted: 13 June 2013
Published: 30 May 2014
Jean Dolbeault
Ceremade CNRS UMR 7534
Université Paris-Dauphine
Place de Lattre de Tassigny
75775 Paris 16
Maria J. Esteban
Ceremade CNRS UMR 7534
Université Paris-Dauphine
Place de Lattre de Tassigny
75775 Paris 16
Ari Laptev
Department of Mathematics
Imperial College London
Huxley Building, 180 Queen’s Gate
London SW7 2AZ
United Kingdom