#### 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
##### Abstract

In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the $d$-dimensional unit sphere. These estimates depend on ${L}^{p}$ 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
##### Mathematical Subject Classification 2010
Primary: 35P15, 58J50, 81Q10, 81Q35
Secondary: 47A75, 26D10, 46E35, 58E35, 81Q20