Analysis & PDE
- Anal. PDE
- Volume 7, Number 2 (2014), 435-460.
Spectral estimates on the sphere
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.
Anal. PDE, Volume 7, Number 2 (2014), 435-460.
Received: 7 January 2013
Accepted: 13 June 2013
First available in Project Euclid: 20 December 2017
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 81Q35: Quantum mechanics on special spaces: manifolds, fractals, graphs, etc.
Secondary: 47A75: Eigenvalue problems [See also 47J10, 49R05] 26D10: Inequalities involving derivatives and differential and integral operators 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 58E35: Variational inequalities (global problems) 81Q20: Semiclassical techniques, including WKB and Maslov methods
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
Dolbeault, Jean; Esteban, Maria; Laptev, Ari. Spectral estimates on the sphere. Anal. PDE 7 (2014), no. 2, 435--460. doi:10.2140/apde.2014.7.435. https://projecteuclid.org/euclid.apde/1513731495