Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 2 (2014), 435-460.

Spectral estimates on the sphere

Jean Dolbeault, Maria Esteban, and Ari Laptev

Full-text: Open access

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 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.

Article information

Source
Anal. PDE, Volume 7, Number 2 (2014), 435-460.

Dates
Received: 7 January 2013
Accepted: 13 June 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731495

Digital Object Identifier
doi:10.2140/apde.2014.7.435

Mathematical Reviews number (MathSciNet)
MR3218815

Zentralblatt MATH identifier
1293.35183

Subjects
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

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

Citation

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


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