Local $L^p$ norms of Schr{ö}dinger eigenfunctions on ${\mathbb{S}}^2$

Abstract

On the canonical $2$-sphere and for Schr{ö}dinger eigenfunctions, we obtain a simple geometric criterion on the potential under which we can improve, near a given point and for every $p\ne 6$, Sogge's estimates by a power of the eigenvalue. This criterion can be formulated in terms of the critical points of the Radon transform of the potential and it is independent of the choice of eigenfunctions.

arXiv comment28 pages, minor modifications. To appear in Annales math{é}matiques du Qu{é}bec (special volume in honor of Alexander Shnirelman)

References in corpus (8)

• Lower bounds for nodal sets of eigenfunctions
• Control of eigenfunctions on surfaces of variable curvature
• L^p norms of eigenfunctions in the completely integrable case
• Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions
• Local and global analysis of eigenfunctions
• Schrödinger equations on compact globally symmetric spaces
• Observability and quantum limits for the Schrödinger equation on the sphere
• $L^p$ estimates for joint quasimodes of semiclassical pseudodifferential operators whose characteristic sets have $k$th order contact