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This is a Quantitative Study study.

September 18, 2025

L q estimates on the restriction of Schrödinger eigenfunctions with singular potentials

Key Points

Eigenfunction estimates demonstrate significant behavior of schrödinger operators with singular potentials.
Prior work established eigenfunction estimates for the case where the potential is zero or critically singular.
This article extends previous results to analyze the impact of singular potentials on submanifolds of compact riemannian manifolds.
Handling these estimates may lead to further insights into the nature of eigenfunctions in mathematical physics.

Abstract

We consider eigenfunction estimates in Lq for Schrödinger operators, HV=−Δg+V(x), on compact Riemannian manifolds (M, g). Eigenfunction estimates over the full manifolds were already obtained by Sogge for V≡0, and by the first author, Sire, and Sogge and the first author, Huang, Sire, and Sogge for critically singular potentials V. For the corresponding restriction estimates for submanifolds, the case V≡0 was considered in Burq, Gérard, and Tzvetkov, and in Hu. In this article, we will handle eigenfunction restriction estimates for some submanifolds Σ on compact Riemannian manifolds (M, g) with n:=dim⁡M≥2, where V is a singular potential.

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