Uncertainty principles and singular potentials

TL;DR

This paper develops uncertainty principles on compact Riemannian manifolds with singular potentials, linking spectral conditions to stability of inequalities and quantifying the effects of spectral inhomogeneity.

Contribution

It introduces spectral conditions replacing homogeneity assumptions, providing sharp, stability-enhanced uncertainty inequalities on manifolds with singular potentials.

Findings

Established uncertainty principles on manifolds with singular potentials.

Quantified deterioration of classical bounds due to spectral inhomogeneity.

Proved automatic homogeneity in one dimension and extended results to higher dimensions.

Abstract

We establish uncertainty principles on compact Riemannian manifolds without boundary in the setting of Laplace-Beltrami operators, including the case of real-valued singular potentials. We replace the classical homogeneity assumption by a quantitative spectral condition and obtain corresponding stability versions of uncertainty inequalities. In particular, we prove that \[ (1-\epsilon-\epsilon')^2 \leq \frac{|E|}{|M|}\cdot \# X_S \cdot \sup_{x\in E} \frac{A_S(x)}{\frac{\# X_S}{|M|}}, \] which recovers the classical bound in the homogeneous case, quantifies its deterioration in the presence of spectral inhomogeneity, and is shown to be sharp in general. In {\it dimension one}, we show that the homogeneity condition holds automatically, and we complement this rigidity by incorporating Fourier-ratio complexity bounds, yielding a quantitative relationship between spectral complexity and spatial support. In higher dimensions, we derive analogous results using pointwise Weyl laws and the eigenfunction restriction estimates on submanifolds.