Quantitative unique continuation principle for Schr\"odinger Operators with Singular Potentials

TL;DR

This paper establishes a quantitative unique continuation principle for Schrödinger operators with singular potentials, providing new insights into their spectral properties and extending classical results to more general potential classes.

Contribution

It introduces a novel quantitative unique continuation principle applicable to Schrödinger operators with singular potentials, broadening the scope of spectral analysis techniques.

Findings

Proves a quantitative unique continuation principle for Schrödinger operators with singular potentials.

Derives a unique continuation principle for spectral projections of such operators.

Extends classical unique continuation results to potentials in L^∞ + L^p spaces.

Abstract

We prove a quantitative unique continuation principle for Schr\"odinger operators $H=-\Delta+V$ on $\mathrm{L}^2(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^d$ and $V$ is a singular potential: $V \in \mathrm{L}^\infty(\Omega) + \mathrm{L}^p(\Omega)$. As an application, we derive a unique continuation principle for spectral projections of Schr\"odinger operators with singular potentials.