Local behavior of solutions of the stationary Schr\" odinger equation with singular potentials and bounds on the density of states of Schr\"{o}dinger operators

TL;DR

This paper investigates the local behavior of solutions to the stationary Schrödinger equation with singular potentials, deriving a decomposition and establishing continuity properties of the density of states in low dimensions.

Contribution

It introduces a local decomposition of solutions into harmonic and lower order parts and proves log-Hölder continuity of the density of states measure for singular potentials.

Findings

Decomposition of solutions into harmonic polynomial and lower order term

Log-Hölder continuity of the density of states measure in 1-3 dimensions

Results applicable to Schrödinger operators with singular potentials

Abstract

We study the local behavior of solutions of the stationary Schr\" od\-inger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term. Combining a corollary to this result with a quantitative unique continuation principle for singular potentials we obtain log-H\"older continuity for the density of states outer-measure in one, two, and three dimensions for Schr\" odinger operators with singular potentials, results that hold for the density of states measure when it exists.