$L^p$-approximation of the integrated density of states for Schr\"odinger operators with finite local complexity

TL;DR

This paper proves that the integrated density of states for Schrödinger operators with locally determined electromagnetic potentials can be approximated in $L^p$ spaces by finite volume eigenvalue counts, ensuring convergence in spectral analysis.

Contribution

It establishes $L^p$-approximation of the integrated density of states for Schrödinger operators with finite local complexity potentials, extending previous results to a broader function space.

Findings

Convergence of eigenvalue counting functions in $L^p$ spaces for Schrödinger operators.

Applicability to potentials determined by lattice colorings with finite pattern frequencies.

Extension of spectral approximation techniques to more general $L^p$ settings.

Abstract

We study spectral properties of Schr\"odinger operators on $\RR^d$. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in $\ZZ^d$, with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e.the normalised eigenvalue counting functions. The convergence holds in the space $L^p(I)$ where $I$ is any finite energy interval and $1\leq p< \infty$ is arbitrary.