CLT for the trace functional of the IDS of magnetic random Schr\"{o}dinger operators

TL;DR

This paper establishes a central limit theorem for the fluctuations of trace functionals of the integrated density of states (IDS) of magnetic random Schrödinger operators, extending the probabilistic understanding of spectral properties.

Contribution

It introduces a CLT for trace functionals of the IDS for magnetic Schrödinger operators with random potentials, covering a broad class of test functions.

Findings

Proves a CLT for trace functionals of the IDS.

Describes fluctuations for a class of decaying test functions.

Extends probabilistic spectral analysis to magnetic operators.

Abstract

We consider the existence of the integrated density of states (IDS) of the magnetic Schr\"{o}dinger operator with a random potential on the Hilbert space \( L^2(\mathbb{R}^d) \), as an analogue of the law of large numbers (LLN) for trace functionals. In this work, we establish an analogue of the central limit theorem (CLT), which describes the fluctuations of the trace functionals of the IDS, for a class of test functions denoted by \( C^1_{d,0}(\mathbb{R}) \). This class consists of real-valued, continuously differentiable functions on \( \mathbb{R} \) that decay at the rate \( O(|x|^{-m}) \) as \( |x| \to \infty \), where \( m > d + 1 \).