CLT for the trace functional of the IDS of magnetic random Schrödinger operators
Dhriti Ranjan Dolai, Naveen Kumar
TL;DR
The paper proves a central limit theorem for fluctuations of trace functionals of the integrated density of states of a magnetic random Schrödinger operator with alloy-type randomness. It introduces a continuum framework with a constant magnetic field, defines the IDS via a thermodynamic limit, and analyzes fluctuations of traces Tr$(f(H^ req_{L,X}))$ for test functions $f$ in the class $C^1_{d,0}[-\|V\|_\infty,\infty)$, under decays $|f(x)|,|f'(x)|=O(|x|^{-m})$ with $m>d+1$. The approach begins with a CLT for Laurent polynomial test functions using resolvent powers and a carefully designed box decomposition to obtain asymptotic independence of local contributions, then extends to general $f$ by approximation, via a probabilistic convergence framework. A key outcome is that for all such $f$, the centered trace fluctuations, scaled by $|\Lambda_L|^{-1/2}$, converge in distribution to $\mathcal{N}(0,\sigma^2_f)$, with a boundary-condition–independent variance that is explicitly given and strictly positive under natural monotonicity conditions on $f$ and $u$. This advances the understanding of IDS fluctuations in continuum magnetic random media, complementing known discrete and one-dimensional results.
Abstract
We consider the existence of the integrated density of states (IDS) of the magnetic Schrödinger 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 \( m > d + 1 \).