CLT for the trace functional of the IDS of magnetic random Schrödinger operator

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 a trace functional. 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 ).