Central limit theorem for the random variables associated with the IDS of the Anderson model on the lattice

We consider the existence of the integrated density of states (IDS) for the Anderson model on the Hilbert space [Formula: see text] as an analogue of the law of large numbers (LLN). In this work, we prove an analogue of the central limit theorem (CLT) for a collection of random variables associated with the integrated density of states, for the class of test functions [Formula: see text], consisting of all once-differentiable functions on the real line whose derivative is continuous and has at most polynomial growth. Our result extends the work of Grinshpon-White (J. Spectr. Theory 12 (2022) 591–615), where the CLT was established for polynomial test functions.