Trace asymptotics formulas and some applications

Here we give results on trace asymptotics with small remainder estimates. We treat situations where the spectral parameter is implicit, and where there is no really natural associated evolution equation. We apply these results to a periodic Schrödinger operator with two different types of perturbations: slowly varying and strong. In both cases we get precise remainder estimates for the counting function of eigenvalues of the perturbed periodic Schrödinger operator in a gap of the non‐perturbed one.