Introduce the family of operators



The Schrödinger operator is unitarily equivalent to the direct integral:



When d = 3, the operator-valued function  is real analytic and each operator
has a compact resolvent as the domain  is compact;
hence, it has only discrete spectrum.

So, to show that the spectrum of the periodic Schrödinger is absolutely continuous, it 
suffices to show that none of the eigenvalues of  is constant.
This is done using perturbation theory in one of its various forms.