I am learning Ashcroft's Solid State Physics. In the Electrons in a Weak Periodic Potential, I got some problems. 1. Ashcroft mentioned in the footnote: The reader familiar with stationary perturbation theory may think that if there is no exact degeneracy, we can always make all level differences large compared with U by considering sufficiently small U. That is indeed true for any given k. However, once we are given a definite U, no matter how small, we want a procedure valid for all k in the first Brillouin zone. We shall see that no matter how small U is we can always find some values of k for which the unperturbed levels are closer together than U. Therefore what we are doing is more subtle than conventional degenerate perturbation theory. I don't quite understand this footnote. I thought that what we are doing is just degenerate and non-degenerate stationary perturbation theory. 2. In the previous section, when Ashcroft was deducing Bloch's theorem, he assumed that the crystal has inversion symmetry: U(r)=U(-r). However, he mentioned in the footnote: The reader is invited to pursue the argument of this section without the assumption of inversion symmetry, which is made solely to avoid inessential complications in the notation. How far we can go without the assumption of inversion symmetry and why? Can we see this problem quantum mechanically? Thank you for your help!