I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11:
It is well-known the following statements about the spectral properties of $L_{t}(q)$ and $L(q)$ [two particular operators]: (...)
Note that the rigorous proof of this theorem can be found in "The Spectral Theory of Periodic Differential Equations" (M.S.P. Eastham, 1973)
The theorem the author is referring to is about the band structure of the absolutely continuous spectrum of a particular periodic operator. Actually, it holds for the periodic Schrödinger operator:
$$H=-\Delta + V_{\text{per}}$$
where $V_{\text{per}}$ is a periodic potential.
My question is a bit vague, but aims to ask people knowing the book of M.S.P. Eastham mentioned above. I have access to this book from my institution but I cannot see where the proof of the mentioned theorem is. All I can see are proofs about spectral gaps (i.e. $\mathbb{R}\setminus \sigma(H)$ where $\sigma(H)$ denotes the spectrum of the operator $H$).
Are they equivalent ? I mean, is the existence of spectral gaps equivalent to a band structure of the spectrum ? I guess it is but cannot understand why all books I have seen so far consider the theorem in the "complementary" way it is proved.
I hope my question is not inappropriate for this site. If so, please close it.