I was looking at the derivation of Bloch's theorem in Griffith's QM:
If $V(x+a)=V(x)$ for any $x$ and some $a$, and
$\psi$ solves
$$
H\psi =\lambda \psi
$$
for $H=-\frac{\hbar^2}{2\pi}\frac{d^2}{dx^2}+V(x)$
then
$$
\psi(x+a)=e^{ika}\psi(x)
$$
The proof relies on the fact that since the shift operator $D\psi(x)=\psi(x+a)$ commutes with the Hamiltonian, we may take an eigenfunction of $H$ to serve as an eigenfunction for $D$. The wikipedia article on the theorem (on a 3 dimensional crystal lattice but the principle is the same) once again asserts the existence of a simultaneous eigenbasis for the two operators.
What am I missing here? I know that this is a fact for matrices, but we are working over $L^2(\mathbb{R})$ here, where I don't believe this is true for unbounded self adjoint operators, and it is not clear to me why it should be true with this particular operator (indeed I think it is not).
Thanks for any help and I apologize if I am being naive.
The statement, if considered for a Hamiltonian with periodic potential which acts as a densely defined selfadjoint operator on an L^2 of the full space R^n, is wrong. Therefore, you won't find "Bloch's theorem" in this form in Reed/Simon. In vol 4., Reed and Simon treat Schroedinger operators with periodic potentials in chapter XIII.16.
You find there a theorem saying that under reasonable assumptions about the regularity of the periodic potential, the Hamiltonian operator has purely absolutely continuous spectrum (which gives rise to the physicist's "band structure") and therefore no eigenvalues or eigenfunctions would exist as elements in the Hilbert space. (Theorem XII.90 for 1 dimension and Theorem XIII.100 for dimensions at least 2.)
Physicists switch repeatedly between a treatment of finitely many stacked/translated primitive cells and the case of the full space with infinitely many translations of a primitive cell. Reed and Simon comment on this situation when they explain the "density of states measure" near the end of this chapter in Theorem XIII.101. Finitely many cells give an L^2-space over a compact domain. In this case, the Hamiltonian usually will have compact resolvents and a complete orthonormal set of eigenstates. There will be no continuous "band structure", but only discrete spectrum. In the case of the full space, the direct integral decomposition of Reed/Simon explains how the continuous band-structure is obtained from combining the discrete spectra of operators of a family H(q) (in the "p-space version") or H(theta) (in the "x-space version"). A sort of "eigenfunction expansion" is provided in Thereom XIII.98 (b).
As for Floquet's theorem for ODEs (i.e. one space dimension, no L^2-space...) you can find a nice explanation of the monodromy matrix and of the charateristic exponents in Vladimir Arnol'd's book about ordinary differential equations, and a "modern" proof in Hale's or Amann's or Walter's book about ordinary differential equations.