I am concerned about the Schrödinger equation
$-x''(t)+q(t)x(t)=Ex(t).$
Here, the potential $q$ is real and quasiperiodic with frequency vector $\omega$. That is, we let $T^d$ be the $d$-dimensional torus and $\omega=(\omega_1,\ldots, \omega_d)\in T^d$ with the $\{\omega_j\}$ rationally independent, and $q(t)=f(\omega_1t,\omega_2t,\ldots,\omega_d t)$ for some continuous $f:T^d\to\mathbb R$,
Equivalently, we may formulate this equation as a matrix equation instead,
$X'(t)=\begin{pmatrix} 0&1\\ q(t)-E& 0 \end{pmatrix}X(t)$,
where $X(t)=\begin{pmatrix} x_1(t)& x_2(t)\\ x_1'(t)& x_2'(t) \end{pmatrix}.$
There are two equivalent descriptions of a Floquet-Bloch solution, which I am having difficulty reconciling.
First, we say a solution of the form $x(t)=e^{kt}(p_1(t)+tp_2(t))$ is a Floquet-Bloch solution, where $k$ is a constant and $p_1, p_2$ are quasiperiodic functions with frequency vector $\omega$ or $\omega/2$.
Second, we say $X(t)=Y(\omega t/2)e^{At}$ is a Floquet-Bloch solution, where $A\in SL(2,R)$ and $Y:T^d\to GL(2,R)$.
I don't see why these two are the same thing?