Let $H=-\dfrac{d^2}{dx^2}+V(x)$, where $V(x)$ is a $2\pi$-periodic and $V\in L^{\infty}(\mathbb R)$.
If $\mathcal H':=L^{2}(0,2\pi)$, we have the decomposition
$\mathcal H=\int_{\mathbb [0,2\pi)}^{\oplus}\mathcal H'\dfrac{d\theta}{2\pi}\cong L^2([0,2\pi),\mathcal H')$. Then defining the Floquet transformation $U:L^2(\mathbb R)\to\mathcal H$ as $(Uf)_{\theta}(x)=\sum_{n\in\mathbb Z}f(x+2\pi n)e^{-i\theta n}$,
we can wirte $UHU^{-1}=\int_{\mathbb [0,2\pi)}^{\oplus}H_{\theta}\dfrac{d\theta}{2\pi}$ and the operators on the fiber have compact resolvent, hence $\sigma(H_{\theta})$ consists of eigenvalues of the form $\lambda_1(\theta)\le \lambda_2(\theta)\le\dots$.
Could you help me finding a reference that explicitily shows the analytic dependence on $\theta$ of the eigenvalues?