I'm searching for quasiperiodic solutions of the Hill equation
$$ H \psi := \frac{d^2 \psi}{d z^2} + V(z) \psi (z) = 0 $$
where
$$ V(z) := \theta_0 + 2\sum_{n=1}^{\infty} \theta_n \cos{(2 n z)} $$
in the special case of $\theta_1 \neq 0$, some $\theta_n = \theta_1 / n$ and all other $\theta_j = 0$ for $j>1 , j \neq n$. The coefficients $\theta_1$ and thus $\theta_n$ are fixed. I'm trying to find $\theta_0$ for that such quasiperiodic solutions with $\psi(z+\pi) = e^{i\mu z}\psi(z)$ with $\mu \in \mathbb{R}$ exist.
This equation is the time-independent Schrödinger equation for a superconducting qubit called the fluxonium qubit (after some simplifications).