A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$

In this paper, we study the spectrum of the complex Hill operator $L=\frac{d^2}{dx^2}+q(x;\tau)$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential $$q(x;\tau):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{\omega_{k}}{2};\tau \right),$$ where $n_k\in\mathbb{Z}_{\geq 0}$ with $\max n_k\geq 1$ and $z_0\in\mathbb{C}$ is chosen such that $q(x;\tau)$ has no singularities on $\mathbb{R}$. For any fixed $\tau\in i\mathbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $\sigma(L)$ is $$\sigma(L)=(-\infty, E_{2g}]\cup[E_{2g-1}, E_{2g-2}]\cup \cdots \cup[E_{1}, E_{0}],\quad E_j\in \mathbb{R},$$ and hence generalizes Ince's remarkable result in 1940 for the Lam\'{e} potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval $(E_{2j-1}$, $E_{2j-2})$, which generalizes the recent result in \cite{HHV} where the Lam\'{e} case $n_1=n_2=n_3=0$ was studied.