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 $L=\frac{d^2}{d{x}^2}+q(x;\tau )$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential $$q(x;\tau ):=-\sum _{k=0}^3{n}_k(n_k+1)\mathrm{\wp }(x+z_0+\frac{{\omega }_k}{2};\tau ),$$ where $n_k\in {\mathbb{Z}}_{\ge 0}$ with $\max n_k\ge 1$ 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 $$\sigma (L)=(-\mathrm{\infty },E_{2g}]\cup [E_{2g-1},E_{2g-2}]\cup \cdots \cup [E_1,E_0],E_j\in \mathbb{R},$$ 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.