A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in ℝ

abstract:In this paper, we study the spectrum of the complex Hill operator $L={d^2\over dx^2}+q(x;\tau)$ in $L^2(\Bbb{R},\Bbb{C})$ with the Darboux-Treibich-Verdier potential $$ q(x;\tau):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp\left(x+z_0+{\omega_k\over 2};\tau\right), $$ where $n_k\in\Bbb{Z}_{\geq 0}$ with $\max n_k\geq 1$ and $z_0\in\Bbb{C}$ is chosen such that $q(x;\tau)$ has no singularities on $\Bbb{R}$. For any fixed $\tau\in i\Bbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $\sigma(L)$ is $$ \sigma(L)=\big(-\infty, E_{2g}\big]\cup\big[E_{2g-1},E_{2g-2}\big]\cup\cdots\cup[E_1,E_0],\quad E_j\in\Bbb{R}, $$ 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 by Haese-Hill et al., who studied the Lam\'{e} case $n_1=n_2=n_3=0$.