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

TL;DR

This paper establishes a precise criterion for the spectrum of the Darboux-Treibich-Verdier potential in complex Hill operators to be real, extending classical results and analyzing eigenvalue distributions in this context.

Contribution

It provides a necessary and sufficient condition for the spectrum to be real and describes the eigenvalue structure, generalizing classical Lamé results to a broader class of potentials.

Findings

Spectrum is real under specific parameter conditions.

Explicit intervals containing the spectrum are identified.

Number of (anti)periodic eigenvalues in each interval is determined.

Abstract

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.