On the Spectra of Real and Complex Lamé Operators

Abstract

We study Lamé operators of the form $$L=-\frac{d^2}{d{x}^2}+m(m+1){\omega }^2\wp (\omega x+z_0),$$ with $m\in \mathbb{N}$ and $\omega$ a half-period of $\wp (z)$. For rectangular period lattices, we can choose $\omega$ and $z_0$ such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a band structure with not more than $m$ gaps. In the first part of the paper, we prove that the opened gaps are precisely the first $m$ ones. In the second part, we study the Lamé spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the $m=1$ case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the $m=2$ case, paying particular attention to the rhombic lattices.

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