On the Spectra of Real and Complex Lam\'e Operators

TL;DR

This paper analyzes the spectral properties of Lamé operators with elliptic potentials, establishing the precise number of spectral gaps for real potentials and exploring the complex case, especially for the case m=1.

Contribution

It proves that the first m gaps are the only open gaps for real Lamé operators and investigates the spectral structure for complex potentials, focusing on the m=1 case.

Findings

The first m spectral gaps are the only ones that open for real Lamé operators.

For complex potentials, the spectrum for m=1 consists of two analytic arcs, one extending to infinity.

Brief discussion on the spectral structure for m=2 and rhombic lattices.

Abstract

We study Lam\'e operators of the form $$L = -\frac{d^2}{dx^2} + m(m+1)\omega^2\wp(\omega x+z_0),$$ 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 Lam\'e 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 Lam\'e 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.