### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 049, 23 pages      arXiv:1609.06247      https://doi.org/10.3842/SIGMA.2017.049

### On the Spectra of Real and Complex Lamé Operators

William A. Haese-Hill a, Martin A. Hallnäs b and Alexander P. Veselov a
a) Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
b) Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden

Received April 04, 2017, in final form June 21, 2017; Published online July 01, 2017

Abstract
We study Lamé 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 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.

Key words: Lamé operators; finite-gap operators; spectral theory; non-self-adjoint operators.

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