On the Spectra of Real and Complex Lamé Operators

• Published in: arXiv.org
• Main Authors: Haese-Hill, William A, Hallnäs, Martin A, Veselov, Alexander P
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 01.07.2017
• Subjects: Lattices; Operators
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1609.06247

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.