On the spectra of real and complex lamé operators
Journal article, 2017

We study Lamé operators of the form L = − dx/dx 2 2 + m(m + 1)ω 2 ℘(ωx + z 0 ), with m ∈ N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω 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.

Lamé operators

Spectral theory

Finite-gap operators

Non-self-adjoint operators


W.A. Haese-Hill

Loughborough University

Martin Hallnäs

University of Gothenburg

Chalmers, Mathematical Sciences, Analysis and Probability Theory

A.P. Veselov

Loughborough University

Symmetry, Integrability and Geometry - Methods and Applications

18150659 (eISSN)

Vol. 13 Article no. 049 - 049

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