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On the Spectra of Real and Complex Lamé Operators

Journal article
Authors W. A. Haese-Hill
Martin Hallnäs
A. P. Veselov
Published in Symmetry Integrability and Geometry-Methods and Applications
Volume 13
ISSN 1815-0659
Publication year 2017
Published at Department of Mathematical Sciences
Language en
Links dx.doi.org/10.3842/SIGMA.2017.049
Keywords Lame operators, finite-gap operators, spectral theory, non-self-adjoint operators, hill equation, quantization, potentials, Physics
Subject categories Mathematics

Abstract

We study Lame operators of the form with m is an element of N and omega a half- period of P(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 Lame 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 Lame 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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