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On the spectra of real and complex lamé operators

W.A. Haese-Hill ; Martin A. Hallnäs (Institutionen för matematiska vetenskaper, Analys och sannolikhetsteori) ; A.P. Veselov
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) Vol. 13 (2017), p. Article no. 049 .
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Finite-gap operators, Lamé operators, Non-self-adjoint operators, Spectral theory

Denna post skapades 2017-07-27. Senast ändrad 2017-08-16.
CPL Pubid: 250814


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Institutionen för matematiska vetenskaper, Analys och sannolikhetsteoriInstitutionen för matematiska vetenskaper, Analys och sannolikhetsteori (GU)



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