On the Spectra of Real and Complex Lamé Operators

Abstract

We study Lamé operators of the form L=−d²/dx²+m(m+1)ω²℘(ωx+z₀), with m∈N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z0 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.