On the Spectra of Real and Complex Lamé Operators

• 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.

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Keyword

Lame operators

finite-gap operators

spectral theory

non-self-adjoint operators

hill equation

quantization

potentials

Physics

Lamé operators

Publication and Content Type

ref (subject category)

art (subject category)