On the shape of spectra for non-self-adjoint periodic Schrödinger operators

Details On the shape of spectra for non-self-adjoint periodic Schrödinger operators On the shape of spectra for non-self-adjoint periodic Schrödinger operators

2004-04-06

arxiv.org/abs/math-ph/0404015v1

J.Phys. A37 (2004) 8287-8292

Physics

Mathematical Physics

5 pages, 2 figures

Scientific paper

The spectra of the Schr\"odinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the potential is complex and periodic, the spectrum consists of at most countably many analytic arcs in the complex plane. In some recent papers, such operators with complex $\mathcal{PT}$-symmetric periodic potentials are studied. In particular, the authors argued that some energy bands would appear and disappear under perturbations. Here, we show that appearance and disappearance of such energy bands imply existence of nonreal spectra. This is a consequence of a more general result, describing the local shape of the spectrum.

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