On a Generic Topological Structure of the Spectrum to One-dimensional Schrödinger Operators with Complex Limit-periodic Potentials

Abstract.

We prove that for a massive set of limit-periodic complex-valued potentials V(x) $(x \in \mathbb{R})$ of Stepanov class, the spectra σ(H_v) of the corresponding one-dimensional Schrödinger operators H_v are 0-dimensional topological subspaces of the complex plane $\mathbb{C}$ . This is a generalization of a similar result obtained by J. Avron and B. Simon for the case of real-valued limit-periodic potentials. Our technique is based upon results of V. Tkachenko concerning an inverse spectral problem for the Hill operator with a complex-valued potential. We also make use of some facts from Differential Topology.