Construction of the essential spectrum for a multidimensional non-self-adjoint Schrödinger operator via the spectra of operators with periodic potentials, II

We investigate the topological structure of the essential spectrum σ_e(H) of a multidimensional Schrödinger operator H with a complex-valued potential V(x) using its description in terms of a family of Schrödinger operators {H^y}_y∈ℝm with periodic potentials V_y(x) which approximate the potential V(x) at infinity in a sense. Under some assumptions on a family of approximating potentials {V_y(x)}_y∈ℝm, we prove that any compact isolated part of the set σ_e(H) consists of a finite number of connected components and for real-valued potentials V_y(x) the set σ_e(H) consists of at most a countable number of segments. For the proof of the last results we develop the theory of awnings. These new topological objects are some kind of fiber bundle, whose fibers are discrete "multiple" sets. We consider several examples of construction of the essential spectrum by using the method developed in this paper.