## Abstract

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.

Original language | English |
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Pages (from-to) | 69-124 |

Number of pages | 56 |

Journal | Integral Equations and Operator Theory |

Volume | 46 |

Issue number | 1 |

DOIs | |

State | Published - 2003 |

### Bibliographical note

Funding Information:Supported by Kamea Project for Scientific Absorption in Israel and partially by a grant from the Israel Science Foundation.

## Keywords

- Boundary problem
- Essential spectrum
- Periodic potential
- Schrödinger operator

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory