## Abstract

We describe the essential spectrum σ_{e}(H) of a multidimensional Schrödinger operator H with a complex-valued potential V(x) 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 conditions we prove that the set σ_{e}(H) coincides with the set Γ{V_{y}} of such points λ ∈ ℂ for which the family of norms {||R_{λ}(H^{y})||}_{y∈ℝ} is unbounded at infinity. Sometimes the set Γ{V_{y}} coincides with the set Σ{V_{y}} of limit points of the spectra σ(H^{y}) of the operators H^{y} for |y| → ∞. In this case we call the family {V_{y}(x)}_{x∈ℝm} spectrally non-degenerate. We find some conditions of the spectral non-degeneracy. To this end we carry out an estimation of resolvents of the operators H^{y} with the help of generalized perturbation determinants for the corresponding cyclic boundary problems on the lattices of the periodicity.

Original language | English |
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Pages (from-to) | 11-68 |

Number of pages | 58 |

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