## Abstract

In the space L2(Rd)(d≤3) we consider the Schrödinger operator H_{γ}= - Δ + V(x) · + γW(x) · , where V(x) = V(x_{1}, x_{2}, ⋯ , x_{d}) is a periodic function with respect to all the variables, γ is a small real coupling constant and the perturbation W(x) tends to zero sufficiently fast as | x| → ∞. We study so called virtual bound levels of the operator H_{γ}, i.e., those eigenvalues of H_{γ} which are born at the moment γ= 0 in a gap (λ-,λ+) of the spectrum of the unperturbed operator H_{0}= - Δ + V(x) · from an edge of this gap while γ increases or decreases. We assume that the dispersion function of H_{0}, branching from an edge of (λ_{-}, λ_{+}) , is non-degenerate in the Morse sense at its extremal set. For a definite perturbation (W(x) ≥ 0) we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as γ→ 0. For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap (λ-,λ+) are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., σ(Hγ)∩(λ-,λ+)=∅ for a small enough | γ|.

Original language | English |
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Pages (from-to) | 307-345 |

Number of pages | 39 |

Journal | Integral Equations and Operator Theory |

Volume | 85 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jul 2016 |

### Bibliographical note

Publisher Copyright:© 2016, Springer International Publishing.

## Keywords

- Asymptotic behavior of virtual eigenvalues
- Coupling constant
- Perturbed periodic potential
- Schrödinger operator
- Virtual eigenvalues

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory