## Abstract

We consider the Schrödinger operator H _{γ} =(-Δ) ^{l} +γ V(x)• acting in the space L_{2} (double scprit R sign^{d} ), where 2l ≥d, V (x)≥ 0,V (x) is continuous and is not identically zero, and lim |x| →∞ V(x) = 0. We study the asymptotic behavior as γ,↑,0 of the non-bottom negative eigenvalues of H _{γ}, which are born at the moment γ=0 from the lower bound λ=0 of the spectrum σ(H _{0}) of the unperturbed operator H _{0}=(-Δ) ^{l} (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.

Original language | English |
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Pages (from-to) | 305-345 |

Number of pages | 41 |

Journal | Integral Equations and Operator Theory |

Volume | 55 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2006 |

### Bibliographical note

Funding Information:Both authors were partially supported from the Israel Science Foundation (ISF), grant number 585/00, and from the German-Israeli Foundation (GIF), grant number I-619-17.6/2001. The second author was partially supported also by the KAMEA Project for Scientific Absorption in Israel.

## Keywords

- Asymptotic behavior of virtual eigenvalues
- Coupling constant
- Lieb-Thirring estimates
- Puiseux-Newton diagram
- Schrödinger operator
- Virtual eigenvalues

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory