## Abstract

We consider the Schrödinger operator H_{γ} = (-Δ)^{ι} + _{γ}V(x)• acting in the space L_{2}(IR^{d}), where 2ι ≥ d, V(x) ≥ 0, V(x) is continuous and is not identically zero, and lim_{|x|→∞} V(x) = 0. We obtain an asymptotic expansion as γ ↑ 0 of the bottom negative eigenvalue of H_{γ}, which is 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)^{ι} (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H_{0}. Furthermore, we extract a finite-rank portion φ(λ) from the Birman- Schwinger operator X _{V}(λ) = V^{1/2}R_{λ}(H _{0})V^{1/2}, which yields the leading terms for the desired asymptotic expansion.

Original language | English |
---|---|

Pages (from-to) | 189-231 |

Number of pages | 43 |

Journal | Integral Equations and Operator Theory |

Volume | 55 |

Issue number | 2 |

DOIs | |

State | Published - Jun 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
- Birman-Schwinger principle
- Coupling constant
- Schrödinger operator
- Virtual eigenvalues

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory