We consider the Schrödinger operator H γ =(-Δ) l +γ V(x)• acting in the space L2 (double scprit R signd ), 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.
Bibliographical noteFunding 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.
- Asymptotic behavior of virtual eigenvalues
- Coupling constant
- Lieb-Thirring estimates
- Puiseux-Newton diagram
- Schrödinger operator
- Virtual eigenvalues
ASJC Scopus subject areas
- Algebra and Number Theory