Virtual eigenvalues of the high order schrödinger operator II

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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.

Original languageEnglish
Pages (from-to)305-345
Number of pages41
JournalIntegral Equations and Operator Theory
Issue number3
StatePublished - 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.


  • 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


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