We consider the Schrödinger operator Hγ = (-Δ)ι + γV(x)• acting in the space L2(IRd), 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 Hγ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H0) of the unperturbed operator H0 = (-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 H0. Furthermore, we extract a finite-rank portion φ(λ) from the Birman- Schwinger operator X V(λ) = V1/2Rλ(H 0)V1/2, which yields the leading terms for the desired asymptotic expansion.
|Number of pages||43|
|Journal||Integral Equations and Operator Theory|
|State||Published - Jun 2006|
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
- Birman-Schwinger principle
- Coupling constant
- Schrödinger operator
- Virtual eigenvalues
ASJC Scopus subject areas
- Algebra and Number Theory