caa a22 4500 467931666 CHVBK 20180406152945.0 cr unu---uuuuu 170328e20060601xx s 000 0 eng 10.1007/s00020-005-1389-x doi (NATIONALLICENCE)springer-10.1007/s00020-005-1389-x Virtual Eigenvalues of the High Order Schrödinger Operator I [Elektronische Daten] [Jonathan Arazy, Leonid Zelenko] Abstract.: We consider the Schrödinger operator Hγ=(−Δ)l+γ V(x)· acting in the space $$L_2 (\mathbb{R}^d ),$$ where 2l≥d,V(x)≥0,V(x) is continuous and is not identically zero, and $$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$ We obtain an asymptotic expansion as $$\gamma \uparrow 0$$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)l (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 $$X_V (\lambda ) = V^{\frac{1} {2}} R_\lambda (H_0 )V^{\frac{1}{2}} ,$$ which yields the leading terms for the desired asymptotic expansion. Birkhäuser Verlag, Basel, 2006 Schrödinger operator nationallicence virtual eigenvalues nationallicence coupling constant nationallicence asymptotic behavior of virtual eigenvalues nationallicence Birman-Schwinger principle nationallicence Arazy Jonathan Department of Mathematics, University of Haifa, 31905, Haifa, Israel aut Zelenko Leonid Department of Mathematics, University of Haifa, 31905, Haifa, Israel aut Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 55/2(2006-06-01), 189-231 0378-620X 55:2<189 2006 55 20 https://doi.org/10.1007/s00020-005-1389-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00020-005-1389-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Arazy Jonathan Department of Mathematics, University of Haifa, 31905, Haifa, Israel aut NATIONALLICENCE 700 1- Zelenko Leonid Department of Mathematics, University of Haifa, 31905, Haifa, Israel aut NATIONALLICENCE 773 0- Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 55/2(2006-06-01), 189-231 0378-620X 55:2<189 2006 55 20 Metadata rights reserved Springer special CC-BY-NC licence nationallicence SWISSBIB 467931569 BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer