Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schrödinger operator Hγ  =  ( − Δ)l  +  γ V(x)· acting in the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2 (\mathbb{R}^d ),$$\end{document} where 2l  ≥  d,  V (x)  ≥   0,  V (x) is continuous and is not identically zero, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$\end{document} We study the asymptotic behavior as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma\,\uparrow\,0$$\end{document} of the non-bottom negative eigenvalues of Hγ, which are born at the moment γ  =  0 from the lower bound λ  =  0 of the spectrum σ(H0) of the unperturbed operator H0  =  ( − 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.