Virtual Eigenvalues of the High Order Schrödinger Operator II

Abstract

We consider the Schrodinger 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 study the asymptotic behavior as \(\gamma\,\uparrow\,0\) 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.