Virtual Eigenvalues of the High Order Schrödinger Operator II

Abstract.

We consider the Schrödinger operator H_γ  =  ( − 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 H_γ, 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.