Virtual Eigenvalues of the High Order Schrödinger Operator I

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.