Virtual Bound Levels in a Gap of the Essential Spectrum of the Weakly Perturbed Periodic Schrödinger Operator

Abstract

In the space ${L_{2}(\mathbf{R}^{d}) (d \le 3)}$ we consider the Schrödinger operator ${H_{\gamma}=-{\Delta}+ V(\mathbf{x})\cdot+\gamma W(\mathbf{x})\cdot}$, where ${V(\mathbf{x})=V(x_{1}, x_{2}, \dots, x_{d})}$ is a periodic function with respect to all the variables, ${\gamma}$ is a small real coupling constant and the perturbation ${W(\mathbf{x})}$ tends to zero sufficiently fast as ${|\mathbf{x}|\rightarrow\infty}$. We study so called virtual bound levels of the operator ${H_\gamma}$, i.e., those eigenvalues of ${H_\gamma}$ which are born at the moment ${\gamma=0}$ in a gap ${(\lambda_-,\,\lambda_+)}$ of the spectrum of the unperturbed operator ${H_0=-\Delta+ V(\mathbf{x})\cdot}$ from an edge of this gap while ${\gamma}$ increases or decreases. We assume that the dispersion function of H ₀, branching from an edge of ${(\lambda_-,\lambda_+)}$, is non-degenerate in the Morse sense at its extremal set. For a definite perturbation ${(W(\mathbf{x})\ge 0)}$ we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as ${\gamma\rightarrow 0}$. For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap ${(\lambda_-,\,\lambda_+)}$ are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., ${\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset}$ for a small enough ${|\gamma|}$.