Virtual bound levels in a gap of the essential spectrum of the Schroedinger operator with a weakly perturbed periodic potential

Abstract: In the space $L_2(R^d)$ we consider the Schr\"odinger operator $H_\gamma=-\Delta+ V(x)\cdot+\gamma W(x)\cdot$, where $V(x)=V(x_1,x_2,\dots,x_d)$ is a periodic function with respect to all the variables, $\gamma$ is a small real coupling constant and the perturbation $W(x)$ tends to zero sufficiently fast as $|x|\rightarrow\infty$. We study so called virtual bound levels of the operator $H_\gamma$, that is 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 $H_0=-\Delta+ V(x)\cdot$ from an edge of this gap while $\gamma$ increases or decreases. For a definite perturbation $(W(x)\ge 0)$ we investigate the number of such levels and an asymptotic behavior of them and of the corresponding eigenfunctions as $\gamma\rightarrow 0$ in two cases: for the case where the dispersion function of $H_0$, branching from an edge of $(\lambda_-,\lambda_+)$, is non-degenerate in the Morse sense at its extremal set and for the case where it has there a non-localized degeneration of the Morse-Bott type. In the first case in the gap there is a finite number of virtual eigenvalues if $d<3$ and we count the number of them, and in the second case in the gap there is an infinite number of ones, if the codimension of the extremal manifold is less than $3$. For an indefinite perturbation we estimate the multiplicity of virtual bound levels. Furthermore, we show that if the codimension of the extremal manifold is at least $3$ at both edges of the gap $(\lambda_-,\,\lambda_+)$, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, that is $\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset$ for a small enough |\gamma|.