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

In the space we consider the Schrodinger operator , where is a periodic function with respect to all the variables, is a small real coupling constant and the perturbation tends to zero sufficiently fast as . We study so called virtual bound levels of the operator , i.e., those eigenvalues of which are born at the moment in a gap of the spectrum of the unperturbed operator from an edge of this gap while increases or decreases. We assume that the dispersion function of H (0), branching from an edge of , is non-degenerate in the Morse sense at its extremal set. For a definite perturbation we show that if d a parts per thousand currency sign 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 . 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 are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., for a small enough .