Levinson formula for perturbed Hill operator

For a one-dimensional perturbed Hill operator $H$ for which the impurity potential has a finite first moment, a “Levinson series” is obtained. This series of relationships generalizes the well-known Levinson formula to the case when there is a periodic potential. The “Levinson series” is an effective tool for investigating the discrete spectrum in gaps (forbidden bands). In particular, it is shown that in the case of a reflectionless impurity potential with finite second moment there are no eigenvalues of the operator $H$ in the distant gaps of the spectrum.