We consider the Schrödinger operator Lα on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner-von Neumann potential c sin(2ωx+δ)x γ , where γ ∈ ( 1 2 , 1). The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point νcr is an eigenvalue of the operator Lα for some value of the boundary parameter α = αcr, specific to that particular point. We prove that for α = αcr the spectral density of the operator Lα has a zero of the exponential type at νcr.