We consider the Schrödinger operator L_α 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 csin(2ωx+δ)xγ, where γ∈(12,1). 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 ν_{c r} is an eigenvalue of the operator L_α for some value of the boundary parameter α= α_{c r}, specific to that particular point. We prove that for α≠ α_{c r} the spectral density of the operator L_α has a zero of the exponential type at ν_{c r}.