  We consider the Schr\"odinger operator $\mathcal L_{\alpha}$ 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 $\frac{c\sin(2\omega x+\delta)}{x^{\gamma}}$, where
$\gamma\in(\frac12,1)$. 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
$\nu_{cr}$ is an eigenvalue of the operator $\mathcal L_{\alpha}$ for some
value of the boundary parameter $\alpha=\alpha_{cr}$, specific to that
particular point. We prove that for $\alpha\neq\alpha_{cr}$ the spectral
density of the operator $\mathcal L_{\alpha}$ has a zero of the exponential
type at $\nu_{cr}$.
