The spectral density of a product of spectral projections

We consider the product of spectral projections $${\mathrm{\Pi }}_{ϵ}(\lambda )=1_{(-\mathrm{\infty },\lambda -ϵ)}(H_0)1_{(\lambda +ϵ,\mathrm{\infty })}(H)1_{(-\mathrm{\infty },\lambda -ϵ)}(H_0)$$ where $H_0$ and $H$ are the free and the perturbed Schr"odinger operators with a short range potential, $\lambda >0$ is fixed and $ϵ\to 0$. We compute the leading term of the asymptotics of $\mathrm{T}\mathrm{r}f({\mathrm{\Pi }}_{ϵ}(\lambda ))$ as $ϵ\to 0$ for continuous functions $f$ vanishing sufficiently fast near zero. Our construction elucidates calculations that appeared earlier in the theory of “Anderson’s orthogonality catastrophe” and emphasizes the role of Hankel operators in this phenomenon.