The spectral density of a product of spectral projections

We consider the product of spectral projections $$ \Pi_\epsilon(\lambda) = 1_{(-\infty,\lambda-\epsilon)}(H_0) 1_{(\lambda+\epsilon,\infty)}(H) 1_{(-\infty,\lambda-\epsilon)}(H_0) $$ where $H_0$ and $H$ are the free and the perturbed Schr\"odinger operators with a short range potential, $\lambda>0$ is fixed and $\epsilon\to0$. We compute the leading term of the asymptotics of $\mathrm{Tr}\ f(\Pi_\epsilon(\lambda))$ as $\epsilon\to0$ 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.