  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.
