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The spectral density of a product of spectral projections

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