The spectral density of a product of spectral projections

We consider the product of spectral projectionsΠε(λ)=1(−∞,λ−ε)(H0)1(λ+ε,∞)(H)1(−∞,λ−ε)(H0)Πε(λ)=1(−∞,λ−ε)(H0)1(λ+ε,∞)(H)1(−∞,λ−ε)(H0) where H0H0 and H   are the free and the perturbed Schrödinger operators with a short range potential, λ>0λ>0 is fixed and ε→0ε→0. We compute the leading term of the asymptotics of Trf(Πε(λ)) as ε→0ε→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.