The spectral density of a product of spectral projections

We consider the product of spectral projections Π_ε(λ)=1_((−∞,λ−ε))(H_0)1_((λ+ε,∞))(H)1_((−∞,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 → 0. We compute the leading term of the asymptotics of Trƒ(Π_ε(λ)) as ε → 0 for continuous functions ƒ 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