The spectral density of a product of spectral projections

Abstract 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 Tr f ( Π ε ( λ ) ) as ε → 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.