The Spectral Density of a Difference of Spectral Projections

Let H 0 and H be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if λ belongs to the absolutely continuous spectrum of H 0 and H , then the difference of spectral projections D ( λ ) = 1 ( - ∞ , 0 ) ( H - λ ) - 1 ( - ∞ , 0 ) ( H 0 - λ ) in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations D ε ( λ ) of D ( λ ) , given by D ε ( λ ) = ψ ε ( H - λ ) - ψ ε ( H 0 - λ ) , where ψ ε ( x ) = ψ ( x / ε ) and ψ ( x ) is a smooth real-valued function which tends to ∓ 1 / 2 as x → ± ∞ . We prove that the eigenvalues of D ε ( λ ) concentrate to the absolutely continuous spectrum of D ( λ ) as ε → + 0 . We show that the rate of concentration is proportional to | log ε | and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of ψ . The proof relies on the analysis of Hankel operators.