The Spectral Density of a Difference of Spectral Projections

Let H₀ 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 H₀ and H, then the difference of spectral projections D(λ)=1(-∞,0)(H-λ)-1(-∞,0)(H₀-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(H₀-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 |logε| 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.