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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 $$D(λ)=1_{(-\infty,0)}(H-λ)-1_{(-\infty,0)}(H_0-λ)$$ in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations $D_\varepsilon(λ)$ of $D(λ)$, given by $$D_\varepsilon(λ)=ψ_\varepsilon(H-λ)-ψ_\varepsilon(H_0-λ),$$ where $ψ_\varepsilon(x)=ψ(x/\varepsilon)$ and $ψ(x)$ is a smooth real-valued function which tends to $\mp1/2$ as $x\to\pm\infty$. We prove that the eigenvalues of $D_\varepsilon(λ)$ concentrate to the absolutely continuous spectrum of $D(λ)$ as $\varepsilon\to+0$. We show that the rate of concentration is proportional to $|\log\varepsilon|$ 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.