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 $\lambda$ belongs to the absolutely continuous spectrum of $H_0$ and $H$, then the difference of spectral projections $$D(\lambda )=1_{(-\mathrm{\infty },0)}(H-\lambda )-1_{(-\mathrm{\infty },0)}(H_0-\lambda )$$ in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations $D_{\epsilon }(\lambda )$ of $D(\lambda )$, given by $$D_{\epsilon }(\lambda )={\psi }_{\epsilon }(H-\lambda )-{\psi }_{\epsilon }(H_0-\lambda ),$$ where ${\psi }_{\epsilon }(x)=\psi (x\mathrm{/}\epsilon )$ and $\psi (x)$ is a smooth real-valued function which tends to $\mp 1\mathrm{/}2$ as $x\to \pm \mathrm{\infty }$. We prove that the eigenvalues of $D_{\epsilon }(\lambda )$ concentrate to the absolutely continuous spectrum of $D(\lambda )$ as $\epsilon \to +0$. We show that the rate of concentration is proportional to $\mathrm{\mid }\log \epsilon \mathrm{\mid }$ and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of $\psi$. The proof relies on the analysis of Hankel operators.