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 $��$ belongs to the absolutely continuous spectrum of $H_0$ and $H$, then the difference of spectral projections $$D(��)=1_{(-\infty,0)}(H-��)-1_{(-\infty,0)}(H_0-��)$$ in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations $D_\varepsilon(��)$ of $D(��)$, given by $$D_\varepsilon(��)=��_\varepsilon(H-��)-��_\varepsilon(H_0-��),$$ where $��_\varepsilon(x)=��(x/\varepsilon)$ and $��(x)$ is a smooth real-valued function which tends to $\mp1/2$ as $x\to\pm\infty$. We prove that the eigenvalues of $D_\varepsilon(��)$ concentrate to the absolutely continuous spectrum of $D(��)$ 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 $��$. The proof relies on the analysis of Hankel operators.

Final version; to appear in Commun. Math. Physics