The spectral density of a difference of spectral projections

Abstract

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 $$D(\lambda )=1_{(-\infty ,0)}(H-\lambda )-1_{(-\infty ,0)}(H_0-\lambda )$$ 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 ${\psi }_{\epsilon }(x)=\psi (x/\epsilon )$ and $\psi (x)$ is a smooth real-valued function which tends to $\mp 1/2$ as $x\to \pm \infty$. 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 $|\log \epsilon |$ 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.

Final version; to appear in Commun. Math. Physics

References in corpus (1)

• The spectral density of a product of spectral projections