The spectral density of a difference of spectral projections

TL;DR

This paper studies the spectral density of a difference of spectral projections for self-adjoint operators, showing how eigenvalues of compact approximations concentrate on the absolutely continuous spectrum with a specific asymptotic rate.

Contribution

It introduces a method to analyze the eigenvalue concentration of compact approximations of spectral projection differences using Hankel operators, providing explicit asymptotic formulas.

Findings

Eigenvalues concentrate on the absolutely continuous spectrum as epsilon approaches zero.

The concentration rate is proportional to the logarithm of epsilon.

The asymptotic density of eigenvalues is explicitly computed and independent of the smoothing function.

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 $D_\varepsilon(\lambda)$ of $D(\lambda)$, given by $$D_\varepsilon(\lambda)=\psi_\varepsilon(H-\lambda)-\psi_\varepsilon(H_0-\lambda),$$ where $\psi_\varepsilon(x)=\psi(x/\varepsilon)$ and $\psi(x)$ is a smooth real-valued function which tends to $\mp1/2$ as $x\to\pm\infty$. We prove that the eigenvalues of $D_\varepsilon(\lambda)$ concentrate to the absolutely continuous spectrum of $D(\lambda)$ 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 $\psi$. The proof relies on the analysis of Hankel operators.