Spectral theory of piecewise continuous functions of self-adjoint operators

TL;DR

This paper characterizes the absolutely continuous spectrum of certain self-adjoint operator differences using scattering matrices, proving the absence of singular continuous spectrum and finite-multiplicity eigenvalues, via explicit model operators.

Contribution

It provides an explicit spectral description of piecewise continuous functions of self-adjoint operators using symmetrised Hankel operators and multichannel scattering theory.

Findings

Explicit description of the absolutely continuous spectrum involving scattering matrices.

Proof that the singular continuous spectrum is empty.

Eigenvalues have finite multiplicities and only accumulate at spectrum thresholds.

Abstract

Let $H_0$, $H$ be a pair of self-adjoint operators for which the standard assumptions of the smooth version of scattering theory hold true. We give an explicit description of the absolutely continuous spectrum of the operator $\mathcal{D}_\theta=\theta(H)-\theta(H_0)$ for piecewise continuous functions $\theta$. This description involves the scattering matrix for the pair $H_0$, $H$, evaluated at the discontinuities of $\theta$. We also prove that the singular continuous spectrum of $\mathcal{D}_\theta$ is empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the "thresholds" of the absolutely continuous spectrum of $\mathcal{D}_\theta$. Our approach relies on the construction of "model" operators for each jump of the function $\theta$. These model operators are defined as certain symmetrised Hankel operators which admit explicit spectral analysis. We develop the multichannel scattering theory for the set of model operators and the operator $\theta(H)-\theta(H_0)$. As a by-product of our approach, we also construct the scattering theory for general symmetrised Hankel operators with piecewise continuous symbols.