Scattering matrix and functions of self-adjoint operators

TL;DR

This paper investigates the spectral behavior of operator differences in scattering theory, linking the spectrum to the scattering matrix and Hankel operator singular values as the parameter tends to zero.

Contribution

It provides an explicit description of the limiting spectrum of operator differences in terms of scattering matrix eigenvalues and Hankel operator singular values.

Findings

Spectral convergence to a set described by scattering matrix eigenvalues.

Explicit characterization of the spectrum in the limit as delta approaches zero.

Connection between spectral differences and Hankel operator singular values.

Abstract

In the scattering theory framework, we consider a pair of operators $H_0$, $H$. For a continuous function $\phi$ vanishing at infinity, we set $\phi_\delta(\cdot)=\phi(\cdot/\delta)$ and study the spectrum of the difference $\phi_\delta(H-\lambda)-\phi_\delta(H_0-\lambda)$ for $\delta\to0$. We prove that if $\lambda$ is in the absolutely continuous spectrum of $H_0$ and $H$, then the spectrum of this difference converges to a set that can be explicitly described in terms of (i) the eigenvalues of the scattering matrix $S(\lambda)$ for the pair $H_0$, $H$ and (ii) the singular values of the Hankel operator $H_\phi$ with the symbol $\phi$.