Spectral and scattering theory for perturbations of the Carleman operator

TL;DR

This paper analyzes the spectral properties of the Carleman Hankel operator and its perturbations, developing a scattering theory framework, and establishing conditions for the spectrum and eigenvalues of the perturbed operator.

Contribution

It introduces a scattering theory for perturbed Carleman operators, providing explicit formulas, spectral analysis, and eigenvalue estimates for these Hankel operators.

Findings

Explicit resolvent formula for the Carleman operator

Conditions for the absence of singular continuous spectrum

Estimates on the number of eigenvalues above the spectrum

Abstract

We study spectral properties of the Carleman operator (the Hankel operator with kernel $h_{0}(t)=t^{-1}$) and, in particular, find an explicit formula for its resolvent. Then we consider perturbations of the Carleman operator $H_{0}$ by Hankel operators $V$ with kernels $v(t)$ decaying sufficiently rapidly as $t\to\infty$ and not too singular at t=0. Our goal is to develop scattering theory for the pair $H_{0}$, $H=H_{0} +V $ and to construct an expansion in eigenfunctions of the continuous spectrum of the Hankel operator $H$. We also prove that under general assumptions the singular continuous spectrum of the operator $H$ is empty and that its eigenvalues may accumulate only to the edge points 0 and $\pi$ in the spectrum of $H_{0}$. We find simple conditions for the finiteness of the total number of eigenvalues of the operator $H$ lying above the (continuous) spectrum of the Carleman operator $H_{0}$ and obtain an explicit estimate of this number. The theory constructed is somewhat analogous to the theory of one-dimensional differential operators.