Spectral and scattering theory for perturbations of the Carleman operator

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.