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.

dedicated to the memory of Vladimir Savel'evich Buslaev there are some minor and editorial changes compared to 1210.5709 of 21 oct