Spectral and scattering theory for perturbations of the Carleman operator

Abstract:

The spectral properties of the Carleman operator (the Hankel operator with the kernel $h_{0}(t)=t^{-1}$) are studied; in particular, an explicit formula for its resolvent is found. Then, perturbations are considered 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 $t=0$. The 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 $H$. Also, it is proved 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 $H_{0}$. Simple conditions are found 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 an explicit estimate of this number is obtained. The theory constructed is somewhat analogous to the theory of one-dimensional differential operators.