Spectral and scattering theory for perturbations of the Carleman operator

الأصل · EN

We study spectral properties of the Carleman operator (the Hankel operator with kernel h₀(t)=t⁻¹) and, in particular, find an explicit formula for its resolvent. Then we consider perturbations of the Carleman operator H₀ by Hankel operators V with kernels v(t) decaying sufficiently rapidly as t→∞ and not too singular at t=0. Our goal is to develop scattering theory for the pair H₀, H=H₀ +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 π in the spectrum of H₀. 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 H₀ and obtain an explicit estimate of this number. The theory constructed is somewhat analogous to the theory of one-dimensional differential operators.