Spectral properties of Jacobi matrices and sum rules of special form

In this article, we relate the properties of elements of a Jacobi matrix from certain class to the properties of its spectral measure. The main tools we use are the so-called sum rules introduced by Case in [Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys. 15 (1974) 2166-2174; Orthogonal polynomials, II. J. Math. Phys. 16 (1975) 1435-1440]. Later, the sum rules were efficiently applied by Killip-Simon [Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158 (2003) 253-321] to the spectral analysis of Jacobi matrices. We use a modification of the method that permits us to work with sum rules of higher orders. As a corollary of the main theorem, we obtain a counterpart of a result of Molchanov-Novitskii-Vainberg [First KdV integrals and absolutely continuous spectrum for 1-D Schrödinger operator, Comm. Math. Phys. 216 (2001) 195-213] for a “continuous” Schrödinger operator on a half-line.