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One of the first theorems in perturbation theory claims that for an arbitraryself-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class,which destroys completely the absolutely continuous spectrum of A (vonNeumann). However, if A is the discrete free 1-D Schr\"odinger operator and Bis a Jacobi matrix the a.c. spectrum remains perfectly the same. Moreover,Killip and Simon described explicitly the spectral properties for such A+B.Jointly with Damanik they generalized this result to the case of perturbationsof periodic Jacobi matrices. Recall that the spectrum of a periodic Jacobimatrix is a system of intervals of a very specific nature. Christiansen, Simonand Zinchenko posed the following question: "is there an extension of theDamanik-Killip-Simon theorem to the general finite system of intervals case?"Here we solve this problem completely. Our method deals with the Jacobi flow onGMP matrices. GMP (an abbreviation for Generalized Moment Problem) matrices areprobably a new object in the spectral theory (a very close relative of Jacobiand CMV matrices). The Jacobi flow on them is also a probably new member of therich family of integrable systems. An ideology of analytic vector bundles playsan essential role in our construction.

Killip-Simon problem and Jacobi flow on GMP matrices