Killip-Simon problem and Jacobi flow on GMP matrices

TL;DR

This paper extends spectral theory results to general finite interval spectra using Jacobi flow on GMP matrices, revealing new integrable systems and solving a longstanding open problem.

Contribution

It introduces a novel approach using Jacobi flow on GMP matrices to generalize Killip-Simon theorems to finite interval spectra.

Findings

Extended Killip-Simon theorem to finite interval spectra

Introduced Jacobi flow on GMP matrices as a new spectral analysis tool

Connected GMP matrices with integrable systems

Abstract

One of the first theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class, which destroys completely the absolutely continuous spectrum of A (von Neumann). However, if A is the discrete free 1-D Schr\"odinger operator and B is 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 perturbations of periodic Jacobi matrices. Recall that the spectrum of a periodic Jacobi matrix is a system of intervals of a very specific nature. Christiansen, Simon and Zinchenko posed the following question: "is there an extension of the Damanik-Killip-Simon theorem to the general finite system of intervals case?" Here we solve this problem completely. Our method deals with the Jacobi flow on GMP matrices. GMP (an abbreviation for Generalized Moment Problem) matrices are probably a new object in the spectral theory (a very close relative of Jacobi and CMV matrices). The Jacobi flow on them is also a probably new member of the rich family of integrable systems. An ideology of analytic vector bundles plays an essential role in our construction.