Killip–Simon problem and Jacobi flow on GMP matrices

P. Yuditskii

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert–Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuous (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is an arbitrary Jacobi matrix (of Hilbert–Schmidt class) the a.c. spectrum remains perfectly the same, that is, the interval [−2,2]. 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 in the non-degenerated case. Recall that the spectrum of a periodic Jacobi matrix is a system of intervals of a very specific nature. Christiansen, Simon and Zinchenko posed in a review dedicated to F. Gesztesy (2013) the following question: “is there an extension of the Damanik–Killip–Simon theorem to the general finite system of intervals case?” In this paper we solve this problem completely. Our method deals with the Jacobi flow on GMP matrices. GMP matrices are probably a new object in the spectral theory. They form a certain Generalization of matrices related to the strong Moment Problem, the latter ones are 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. Finally, related to Jacobi matrices of Killip–Simon class, analytic vector bundles and their curvature play a certain role in our construction and, at least on the level of ideology, this role is quite essential.

Original languageEnglish
Pages (from-to)811-865
Number of pages55
JournalAdvances in Mathematics
Volume323
DOIs
StatePublished - 7 Jan 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

  • Generalized Szegö theorems
  • Hardy spaces on Riemann surfaces
  • Hilbert–Schmidt perturbations
  • Integrable systems
  • Jacobi matrices

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