Killip–Simon problem and Jacobi flow on GMP matrices
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 SchrOdinger 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.
유료 다운로드의 경우 해당 사이트의 정책에 따라 신규 회원가입, 로그인, 유료 구매 등이 필요할 수 있습니다. 해당 사이트에서 발생하는 귀하의 모든 정보활동은 NDSL의 서비스 정책과 무관합니다.
원문복사신청을 하시면, 일부 해외 인쇄학술지의 경우 외국학술지지원센터(FRIC)에서
무료 원문복사 서비스를 제공합니다.
NDSL에서는 해당 원문을 복사서비스하고 있습니다. 위의 원문복사신청 또는 장바구니 담기를 통하여 원문복사서비스 이용이 가능합니다.
- 이 논문과 함께 출판된 논문 + 더보기