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Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique

Volume 242 / 2018

Edmund Judge, Sergey Naboko, Ian Wood Studia Mathematica 242 (2018), 179-215 MSC: Primary 47B36; Secondary 47A75. DOI: 10.4064/sm170325-23-8 Published online: 26 January 2018

Abstract

For an arbitrary Hermitian period-$T$ Jacobi operator, we assume a perturbation by a Wigner–von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, $S$, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solutions. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an $l^1$-perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.

Authors

  • Edmund JudgeSchool of Mathematics, Statistics
    and Actuarial Science
    Sibson Building
    University of Kent
    Canterbury, Kent, CT2 7FS, United Kingdom
    e-mail
  • Sergey NabokoDepartment of Mathematical Physics
    Institute of Physics
    St. Petersburg State University
    1 Ulianovskaia St., Petergoff
    St. Petersburg, 198504, Russia
    e-mail
  • Ian WoodSchool of Mathematics, Statistics, and Actuarial Science
    Sibson Building
    University of Kent
    Canterbury, Kent, CT2 7FS, United Kingdom
    e-mail

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