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Eigenvalues for Perturbed Periodic Jacobi Matrices by the Wigner-von Neumann Approach

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Abstract

The Wigner-von Neumann method, which has previously been used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary \({T}\)-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, \({\lambda}\), into the operator’s absolutely continuous spectrum. Introducing a new rational function, \({C(\lambda; T)}\), related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of \({C(\lambda; T)}\)); in particular showing that there are only finitely many of them.

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Authors and Affiliations

  1. SMSAS, Cornwallis Building, University of Kent, Canterbury, Kent, CT2 7NF, UK

    Edmund Judge & Ian Wood

  2. Department of Math. Physics, Institute of Physics, St. Petersburg University, St. Petergoff, St. Petersburg, 198904, Russia

    Sergey Naboko

Authors
  1. Edmund Judge
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  2. Sergey Naboko
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  3. Ian Wood
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Corresponding author

Correspondence to Ian Wood.

Additional information

Edmund Judge was supported by the Engineering and Physical Sciences Research Council (Grant EP/M506540/1). Sergey Naboko was supported by the Russian Science Foundation (Grant 15-11-30007), NCN 2013/09/BST1/04319 and Marie Curie Grant (2013-2014 yy) PIIF-GA-2011-299919.

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Judge, E., Naboko, S. & Wood, I. Eigenvalues for Perturbed Periodic Jacobi Matrices by the Wigner-von Neumann Approach. Integr. Equ. Oper. Theory 85, 427–450 (2016). https://doi.org/10.1007/s00020-016-2302-5

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  • DOI: https://doi.org/10.1007/s00020-016-2302-5

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