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Advances in Mathematical Physics
Volume 2018 (2018), Article ID 4696790, 5 pages
Research Article

A Note on the Asymptotic and Threshold Behaviour of Discrete Eigenvalues inside the Spectral Gaps of the Difference Operator with a Periodic Potential

Department of Mathematics, University of Zimbabwe, MP 167, Mt Pleasant, Harare, Zimbabwe

Correspondence should be addressed to Gift Muchatibaya

Received 1 August 2017; Revised 10 January 2018; Accepted 28 January 2018; Published 22 February 2018

Academic Editor: Antonio Scarfone

Copyright © 2018 Gift Muchatibaya and Josiah Mushanyu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The asymptotic and threshold behaviour of the eigenvalues of a perturbed difference operator inside a spectral gap is investigated. In particular, applications of the Titchmarsh-Weyl -function theory as well as the Birman-Schwinger principle is performed to investigate the existence and behaviour of the eigenvalues of the operator inside the spectral gap of in the limits and