Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von Neumann approach

The Wigner-von Neumann method, which was previously 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, $λ$, into the operator's absolutely continuous spectrum. Introducing a new rational function, $C(λ;T)$, related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of $C(λ;T)$); in particular showing that there are only finitely many of them.

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