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