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, $\lambda$, 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.

Cited by in corpus (4)

• Topics on Fermi varieties of discrete periodic Schrödinger operators
• Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators
• Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach
• Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators