Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique

Abstract

For an arbitrary Hermitian period-T Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, S, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solution. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an l 1 -perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.2010 Mathematics Subject Classification. Primary 47B36 ; Secondary 47A75.

Absence of singular continuous spectrum for perturbed discrete Schrödinger operators

“…The construction of potentials with dense embedded eigenvalues for perturbed periodic operator was known for around 20 years [8]. However, similar results for the discrete case were only done in very recent papers [5,11]. Although the proof of this paper follows the strategy for the continuous case [6], the extension is not completely straightforward.…”

Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators

“…Before that, Wigner–von Neumann type functions can only create one L 2 solution [41]. Recently, there have been several important developments on the problem of embedded eigenvalues for Schrödinger operators, Laplacians on manifolds or other models [12, 14–17, 25, 27, 29–32, 34]. For perturbed Stark type operators, under the rational independence assumption of set , Naboko and Pushnitskii [36] constructed operators with given a set as embedded eigenvalues.…”

“…In [39], Simon used Wigner–von Neumann type functions , to complete his constructions. It turns out that Wigner–von Neumann type function is a good way to create embedded eigenvalues [15, 17, 26, 31–33]. Moreover, Wigner–von Neumann type functions can also be used to achieve the optimal bounds.…”

Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators

“…) , to complete his constructions. It turns out that Wigner-von Neumann type function is a good way to create embedded eigenvalues [14,16,24,[29][30][31]. Moreover, Wigner-von Neumann type functions can also be used to achieve the optimal bounds.…”