English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We continue the investigation of the existence of absolutely continuous(a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on$\ell^2(\Z^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the longrange condition $n_i(V-\tau_i ^{\kappa}V)(n) = O(\ln^{-q}(|n|))$ for some$q>2$, $\kappa \in \N$, and all $1 \leq i \leq d$, as $|n| \to \infty$. $\tau_i^{\kappa} V$ is the potential shifted by $\kappa$ units on the $i^{\text{th}}$coordinate. The difference between this article and \cite{GM2} is that herefinite linear combinations of conjugate operators are constructed leading tomore bands of a.c.\ spectrum being observed. The methodology is backedprimarily by graphical evidence because the linear combinations are built bynumerically implementing a polynomial interpolation. On the other hand aninfinitely countable set of thresholds, whose exact definition is given later,is rigorously identified. Our overall conjecture, at least in dimension 2, isthat the spectrum of $\Delta+V$ is void of singular continuous spectrum, andconsecutive thresholds are endpoints of a band of a.c. spectrum.

Thresholds and more bands of a.c. Spectrum for the discrete Schr{\"o}dinger operator with a more general long range condition