THRESHOLDS AND MORE BANDS OF A.C. SPECTRUM FOR THE DISCRETE SCHRÖDINGER OPERATOR WITH A MORE GENERAL LONG RANGE CONDITION
Language
en
Document de travail - Pré-publication
English Abstract
We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on $\ell^2(\mathbb{Z}^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying ...Read more >
We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on $\ell^2(\mathbb{Z}^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the long range condition $n_i(V-\tau_i ^{\kappa}V)(n) = O(\ln^{-q}(|n|))$ for some $q>2$, $\kappa \in \mathbb{N}$, 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 here \textit{finite} linear combinations of conjugate operators are constructed leading to more bands of a.c.\ spectrum being observed. The methodology is backed primarily by graphical evidence because the linear combinations are built by numerically implementing a polynomial interpolation. On the other hand an infinitely countable set of thresholds, whose exact definition is given later, is rigorously identified. Our overall conjecture, at least in dimension 2, is that the spectrum of $\Delta+V$ is void of singular continuous spectrum, and consecutive thresholds are endpoints of a band of a.c. spectrum.Read less <
English Keywords
2010 Mathematics Subject Classification. 39A70
81Q10
47B25
47A10 discrete Schrödinger operator
long range potential
limiting absorption principle
Mourre theory
Chebyshev polynomials
polynomial interpolation
threshold
2010 Mathematics Subject Classification. 39A70
Origin
Hal imported