Thresholds and more bands of a.c. Spectrum for the discrete Schrödinger operator with a more general long range condition

2022/01/24 by Golénia, Sylvain, Mandich, Marc-Adrien
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2201.09545

Abstract

We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schrödinger operator Δ+V on ℓ²(\Z^d), in dimensions d≥ 2, for potentials V satisfying the long range condition n_i(V-τ_i ^κV)(n) = O(ln^-q(|n|)) for some q>2, κ∈ \N, and all 1 ≤ i ≤ d, as |n| → ∞. τ_i ^κ V is the potential shifted by κ units on the i^th coordinate. The difference between this article and \citeGM2 is that here 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 Δ+V is void of singular continuous spectrum, and consecutive thresholds are endpoints of a band of a.c. spectrum.

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