Unique continuation for fractional Schrödinger operators in three and higher dimensions

Abstract

We prove the unique continuation property for the differential inequality |(−Δ) α/2 u| ≤ |V (x)u|, where 0 < α < n and V ∈ L n/α,∞ loc (R n ), n ≥ 3.

Unique continuation properties for relativistic Schrödinger operators with a singular potential

“…We mention that recently some strong unique continuation properties for fractional laplacian have been proved by several authors, see [9,13,25,24,32]. We note that, due to its critical homogeneity, the singular potential a(x/|x|) |x| 2s…”

“…does not belong to the class of potentials for which Seo in [25] extended to the fractional case the unique continuation results established by Jerison and Kenig in the seminal paper [19]; indeed such potential is not in L N/(2s) loc and does not have small L N/(2s),∞ -norm. We also observe that, arguing as in [12], the results of the present paper can be proved for a more general class of perturbing potentials h satisfying some integrability type conditions instead of the point-wise decay (2).…”

On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates

“…While both the qualitative and quantitative unique continuation properties of classical Schrödinger equations have been intensively studied much less is known about the analogous properties of solutions of fractional Schrödinger equations. Only very recently, first results on the weak and strong unique continuation principles for these nonlocal equation appeared [Seo13a], [Seo13b], [FF13], [Rül14a], [Rül14b]. Our results can be regarded as a quantitative version of this line of thought on compact Riemannian manifolds without boundary.…”

Zero-Energy Bound State Decay for Non-local Schrödinger Operators

“…For d = 3 it is known that provided |V |, |x · ∇V | and |x · ∇(x · ∇V )| are bounded by C(1 + x 2 ) −1/2 , with a small C > 0, jointly imply that (−∆) 1/2 + V has no non-negative eigenvalue [40]. Related work on unique continuation for fractional Schrödinger equations imply further non-existence results [11,44,45,41]. Some further recent work include non-positive potentials with compact support and L chosen to be the massive relativistic operator [34], and a class of generalized Schrödinger operators [6].…”