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

Abstract

Abstract. In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schrödinger equations) on a compact, smooth Riemannian manifold, (M, g), without boundary. Moreover, with only slight modifications these results generalize to equations with C 1 potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary doubling property, we prove upper bounds on the H n−1 -measure of nodal domains of eigenfunctions of the generalized Dirichlet-to-Neumann map on analytic manifolds.

The fractional Calderón problem: Low regularity and stability

“…where > 0 is a small constant, depending only on n, s and the ellipticity constants of the metric g. (ii) If s ∈ (0, 1/2), there is a large constant C M,s,n > 1 depending on M, s, n such that we have ∂ n+1 b = 0 = ∂ n+1 c for y ∈ M × [0, C M,s,n ] . Then in deriving the estimate (50) of Lemma 5.1 in [Rü17], it is possible to argue along the same lines as in the proof of Proposition 5.13. All constants (in particular the constant in (50) in [Rü17]…”

“…We outline the argument for Lemma 5.1 in [Rü17] under the conditions (i), (ii): To this end, we first note that the Carleman estimate in Proposition 5.7 remains valid for operators of the form x 1−2s n+1 ∇ · g∇ + ∂ n+1 x 1−2s n+1 ∂ n+1 , where g is a smooth, uniformly elliptic, symmetric tensor field, which only depends on the tangential directions. This is a consequence of the structure of the proof of Proposition 5.7, in which the tangential and the normal components (of the commutators) decoupled, and the possibility to adjust the constant γ > 0.…”

Doubling Property and Vanishing Order of Steklov Eigenfunctions

“…Let Remark. After having completed the paper, we learn that A. Rüland just considered some similar problems for fractional Schrödinger equations [17] using a different method.…”

Carleman estimates for Baouendi–Grushin operators with applications to quantitative uniqueness and strong unique continuation

“…The authors also established a certain boundary version of the vanishing order estimate. Finally, we refer to the paper [52] for an interesting generalisation to nonlocal equations of the quantitative uniqueness result in [6], and also to [8] for a generalisation to Carnot groups of arbitrary step.…”