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

Rüland, Angkana

doi:10.1090/tran/6758

On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates
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Trans. Amer. Math. Soc. 369 (2017), 2311-2362 Request permission

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 $\mathcal {H}^{n-1}$-measure of nodal domains of eigenfunctions of the generalized Dirichlet-to-Neumann map on analytic manifolds.

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