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Transactions of the American Mathematical Society

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


Author: Angkana Rüland
Journal: Trans. Amer. Math. Soc. 369 (2017), 2311-2362
MSC (2010): Primary 35R11, 35A02
DOI: https://doi.org/10.1090/tran/6758
Published electronically: June 20, 2016
MathSciNet review: 3592513
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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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Additional Information

Angkana Rüland
Affiliation: Mathematical Institute of the University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG Oxford, United Kingdom
Email: ruland@maths.ox.ac.uk

DOI: https://doi.org/10.1090/tran/6758
Keywords: Quantitative unique continuation, fractional Schr\"odinger equations, Carleman estimates, eigenfunctions, nodal domains
Received by editor(s): July 6, 2014
Received by editor(s) in revised form: March 28, 2015
Published electronically: June 20, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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