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The shape of the level sets of the first eigenfunction of a class of two-dimensional Schrödinger operators


Author: Thomas Beck
Journal: Trans. Amer. Math. Soc. 370 (2018), 3197-3244
MSC (2010): Primary 35J10, 35P15
DOI: https://doi.org/10.1090/tran/7049
Published electronically: September 25, 2017
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Abstract: We study the first Dirichlet eigenfunction of a class of Schrödinger operators with a convex, non-negative, potential $ V$ on a convex, planar domain $ \Omega $. In the case where the diameter of $ \Omega $ is large and the potential $ V$ varies on different length scales in orthogonal directions, we find two length scales $ L_1$ and $ L_2$ and an orientation of the domain $ \Omega $ which determine the shape of the level sets of the eigenfunction. As an intermediate step, we also establish bounds on the first eigenvalue in terms of the first eigenvalue of an associated ordinary differential operator.


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Additional Information

Thomas Beck
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
Email: tdbeck@mit.edu

DOI: https://doi.org/10.1090/tran/7049
Received by editor(s): March 10, 2016
Received by editor(s) in revised form: July 22, 2016
Published electronically: September 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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