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

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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.## References

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## Additional Information

**Thomas Beck**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
- MR Author ID: 1039108
- Email: tdbeck@mit.edu
- Received by editor(s): March 10, 2016
- Received by editor(s) in revised form: July 22, 2016
- Published electronically: September 25, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 3197-3244 - MSC (2010): Primary 35J10, 35P15
- DOI: https://doi.org/10.1090/tran/7049
- MathSciNet review: 3766847