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

Beck, Thomas

doi:10.1090/tran/7049

The shape of the level sets of the first eigenfunction of a class of two-dimensional Schrödinger operators
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Trans. Amer. Math. Soc. 370 (2018), 3197-3244 Request permission

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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