Abstract
For a class of one-dimensional Schrödinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit distribution in the complex plane as the eigenvalues tend to infinity. This limit distribution depends only on the degree of the polynomial potential and on the boundary conditions.
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Eremenko, A., Gabrielov, A. & Shapiro, B. High Energy Eigenfunctions of One-Dimensional Schrödinger Operators with Polynomial Potentials. Comput. Methods Funct. Theory 8, 513–529 (2008). https://doi.org/10.1007/BF03321702
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DOI: https://doi.org/10.1007/BF03321702