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arXiv · 1608.06905

Pólya's conjecture fails for the fractional Laplacian

Mateusz Kwaśnicki·Richard S. Laugesen·Bartłomiej A. Siudeja
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Abstract

The analogue of Pólya's conjecture is shown to fail for the fractional Laplacian (-Delta)^{alpha/2} on an interval in 1-dimension, whenever 0 < alpha < 2. The failure is total: every eigenvalue lies below the corresponding term of the Weyl asymptotic. In 2-dimensions, the fractional Pólya conjecture fails already for the first eigenvalue, when 0 < alpha < 0.984.

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Mateusz Kwaśnicki, Richard S. Laugesen, Bartłomiej A. Siudeja. 2016-08-24. Pólya's conjecture fails for the fractional Laplacian. https://arxiv.org/abs/1608.06905

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