P\'olya's conjecture fails for the fractional Laplacian
Spectral Theory
2016-08-26 v1 Analysis of PDEs
Abstract
The analogue of P\'olya'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 P\'olya conjecture fails already for the first eigenvalue, when 0 < alpha < 0.984.
Cite
@article{arxiv.1608.06905,
title = {P\'olya's conjecture fails for the fractional Laplacian},
author = {Mateusz Kwaśnicki and Richard S. Laugesen and Bartłomiej A. Siudeja},
journal= {arXiv preprint arXiv:1608.06905},
year = {2016}
}
Comments
7 pages