Pólya's conjecture fails for the fractional Laplacian

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 $0<\alpha <0.984$.