P\'olya's conjecture fails for the fractional Laplacian

TL;DR

This paper demonstrates that Pólya's conjecture does not hold for the fractional Laplacian in one and two dimensions, showing that all eigenvalues are below the Weyl asymptotic for certain fractional orders.

Contribution

It provides the first known counterexamples to Pólya's conjecture for the fractional Laplacian, highlighting the failure of the conjecture in fractional spectral problems.

Findings

Failure of Pólya's conjecture for fractional Laplacian in 1D for 0<alpha<2

Failure in 2D for the first eigenvalue when 0<alpha<0.984

All eigenvalues lie below Weyl asymptotic in the studied cases

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.