Pólya’s Conjecture Fails for the Fractional Laplacian

The analogue of Pólya’s conjecture is shown to fail for the fractional Laplacian (−∆) on an interval in 1-dimension, whenever 0 < α < 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 < α < 0.984. Introduction. The Weyl asymptotic for the n-th eigenvalue of the Dirichlet Laplacian on a bounded domain of volume V in R says that λn ∼ (nCd/V ) 2/d as n → ∞, where Cd = (2π) /ωd and ωd = volume of the unit ball in R . In 1-dimension, “volume” means length and in 2-dimensions it means area, so that C1 = π, C2 = 4π. Pólya suggested that the Weyl asymptotic provides more than a limiting relation. He conjectured that it gives a lower bound on each eigenvalue: λn ≥ (nCd/V ) , n = 1, 2, 3, . . . . He proved this inequality for tiling domains [18], but it remains open in general. In this note, we deduce from existing results in the literature that the analogue of Pólya’s conjecture fails for the fractional Laplacian (−∆) on the simplest domain imaginable — an interval in 1-dimension. In 2-dimensions we show it fails on the disk and square, at least for some values of α. Fractional Pólya conjecture. The fractional Laplacian (−∆) is a Fourier multiplier operator, with ( (−∆)u ) (̂ξ) = |ξ|û(ξ), α > 0, where the Fourier transform is defined by