Pólya’s conjecture fails for the fractional Laplacian

Mateusz Kwaśnicki; Richard S. Laugesen; Bartłomiej A. Siudeja

doi:10.4171/JST/242

Pólya’s conjecture fails for the fractional Laplacian

Mateusz Kwaśnicki, Richard S. Laugesen, Bartłomiej A. Siudeja

Research output: Contribution to journal › Article › peer-review

Abstract

The analogue of Pólya’s conjecture is shown to fail for the fractional Laplacian ./ ⁼² on an interval in 1-dimension, whenever 0 < < 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 < < 0:984.

Original language	English (US)
Pages (from-to)	127-135
Number of pages	9
Journal	Journal of Spectral Theory
Volume	9
Issue number	1
DOIs	https://doi.org/10.4171/JST/242
State	Published - 2019
Externally published	Yes

Keywords

Berezin–Li–Yau inequality
Fractional Sobolev
Weyl asymptotic

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Geometry and Topology

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10.4171/JST/242

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abstract = " The analogue of P{\'o}lya{\textquoteright}s conjecture is shown to fail for the fractional Laplacian ./ =2 on an interval in 1-dimension, whenever 0 < < 2. The failure is total: every eigenvalue lies below the corresponding term of the Weyl asymptotic. In 2-dimensions, the fractional P{\'o}lya conjecture fails already for the first eigenvalue, when 0 < < 0:984.",

keywords = "Berezin–Li–Yau inequality, Fractional Sobolev, Weyl asymptotic",

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AU - Kwaśnicki, Mateusz

AU - Laugesen, Richard S.

AU - Siudeja, Bartłomiej A.

N1 - Publisher Copyright: © European Mathematical Society

PY - 2019

Y1 - 2019

N2 - The analogue of Pólya’s conjecture is shown to fail for the fractional Laplacian ./ =2 on an interval in 1-dimension, whenever 0 < < 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 < < 0:984.

AB - The analogue of Pólya’s conjecture is shown to fail for the fractional Laplacian ./ =2 on an interval in 1-dimension, whenever 0 < < 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 < < 0:984.

KW - Berezin–Li–Yau inequality

KW - Fractional Sobolev

KW - Weyl asymptotic

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ER -

Pólya’s conjecture fails for the fractional Laplacian

Abstract

Keywords

ASJC Scopus subject areas

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