Pólya’s conjecture fails for the fractional Laplacian

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

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)127-135
Number of pages9
JournalJournal of Spectral Theory
Issue number1
StatePublished - 2019


  • 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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