Planck-scale number of nodal domains for toral eigenfunctions

DOI10.1016/J.JFA.2020.108663arXiv1911.06247MaRDI QIDQ6329187

Publication date: 14 November 2019

Abstract: We study the number of nodal domains in balls shrinking slightly above the Planck scale for "generic" toral eigenfunctions. We prove that, up to the natural scaling, the nodal domains count obeys the same asymptotic law as the global number of nodal domains. The proof, on one hand, uses new arithmetic information to refine Bourgain's de-randomisation technique at Planck scale. And on the other hand, it requires a Planck scale version of Yau's conjecture which we believe to be of independent interest.

Mathematics Subject Classification ID

Random fields (60G60) Gaussian processes (60G15) Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Asymptotic distributions of eigenvalues in context of PDEs (35P20) PDEs on manifolds (35R01)

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