Log-scale equidistribution of nodal sets in Grauert tubes

Robert Chang, Steven Morris Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

Let Mτ0 be the Grauert tube (of some fixed radius τ0) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φλ be a Laplacian eigenfunction on M of eigenvalue −λ2 and let φλ C be its holomorphic extension to Mτ0 . In this article, we prove that on Mτ0 ∖M, there exists a dimensional constant α>0 and a full density subsequence {λjk }k=1 of the spectrum for which the masses of the complexified eigenfunctions φλjk C are asymptotically equidistributed at length scale (log⁡λjk )−α. Moreover, the complex zeros of φλjk C also become equidistributed on this logarithmic length scale.

Original languageEnglish (US)
JournalJournal des Mathematiques Pures et Appliquees
DOIs
StateAccepted/In press - Jan 1 2018

Fingerprint

Equidistribution
Eigenvalues and eigenfunctions
Length Scale
Eigenfunctions
Tube
Holomorphic Extension
Subsequence
Riemannian Manifold
Logarithmic
Radius
Eigenvalue
Zero

Keywords

  • Eigenfunction
  • Grauert tube
  • Nodal set
  • Quantum ergodicity
  • Small-scale

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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Log-scale equidistribution of nodal sets in Grauert tubes. / Chang, Robert; Zelditch, Steven Morris.

In: Journal des Mathematiques Pures et Appliquees, 01.01.2018.

Research output: Contribution to journalArticle

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AU - Zelditch, Steven Morris

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AB - Let Mτ0 be the Grauert tube (of some fixed radius τ0) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φλ be a Laplacian eigenfunction on M of eigenvalue −λ2 and let φλ C be its holomorphic extension to Mτ0 . In this article, we prove that on Mτ0 ∖M, there exists a dimensional constant α>0 and a full density subsequence {λjk }k=1 ∞ of the spectrum for which the masses of the complexified eigenfunctions φλjk C are asymptotically equidistributed at length scale (log⁡λjk )−α. Moreover, the complex zeros of φλjk C also become equidistributed on this logarithmic length scale.

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