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Communications in Mathematical Physics

, Volume 349, Issue 1, pp 425–440 | Cite as

Small Scale Equidistribution of Random Eigenbases

  • Xiaolong HanEmail author
Article

Abstract

We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold \({{\mathbb M}}\). Assume that the group of isometries acts transitively on \({{\mathbb M}}\) and the multiplicity \({m_\lambda}\) of eigenfrequency \({\lambda}\) tends to infinity at least logarithmically as \({\lambda \to \infty}\). We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of \({m_\lambda}\). In particular, this implies that almost surely random eigenbases on the sphere \({{\mathbb S}^n}\) (\({n \ge 2}\)) and the tori \({{\mathbb T}^n}\) (\({n \ge 5}\)) are equidistributed at polynomial scales.

Keywords

Manifold Probability Measure Pseudodifferential Operator Nodal Domain Counting Multiplicity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe Australian National UniversityCanberraAustralia

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