On the spectrum of geometric operators on kähler manifolds

Dmitry Jakobson*, Alexander Strohmaier, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

On a compact Kählermanifold, there is a canonical action of a Liesuperalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator, and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Liesuperalgebra on the eigenspaces of the Laplace - Beltrami operator. Because of the high degree of symmetry, the Laplace - Beltrami operator on forms can not be quantumergodic. We show that, after taking these symmetries into account, quantum ergodicity holds for the Laplace - Beltrami operator and for the Spin-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved Kählermanifolds of odd complex dimension.

Original languageEnglish (US)
Pages (from-to)701-718
Number of pages18
JournalJournal of Modern Dynamics
Volume2
Issue number4
DOIs
StatePublished - 2008

Keywords

  • Dirac operator
  • Eigenfunction
  • Frame flow
  • Kähler manifold
  • Quantum ergodicity

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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