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October  2008, 2(4): 701-718. doi: 10.3934/jmd.2008.2.701

On the spectrum of geometric operators on Kähler manifolds

Dmitry Jakobson 1, , Alexander Strohmaier 2, and  Steve Zelditch 3,

1. 

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6, Canada
2. 

Department of Mathematical Sciences, LoughboroughUniversity, Loughborough, Leicestershire, LE11 3TU, United Kingdom
3. 

Johns Hopkins University, Department of Mathematics, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, United States

Received  May 2008 Revised  September 2008 Published  October 2008

On a compact Kähler manifold, there is a canonical action of a Lie-superalgebra 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 Lie-superalgebra on the eigenspaces of the Laplace--Beltrami operator. Because of the high degree of symmetry, the Laplace--Beltrami operator on forms can not be quantum ergodic. We show that, after taking these symmetries into account, quantum ergodicity holds for the Laplace--Beltrami operator and for the Spin$^\cbb$-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved Kähler manifolds of odd complex dimension.
Keywords: Kähler manifold, frame flow, Dirac operator, quantum ergodicity, eigenfunction.
Mathematics Subject Classification: Primary: 81Q50 Secondary: 35P20, 37D30, 58J50, 81Q005.
Citation: Dmitry Jakobson, Alexander Strohmaier, Steve Zelditch. On the spectrum of geometric operators on Kähler manifolds. Journal of Modern Dynamics, 2008, 2 (4) : 701-718. doi: 10.3934/jmd.2008.2.701
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