Abstract
On a compact K\"ahler 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^c-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved K\"ahler manifolds of odd complex dimension.
Highlights
Properties of the spectrum of the Laplace-Beltrami operator on a manifold are closely related to the properties of the underlying classical dynamical system
This paper deals with a situation in which the frame flow is not ergodic, namely the case of Kahler manifolds
For example our analysis shows that in case of an ergodic U (m)-frame flow for any complete sequence of co-closed primitive (p, q)-forms there is a density one subsequence which converges to a state which is an extension of the Liouville measure and can be explicitly given
Summary
Properties of the spectrum of the Laplace-Beltrami operator on a manifold are closely related to the properties of the underlying classical dynamical system. This paper deals with a situation in which the frame flow is not ergodic, namely the case of Kahler manifolds In this case the conclusions in [JS] do not hold since there is a huge symmetry algebra acting on the space of differential forms. Lt is not an endomorphism of vector bundles, but it acts as a pseudodifferential operator of order zero Guided by this result we tackle the question of quantum ergodicity for the Laplace-Beltrami operator on (p, q)-forms. For example our analysis shows that in case of an ergodic U (m)-frame flow for any complete sequence of co-closed primitive (p, q)-forms there is a density one subsequence which converges to a state which is an extension of the Liouville measure and can be explicitly given. Our analysis shows that there are certain invariant subspaces for the Dirac operator in this case and we prove quantum ergodicity for the Dirac operator restricted to these subspaces provided that the U (m)-frame flow is ergodic
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