On the spectrum of geometric operators on Kähler manifolds (Q998861)

There is a close relation between the eigenfunctions, \(\{\psi_j\}\), of the Laplace-Beltrami operator on a manifold, \(X\), and properties of the underlying dynamical system. For example, when the geodesic flow on the unit tangent bundle, \(T_1X\), of a complete Riemannian manifold is ergodic, the eigenfunctions satisfy quantum ergodicity. That is, there is a density one subsequence \(\psi_{j_k}\) such that for any zero-order pseudodifferential operator \(A\), \[ \lim_{k\to\infty}\langle A\psi_{j_k},\psi_{j_k}\rangle=\int_{T^1X}\sigma_A(\xi)d\mu(\xi), \] where \(\sigma_A\) is the principal symbol of \(A\) and \(d\mu\) is the Liouville measure on the unit cotangent bundle \(T_1^*X\). This paper studies the notion of Quantum ergodicity for systems, i.e., non scalar operators acting on many-component wave functions. In particular it studies the relation between the ergodicity of the frame flow on a Kähler manifold and the eigenfunctions of the Dolbeault Laplacian and Spin\(^\mathbb C\)-Dirac operators on these manifolds. Because of the large symmetry algebra for these systems, ergodicity of the frame flow does not imply quantum ergodicity. Rather, the action of the symmetry algebra creates a finite number of quantum ergodic components in the Hilbert space (of forms or spinors respectively). On each of these components, ergodicity of the frame flow implies a form of quantum ergodicity. However, the limiting distribution depends on the symmetry component, hence, for the full system there are a finite number of quantum limits of positive density.