Local trace formulae for commuting Hamiltonians in Toeplitz quantization

Let $(M, J, \omega)$ be a quantizable compact Kähler manifold, with quantizing Hermitian line bundle $(A, h)$, and associated Hardy space $H(X)$, where $X$ is the unit circle bundle. Given a collection of $r$ Poisson commuting quantizable Hamiltonian functions $f_j$ on $M$, there is an induced Abelian unitary action on $H(X)$, generated by certain Toeplitz operators naturally induced by the $f_j$’s. As a multi-dimensional analogue of the usual Weyl law and trace formula, we consider the problem of describing the asymptotic clustering of the joint eigenvalues of these Toeplitz operators along a given ray, and locally on $M$ the asymptotic concentration of the corresponding joint eigenfunctions. This problem naturally leads to a “directional local trace formula,” involving scaling asymptotics in the neighborhood of certain special loci in $M$. Under natural transversality assumption, we obtain asymptotic expansions related to the local geometry of the Hamiltonian action and flow.

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Published 28 April 2017