Local trace formulae for commuting Hamiltonians in T\"oplitz quantization

TL;DR

This paper develops local trace formulae for commuting Hamiltonians in T"oplitz quantization on compact K"ahler manifolds, describing eigenvalue clustering and eigenfunction concentration asymptotics.

Contribution

It introduces a directional local trace formula for commuting Hamiltonians in T"oplitz quantization, linking asymptotics to local geometric and flow properties.

Findings

Derived asymptotic expansions related to local geometry.

Established clustering behavior of joint eigenvalues.

Analyzed concentration of joint eigenfunctions.

Abstract

Let $(M,J,\omega)$ be a quantizable compact K\"ahler 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 T\"oplitz 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 T\"oplitz operators along a given ray, and locally on $M$ the asymptotic concentration of the corresponding joint eigenfunctions. This problem naturally leads to a \lq directional local trace formula\rq, 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.