Local scaling asymptotics in phase space and time in Berezin-Toeplitz quantization (Q2874687)

This paper is concerned with the local asymptotics of the quantization \(\Phi ^\hbar_\tau\), \(\tau \in \mathbb R\), of a varying Hamiltonian symplectomorphism \(\phi _\tau:M \rightarrow M\) of a symplectic manifold in the Berezin-Toeplitz scheme. Here, the asymptotics are taken in the semiclassical regime \(\hbar \rightarrow 0^+\) and \textit{varying} means that, rather than considering one fixed symplectomorphism \(\phi_\tau\) at time say \(\tau =\tau _0\), this work is concerned with the behavior of \(\Phi^\hbar_\tau\) when \(\tau -\tau _0\rightarrow 0\) at different rates with respect to \(\hbar \rightarrow 0^+\). This means that the asymptotics of the distributional kernels of \(\Phi^\hbar_\tau\) is considered when both ``time'' and ``phase space'' variables are suitably rescaled in terms of \(\hbar\). A very interesting example is considered on \(\mathbb P^1\) with the double of the Fubini-Study form.