Local Scaling Asymptotics for the Gutzwiller Trace Formula in Berezin–Toeplitz Quantization

Let f be a smooth function on a quantizable compact Kähler manifold, with the special property that the associated Hamiltonian flow acts by isometries. We consider a naturally associated Berezin–Toeplitz operator, which generates a unitary action on the Hardy space of the quantization. In this setting, we produce local scaling asymptotics in the semiclassical regime for a Berezin–Toeplitz version of the Gutzwiller trace formula, in the spirit of the near-diagonal scaling asymptotics of Szegö and Toeplitz kernels. More precisely, we consider an analogue of the ‘Gutzwiller–Toeplitz kernel’ previously introduced in this setting by Borthwick, Paul, and Uribe, and study how it asymptotically concentrates along the appropriate classical loci defined by the dynamics, with an explicit description of the exponential decay along normal directions. These local scaling asymptotics probe into the concentration behavior of the eigenfunctions of the quantized Hamiltonian flow. When globally integrated, they yield the analogue of the Gutzwiller trace formula