Resolvents of Bochner Laplacians in the semiclassical limit
Laurent Charles
TL;DR
This work develops a semiclassical Heisenberg pseudodifferential calculus to study the bottom of the spectrum of Bochner Laplacians $\Delta_k$ on sections of $L^k$ as $k\to\infty$. By combining twisted semiclassical pseudodifferential operators with a global Heisenberg framework, the authors construct a resolvent parametrix $Q(z)\in\Psi^{-2}_{Heis}$ and show the spectral projector $1_{[E-1/2,E+1/2]}(k^{-1}\Delta_k)\in\Psi^{-\infty}_{Heis}$, with principal symbols tied to the harmonic oscillator data via $R_{d,z}$ and $\pi_{d,E}$. The calculus captures the curvature-dependence through a fiberwise noncommutative product $\sharp_x$ and yields Weyl-type trace formulas and cluster counts for eigenspaces, including Bergman-kernel-type projections. These results provide a robust, frame-independent tool for analyzing large-curvature limits in complex geometry, geometric quantization, and magnetic Schrödinger-type problems, and extend to auxiliary bundles, enabling broad applications in semiclassical spectral theory.
Abstract
We introduce a new class of pseudodifferential operators, called Heisenberg semiclassical pseudodifferential operators, to study the space of sections of a power of a line bundle on a compact manifold, in the limit where the power is large. This class contains the Bochner Laplacian associated with a connection of the line bundle, and when the curvature is nondegenerate, its resolvent and some associated spectral projections, including generalized Bergman kernels.