Toeplitz Operators with Analytic Symbols

Abstract

We develop a new semiclassical calculus in analytic regularity, and apply these techniques to the study of Berezin–Toeplitz quantization in real-analytic regularity. We provide asymptotic formulas for the Bergman projector and Berezin–Toeplitz operators on a compact Kahler manifold. These objects depend on an integer N and we study, in the limit $$N\rightarrow +\infty $$ , situations in which one can control them up to an error $$O(e^{-cN})$$ for some $$c>0$$ . We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to Kahler manifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to $$O(e^{-cN})$$ on any real-analytic Kahler manifold as $$N\rightarrow +\infty $$ . We also prove that Toeplitz operators with analytic symbols can be composed and inverted up to $$O(e^{-cN})$$ . As an application, we study eigenfunction concentration for Toeplitz operators if both the manifold and the symbol are real-analytic. In this case we prove exponential decay in the classically forbidden region.