Trace asymptotics for Bergman Toeplitz operators via hyperbolic boundary geometry

Release time:2026-10-10

Journal:Submitted

Abstract:Let \(\Omega \Subset \mathbb{D}\) be a finitely connected domain with $C^3$ boundary and \(T_{\Omega}^{(\alpha)}\) be the weighted Bergman Toeplitz operator with symbol \(\chi_{\Omega}\). For admissible continuous functions $f$ on $[0,1]$, we prove that \[ \operatorname{tr}f(T_\Omega^{(\alpha)}) =\frac{s(\partial\Omega)}{2\pi}M(f)\sqrt\alpha+o(\sqrt\alpha), \] where $M(f)$ is a one-dimensional functional determined by the complementary error-function, and $s(\partial\Omega)$ is the hyperbolic length of $\partial \Omega$. This yields a two-term Szeg\H{o}-type asymptotic formula for Bergman Toeplitz operators, with a leading term determined by the hyperbolic area and a second-order term determined by the hyperbolic length of the boundary. As applications of our results, we derive hyperbolic area laws for the von Neumann and R\'enyi entropies, together with corresponding trace asymptotic formulas for Cauchy wavelet localization operators.

Co-author:Wei Dai, Pan Ma, Fugang Yan, Quan Zhao

Translation or Not:no