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Rares Stan

Publications and source records attributed to Rares Stan.

3 recordsLinked to original sources
arXiv2023

Spectral convergence of the Dirac operator on typical hyperbolic surfaces of high genus

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil-Petersson surfaces of large genus $g$ with $o(\sqrt{g})$ cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.

math.SP↗
arXiv2022

The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces

We investigate the spectrum of the spin Dirac operator on families of hyperbolic surfaces where a set of disjoint simple geodesics shrink to $0$, under the hypothesis that the spin structure is non-trivial along each pinched geodesic. The main tool is a trace formula for the Dirac operator on finite area hyperbolic surfaces. We derive a version of Huber's theorem and a non-standard small-time heat trace asymptotic expansion for hyperbolic surfaces with cusps. As a corollary we find a simultaneous Weyl law for the eigenvalues of the Dirac operator which is uniform in the degenerating parameter. The main result is the convergence of the Selberg zeta function associated to the Dirac operator on such families of hyperbolic surfaces. A central role is played by a $\{ \pm 1 \}$-valued class function $\varepsilon$ determined by the spin structure.

math.DG↗