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

TL;DR

This paper develops a trace formula for the spin Dirac operator on degenerating hyperbolic surfaces, analyzing spectral behavior and zeta function convergence as geodesics shrink to zero.

Contribution

It introduces a novel trace formula for the Dirac operator on hyperbolic surfaces with degenerating geodesics, extending Huber's theorem and analyzing spectral limits.

Findings

Derived a version of Huber's theorem for hyperbolic surfaces with cusps.

Established a uniform Weyl law for Dirac eigenvalues during degeneration.

Proved convergence of the associated Selberg zeta function in degenerating families.

Abstract

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.