Counting Resonances on Hyperbolic Surfaces with Unitary Twists

Understand

We present the Laplace operator associated to a hyperbolic surface $\Gamma\setminus\mathbb{H}$ and a unitary representation of the fundamental group $\Gamma$, extending the previous definition for hyperbolic surfaces of finite area to those of infinite area.

• We show that the resolvent of this operator admits a meromorphic continuation to all of $\mathbb{C}$ by constructing a parametrix for the Laplacian, following the approach by Guillop\'e and Zworski.
• We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.