Upper Bounds on the Number of Resonances for Non-compact Riemann Surfaces

Let X be a Riemannian surface of finite geometric type and with hyperbolic ends. The resolvent (ΔX−s(1−s))−1, Re s > 1 of the Laplacian on X extends to a meromorphic family of operators on C and its poles are called resonances. We prove an optimal polynomial bound for their counting function.