Asymptotic Number of Scattering Resonances for Generic Schrödinger Operators

Let −Δ + V be the Schrödinger operator acting on L 2 ( R d , C ) with d ≥ 3 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a . Let n V ( r ) denote the number of resonances of −Δ + V with modulus ≤ r . We show that if the potential V is generic in a sense of pluripotential theory then n V ( r ) = c d a d r d + O ( r d - 3 16 + ϵ ) as r → ∞ for any ε > 0, where c d is a dimensional constant.