Maximal order of growth for the resonance counting functions for generic potentials in even dimensions

We prove that the resonance counting functions for Schr̈odinger operators H_V = -δ+V on L²(ℝ^d), for d ≥ 2 even, with generic, compactly-supported, real- or complex-valued potentials V, have the maximal order of growth d on each sheet Λ_m, m ∈ Znf0g, of the logarithmic Riemann surface. We obtain this result by constructing, for eachm ∈ ℤ\{0}, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet Λ_o. determine the poles on Λ _m. We prove that the order of growth of the counting function is related to a suitable estimate on this function that we establish for generic potentials. We also show that for a potential that is the characteristic function of a ball, the resonance counting function is bounded below by Cmr ^d on each sheet Λ_m, m ∈ ℤ\{0}.