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

Abstract

We prove that the resonance counting functions for Schrodinger operators H v = -Δ + V on L 2 (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 ∈ ℤ {O}, of the logarithmic Riemann surface. We obtain this result by constructing, for each m ∈ ℤ {O}, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet Ao 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 C m r d on each sheet Λ m , m ∈ ℤ {0}.