2008/11/28 by T. J. Christiansen, Christiansen, T. J., Peter D. Hislop +2
Mathematics · Physics and Astronomy · #35P25 #47A10 #47A40 #81U20 #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:35P25 #msc:47A10 #msc:47A40 #msc:81U20
paper · pdf · doi:10.48550/arxiv.0811.4761
33 pages and 1 figure
arxiv created 2008/11/28 · openalex publication_date 2008/11/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the resonance counting functions for Schrödinger operators HV = - Δ+ V on L2 (\Rd), for d ≥ 2 \it even, with generic, compactly-supported, real- or complex-valued potentials V, have the maximal order of growth d on each sheet Λm, m ∈ \Z \backslash \0 \, of the logarithmic Riemann surface. We obtain this result by constructing, for each m ∈ \Z \backslash \0 \, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet Λ0 determine the poles on Λm. 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 Cm rd on each sheet Λm, m ∈ \Z \backslash \0\.