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

Christiansen, T. J.; Hislop, P. D.

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

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Description

We prove that the resonance counting functions for Schrödinger operators $H_V = - Δ+ V$ on $L^2 (\R^d)$, for $d \geq 2$ {\it even}, with generic, compactly-supported, real- or complex-valued potentials $V$, have the maximal order of growth $d$ on each sheet $Λ_m$, $m \in \Z \backslash \{0 \}$, of the logarithmic Riemann surface. We obtain this result by constructing, for each $m \in \Z \backslash \{0 \}$, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet $Λ_0$ 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 $Λ_m$, $m \in \Z \backslash \{0\}$.
33 pages and 1 figure