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

TL;DR

This paper proves that for generic potentials in even-dimensional Schrödinger operators, the resonance counting functions grow at the maximal order of dimension, using complex analysis and scattering theory techniques.

Contribution

It establishes the maximal order of growth for resonance counting functions on each sheet of the Riemann surface for generic potentials in even dimensions.

Findings

Resonance counting functions grow with order d on each sheet for generic potentials.

Constructs a plurisubharmonic function from scattering determinants to analyze resonance poles.

Shows a lower bound for specific potentials, like characteristic functions of a ball.

Abstract

We prove that the resonance counting functions for Schr\"odinger operators $H_V = - \Delta + V$ 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 $\Lambda_m$, $m \in \Z \backslash \{0 \}$, 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 $\Lambda_0$ determine the poles on $\Lambda_m$. 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 $\Lambda_m$, $m \in \Z \backslash \{0\}$.