Resonances for Schrodinger operators with compactly supported potentials

TL;DR

This paper investigates the typical growth behavior of resonance counting functions for Schrödinger operators with compactly supported potentials across different dimensions, highlighting maximal growth rates and their dimensional dependence.

Contribution

It provides a detailed analysis and proof sketch showing that the resonance counting function's growth is generically maximal in both odd and even dimensions, extending previous results.

Findings

Maximal order of resonance counting function in odd dimensions is the space dimension d.

In even dimensions, the maximal growth occurs on each nonphysical sheet of the Riemann surface.

The results generalize and unify previous findings on resonance behavior for compactly supported potentials.

Abstract

We describe the generic behavior of the resonance counting function for a Schr\"odinger operator with a bounded, compactly-supported real or complex valued potential in $d \geq 1$ dimensions. This note contains a sketch of the proof of our main results \cite{ch-hi1,ch-hi2} that generically the order of growth of the resonance counting function is the maximal value $d$ in the odd dimensional case, and that it is the maximal value $d$ on each nonphysical sheet of the logarithmic Riemann surface in the even dimensional case. We include a review of previous results concerning the resonance counting functions for Schr\"odinger operators with compactly-supported potentials.