Lower bounds for resonance counting functions for Schr\"odinger operators with fixed sign potentials in even dimensions

TL;DR

This paper establishes that for Schrödinger operators with fixed sign potentials in even dimensions, the resonance counting functions on each sheet of the logarithmic cover grow at the maximal possible rate, revealing fundamental spectral properties.

Contribution

It proves that the resonance counting functions for fixed sign potentials in even dimensions have maximal order of growth on each sheet of the logarithmic cover.

Findings

Resonance counting functions grow maximally on each sheet.

Results apply to Schrödinger operators with fixed sign potentials.

The work characterizes spectral distribution in even dimensions.

Abstract

If the dimension $d$ is even, the resonances of the Schr\"odinger operator $-\Delta +V$ on ${\mathbb R}^d$ with $V$ bounded and compactly supported are points on $\Lambda$, the logarithmic cover of ${\mathbb C} \setminus \{0\}$. We show that for fixed sign potentials $V$ and for nonzero integers $m$, the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth.