Distribution of scattering resonances for generic Schrodinger operators

TL;DR

This paper studies the asymptotic distribution of scattering resonances for a class of Schrödinger operators with bounded potentials, revealing their distribution pattern in the complex plane as the scaling parameter grows large.

Contribution

It establishes the asymptotic distribution of resonances for a broad class of potentials, extending previous results to more general cases and providing explicit probability measures.

Findings

Resonances are asymptotically distributed according to an explicit probability measure.

Distribution pattern is on the closed lower unit half-disc in the complex plane.

Rate of convergence of the distribution is analyzed for subclasses of potentials.

Abstract

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials introduced by Christiansen, we show that when r goes to infinity, the resonances of -Delta+V, scaled down by the factor r, are asymptotically distributed, with respect to an explicit probability distribution on the closed lower unit half-disc of the complex plane. The rate of convergence is also considered for subclasses of potentials.