Asymptotic number of scattering resonances for generic Schrodinger operators

TL;DR

This paper establishes that for generic potentials, the number of scattering resonances of the Schrödinger operator in odd dimensions grows asymptotically proportionally to the volume of a sphere of radius r, with a specific constant.

Contribution

It proves an asymptotic formula for the number of resonances for generic potentials in odd-dimensional Schrödinger operators, extending understanding of resonance distribution.

Findings

Resonance count grows as a constant times a^d r^d for large r.

The result applies to generic potentials vanishing outside a finite ball.

Provides a precise asymptotic estimate for the distribution of resonances.

Abstract

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of resonances of -Delta+V with modulus less than r is approximatively equal to a constant times a^dr^d.