On the Angular Density of Three Dimensional Scattering Resonances

Lung-Hui Chen

doi:10.1155/2014/460741

Abstract

We apply Cartwright’s theory in integral function theory to describe the angular distribution of scattering resonances in mathematical physics. A quantitative description on the counting function along rays in complex plane is obtained.

Highlights

We study the distribution of the scattering resonances of a certain class of elliptic operators arousing from Schrodinger operator
It is well-known from spectral analysis that the resolvent operator (H − λ2)−1 : L2(R3) → H2(R3) is bounded in P except for some finite set {μ1, . . . , μe} such that {μ12, . . . , μe2} are the pure point spectrum of H
The resolvent (H − λ2)−1 can be meromorphically extended from P to C as an operator: R (λ) := (H − λ2)−1 : Lc2omp (R3) 󳨀→ Hl2oc (R3)

Summary

Introduction

We study the distribution of the scattering resonances of a certain class of elliptic operators arousing from Schrodinger operator. All such meromorphic poles in C are called resolvent resonances in mathematical physics literature. The poles are called the scattering resonances which share the same multiplicity at each pole as resolvent resonances It is a subject of great interest in mathematical physics to describe the scattering resonances approximately inside a disc of radius r or in certain region in complex plane C. In [11, page 278], Zworski studied the resonances using the theory of zeros of certain Fourier transform developed by Cartwright and Titchmarsh. In [13, page 269], Froese computed the indicator function in one dimensional potential scattering and used the fundamental theorem on the distribution of the zeros of a function of completely regular growth [9, page 152] to prove his results. We previously have lower bound [4] that n (r, 0, 2π, s) rl→im∞

Cartwright’s Theory

Proof of Theorem 1