On the Angular Density of Three Dimensional Scattering Resonances

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. 1. Introduction In this paper, we study the distribution of the scattering resonances of a certain class of elliptic operators arousing from Schr？dinger operator. We have where . Let us denote the physical plane by It is well-known from spectral analysis that the resolvent operator is bounded in except for some finite set such that are the pure point spectrum of . The resolvent can be meromorphically extended from to as an operator: with poles of finite rank. All such meromorphic poles in are called resolvent resonances in mathematical physics literature. There are scattering theories in more generalized formalism. We refer to [1–3]. Let all of the meromorphic poles of be denoted as repeated according to the multiplicity such that the only accumulation point is at infinity. The possible infinite set is in the lower half complex plane. The resolvent operator defines a scattering matrix which is of the form , where is of trace class depending meromorphically on . 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 or in certain region in complex plane . Therefore, we count the poles of the meromorphically defined scattering determinant . In any case, we consider the determinant satisfying the following properties [1–4]: ;？ the point set and is symmetric about the imaginary axis;？ there is no pole on the real axis except possibly a double pole at ;？ there are only exceptionally finitely many poles in ; infinitely many poles in ;？ the functional determinant is of order 3, the number of space dimension.The growth estimate on has only upper bound as proved in [2] which is an optimal upper bound. The actual lower bound is unknown to the author. is the most nontrivial hypothesis. Let us define where is chosen minimally such that has no zero at . Surely, is a regular function of order three in . Because of , the zeros of are substitutes in the study of poles of in . As suggested by , there are infinitely many zeros of in . We will use Cartwright’s theory [5–10] to describe the zeros of more precisely. We state the following result. Theorem 1. Let be the number of the zeros of inside the sector and . Let be the generalized indicator