Scattering theory of hyperbolic spaces with potential scatterer

Lung-Hui Chen, Purdue University

Abstract

In this paper, we study the scattering theory on hyperbolic spaces. We first construct the perturbed resolvent operator by Lipmann-Schwinger equation which has a long history in scattering theory or in mathematical physics. From this operator, we can obtain the scattering matrix by observing the asymptotic behavior of perturbed resolvent operator near the boundary. We also see that the perturbed resolvent and scattering matrix have a meromorphic extension over complex plane. From the point of view of Birman-Krein theory, this perturbation is of trace class. This property enables us to compute the determinant of the scattering matrix which is called scattering phase in the literature. The estimation of the order of this determinant as a meromorphic function turns out to be nontrivial. By the method of G. Vodev, M. Zworski and L. Guillopé, we can expect that the order of this meromorphic function is the same as the dimension of its ambient manifold. In this paper, we will provide a non-optimal estimate as an example. Once we can prove the scattering phase is of finite order, we can define its Fourier transform and establish a Birman-Krein type trace formula. From this trace formula, we can prove a Poisson type summation formula. This is fundamental to study the distribution of the poles of the scattering matrix, including their growth inside a disc of a given radius. In this paper, we obtain some lower bounds for the resonance growth. The odd dimensional case turns out to be less satisfactory. These poles also are called resonances in the mathematics or physics literature. We also study the 2-dimensional Euclidean case as an example. A mild lower bound of resonance growth is obtained via Phragmén-Lindelöf Theorem in complex analysis. This example shows the relation between real line as continuous spectrum and the distribution of the resonances on the complex plane.

Degree

Ph.D.

Advisors

Barreto, Purdue University.

Subject Area

Mathematics

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