Mittag-Leffler's function, Vekua transform and an inverse obstacle scattering problem

TL;DR

This paper introduces a novel method using Mittag-Leffler functions and the Vekua transform to extract obstacle shape information from far field data in a 2D inverse scattering problem.

Contribution

It provides an explicit construction and analysis of a new indicator function based on Mittag-Leffler functions for obstacle detection.

Findings

The indicator function's asymptotic behavior reveals obstacle exterior features.

The method utilizes explicit modifications of Mittag-Leffler functions and the Vekua transform.

The approach is effective with fixed wave number data.

Abstract

This paper studies a prototype of inverse obstacle scattering problems whose governing equation is the Helmholtz equation in two dimensions. An explicit method to extract information about the location and shape of unknown obstacles from the far field operator with a fixed wave number is given. The method is based on: an explicit construction of a modification of Mittag-Leffler's function via the Vekua transform and the study of the asymptotic behaviour; an explicit density in the Herglotz wave function that approximates the modification of Mittag-Leffler's function in the bounded domain surrounding unknown obstacles; a system of inequalities derived from Kirsch's factorization formula of the far field operator. Then an indicator function which can be calculated from the far field operator acting on the density is introduced. It is shown that the asymptotic behaviour of the indicator function yields information about the visible part of the exterior of the obstacles.