Integrating the probe and singular sources methods: IV. IPS function for the Schr\"odinger equation

TL;DR

This paper develops an integrated inverse method combining probe and singular sources techniques to detect penetrable obstacles in a Schrödinger equation setting, providing a unified indicator function approach.

Contribution

It introduces an IPS function that unifies probe and singular sources methods, enabling analytical detection of obstacles from boundary measurements in Schrödinger problems.

Findings

Unified indicator functions for obstacle detection

Analytical detection method from Dirichlet-to-Neumann map

Blowing-up property of a sequence related to the IPS method

Abstract

The integrated theory of the probe and singular sources methods (IPS) is developed for an inverse obstacle problem governed by the stationary Schr\"odinger equation in a bounded domain. The unknown obstacles are penetrable, and their surface is modeled by a part of the support of the potential in the governing equation. The main results concern an analytical detection method for these obstacles from the Dirichlet-to-Neumann map. They consist of three parts: a singular sources method via the probe method using a solution with higher-order singularity for the governing equation of the background medium; the discovery of an IPS function whose two ways of decomposition give us the indicator functions for both the probe and singular sources methods; a completely integrated version of both methods, which means their indicator functions coincide. Furthermore, a result on Side B of IPS is also given, concerning the blowing-up property of a sequence calculated from the Dirichlet-to-Neumann map.