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arXiv · 2601.14779

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

Masaru Ikehata
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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.

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Masaru Ikehata. 2026-01-21. Integrating the probe and singular sources methods: IV. IPS function for the Schr\"odinger equation. https://doi.org/10.1515/jiip-2025-0097

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