Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integrating the probe and singular sources methods: IV. IPS function for the Schrödinger equation

Masaru Ikehata

TL;DR

The paper develops and unifies the probe and singular sources approaches (IPS) for an inverse obstacle problem with penetrable obstacles in the stationary Schrödinger equation $- abla^2 u+V(z)u=0$. It introduces the IPS function via two complementary decompositions, derives Side A results that connect the probe indicator ${\\mathcal{I}}(x)$ and the singular-sources indicators through liftings and energy identities, and then presents a completely integrated IPS that coherently blends both ideas into a single indicator ${\\mathcal{I}}^*(x)$ tied to boundary data. A Side B analysis establishes blowing-up behavior of indicator sequences derived from the Dirichlet-to-Neumann map, enabling obstacle characterization, while uniqueness from local probing shows global determination of the IPS indicators from local measurements. The work also compares IPS with Potthast's method, discusses extensions, and outlines future directions for IPS in more general backgrounds and systems.

Abstract

The integrated theory of the probe and singular sources methods (IPS) is developed for an inverse obstacle problem governed by the stationary Schrödinger 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.

Integrating the probe and singular sources methods: IV. IPS function for the Schrödinger equation

TL;DR

The paper develops and unifies the probe and singular sources approaches (IPS) for an inverse obstacle problem with penetrable obstacles in the stationary Schrödinger equation . It introduces the IPS function via two complementary decompositions, derives Side A results that connect the probe indicator and the singular-sources indicators through liftings and energy identities, and then presents a completely integrated IPS that coherently blends both ideas into a single indicator tied to boundary data. A Side B analysis establishes blowing-up behavior of indicator sequences derived from the Dirichlet-to-Neumann map, enabling obstacle characterization, while uniqueness from local probing shows global determination of the IPS indicators from local measurements. The work also compares IPS with Potthast's method, discusses extensions, and outlines future directions for IPS in more general backgrounds and systems.

Abstract

The integrated theory of the probe and singular sources methods (IPS) is developed for an inverse obstacle problem governed by the stationary Schrödinger 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.
Paper Structure (14 sections, 6 theorems, 194 equations)

This paper contains 14 sections, 6 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. IPS and Side A of Singular Sources Method
  3. Completely Integrated IPS
  4. Comparison to Singular Sources Method of Potthast
  5. Side A of IPS
  6. Proof of Theorem 1.2
  7. Remarks on Liftings and Proof of Proposition 1.1
  8. Proof of Proposition 1.1
  9. Proof of Theorem 1.3
  10. Proof of Theorem 1.4
  11. Uniqueness from local probing
  12. Side B of IPS
  13. Remarks on Future Directions of IPS
  14. Appendix

Key Result

Theorem 1.1

(a) Let $x\in\Omega\setminus\overline{D}$ and $\sigma\in N_x$ satisfy $\sigma\cap\overline{D}=\emptyset$. Then, for an arbitrary needle sequence $\{v_n^j\}$ with $j=1,2,3$ for $(x,\sigma)$ we have where the function ${\mathcal{I}}(x)$ of independent variable $x\in\Omega\setminus\overline{D}$ is defined by and the vector valued function $\hbox{\boldmath $w$}_x=\hbox{\boldmath $w$}(z)$ solves (b)

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 3.1