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Inverse scattering for the multipoint potentials of Bethe-Peierls-Thomas-Fermi type

Pei-Cheng Kuo, Roman G. Novikov

Abstract

We consider the Schrödinger equation with a multipoint potential of the Bethe-Peierls-Thomas-Fermi type. We show that such a potential in dimension d=2 or d=3 is uniquely determined by its scattering amplitude at a fixed positive energy. Moreover, we show that there is no non-zero potential of this type with zero scattering amplitude at a fixed positive energy and a fixed incident direction. Nevertheless, we also show that a multipoint potential of this type is not uniquely determined by its scattering amplitude at a positive energy E and a fixed incident direction. Our proofs also contribute to the theory of inverse source problem for the Helmholtz equation with multipoint source.

Inverse scattering for the multipoint potentials of Bethe-Peierls-Thomas-Fermi type

Abstract

We consider the Schrödinger equation with a multipoint potential of the Bethe-Peierls-Thomas-Fermi type. We show that such a potential in dimension d=2 or d=3 is uniquely determined by its scattering amplitude at a fixed positive energy. Moreover, we show that there is no non-zero potential of this type with zero scattering amplitude at a fixed positive energy and a fixed incident direction. Nevertheless, we also show that a multipoint potential of this type is not uniquely determined by its scattering amplitude at a positive energy E and a fixed incident direction. Our proofs also contribute to the theory of inverse source problem for the Helmholtz equation with multipoint source.

Paper Structure

This paper contains 12 sections, 5 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Proof of Proposition \ref{['key lemma complex']}
  5. Proofs of Theorem \ref{['inverse scattering']}, \ref{['transparent']}
  6. Nontriviality of $q_{j}$
  7. Proof of Theorem \ref{['inverse scattering']}
  8. Proof of Theorem \ref{['transparent']}
  9. Proofs of Theorem \ref{['counter']}
  10. Proof of Theorem \ref{['counter']} by fitting parameters.
  11. Proof of Theorem \ref{['counter']} by adding invisible point scatterers
  12. Numerical example

Key Result

Theorem 3.1

A multipoint potential $\nu$ of the form (multipotential), for $d=2$ or $d=3$ under condition (det), is uniquely determined by its scattering amplitude $f^{+}$ at a fixed energy $E>0$.

Figures (1)

  • Figure 1: Level sets of $\Re (\psi^{+})=0$ and $\Im (\psi^{+})=0$ intersect at a point around $(0.994,-4.398)$

Theorems & Definitions (5)

  • Theorem 3.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 5.1