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Inverse scattering at a fixed energy for discrere Schrödinger operators on the square lattice

Hiroshi Isozaki, Hisashi Morioka

Abstract

We study an inverse scattering problem for the discrete Schrödinger operator on the multi-dimensional square lattice, with compactly supported potential. We show that the potential is uniquely reconstructed from a scattering matrix for a fixed energy.

Inverse scattering at a fixed energy for discrere Schrödinger operators on the square lattice

Abstract

We study an inverse scattering problem for the discrete Schrödinger operator on the multi-dimensional square lattice, with compactly supported potential. We show that the potential is uniquely reconstructed from a scattering matrix for a fixed energy.

Paper Structure

This paper contains 34 sections, 37 theorems, 255 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Inverse scattering
  3. Main result
  4. The plan of the proof
  5. Remarks for references
  6. Notation
  7. Acknowledgement
  8. Momentum representation
  9. Discrete Fourier transform
  10. Sobolev and Besov spaces
  11. Resolvent estimate
  12. Spectral representations and S-matrices
  13. Spectral representation on the torus
  14. Spectral representation on the lattice
  15. Scattering matrix
  16. ...and 19 more sections

Key Result

Theorem 1.1

Fix $\lambda\in I_d$ arbitrarily. Then from the S-matrix $\mathcal{S}(\lambda)$, one can uniquely reconstruct the potential $\widehat{V}$.

Figures (10)

  • Figure 1: $d=2$, $\lambda = 0.25$.
  • Figure 2: $d=2$, $\lambda =0.75$.
  • Figure 3: $d=2$, $\lambda=1.25$.
  • Figure 4: $d=3, \lambda = 0.45$.
  • Figure 5: $d=3, \lambda=2.55$.
  • ...and 5 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 28 more