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Relationship between different types of inverse data for the one-dimensional Schrödinger operator on a half-line

A. S. Mikhaylov, V. S. Mikhaylov

Abstract

We consider inverse dynamical, spectral, quantum and acoustical scattering problems for the Schrödinger operator on the half line. The goal of the paper is to establish the connections between different types of inverse data for these problems. The central objects which serve as a source for all formulaes are kernels of so-called connecting operators and Krein equations.

Relationship between different types of inverse data for the one-dimensional Schrödinger operator on a half-line

Abstract

We consider inverse dynamical, spectral, quantum and acoustical scattering problems for the Schrödinger operator on the half line. The goal of the paper is to establish the connections between different types of inverse data for these problems. The central objects which serve as a source for all formulaes are kernels of so-called connecting operators and Krein equations.
Paper Structure (13 sections, 7 theorems, 102 equations)

This paper contains 13 sections, 7 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Inverse data
  3. Dynamical inverse data
  4. Spectral inverse data.
  5. Quantum scattering inverse data.
  6. Inverse acoustical scattering problem
  7. Kernel of the connecting operator $C^T$
  8. The spectral function of Levitan and the kernel of the connecting operator.
  9. On the connection of the spectral, dynamical and scattering data.
  10. Weyl function and response function
  11. Response function and spectral measure
  12. Quantum scattering data and response function
  13. Scattering matrix and acoustical response function.

Key Result

Proposition 1

For $q \in L_{loc}^1(0,T)$ and $T >0$, operator $C^T$ admits the representation where

Theorems & Definitions (9)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof
  • Theorem 6
  • proof