Relationship between different types of inverse data for the one-dimensional Schr\"odinger operator on a half-line

TL;DR

This paper explores the relationships between various inverse scattering problems for the one-dimensional Schrödinger operator on a half-line, focusing on connecting operators and Krein equations as key tools.

Contribution

It establishes new connections between inverse dynamical, spectral, quantum, and acoustical scattering data for the Schrödinger operator.

Findings

Unified framework linking different inverse data types

Derivation of formulas using connecting operators and Krein equations

Enhanced understanding of inverse scattering problems on a half-line

Abstract

We consider inverse dynamical, spectral, quantum and acoustical scattering problems for the Schr\"odinger 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.