On the direct and inverse transmission eigenvalue problems for the Schrodinger operator on the half line

TL;DR

This paper studies the distribution of transmission eigenvalues for the Schrödinger operator on the half line and demonstrates that the potential can be uniquely reconstructed from these eigenvalues and additional boundary data.

Contribution

It provides a detailed asymptotic analysis of transmission eigenvalues and establishes a unique reconstruction method for the potential from spectral data.

Findings

Asymptotic distribution of transmission eigenvalues characterized.

Unique potential reconstruction from eigenvalues and boundary parameters proven.

Reconstruction algorithms are explicitly developed.

Abstract

For the direct problem, we give the asymptotic distribution of the (real and non-real) transmission eigenvalues for the Schrodinger operator on the half line. For the inverse problem, we prove that the potential can be uniquely determined by all transmission eigenvalues, the parameter in the boundary condition and some non-zero value related to the potential (whereas without the certain constant {\gamma}). The reconstruction algorithms are also revealed.