Bound state eigenvalues from transmission coefficients

TL;DR

This paper demonstrates how to extract bound state eigenvalues directly from transmission coefficients in six analytically solvable potential wells, highlighting the connection between scattering data and bound states.

Contribution

It provides explicit methods and illustrations for deriving bound state energies from transmission coefficients, emphasizing the link often overlooked in textbooks.

Findings

Bound state eigenvalues can be obtained from transmission coefficients.

Explicit examples for six solvable potentials are provided.

The connection between scattering data and bound states is clarified.

Abstract

Experts know that bound state energy eigenvalues of a potential well are poles of its transmission amplitude, $t(E)$. The textbooks do well by solving the bound states and scattering states separately, but the connection goes unheeded. Here, we present interesting and instructive illustrations, where we extract bound state eigenvalues from the known transmission coefficients $(T(E)=|t(E)|^2)$ for six analytically solvable potential wells.