Bound state eigenvalues from transmission coefficients

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 $(T(E)=\mathrm{\mid }t(E){\mathrm{\mid }}^2)$ for six analytically solvable potential wells.