The problem of many discrete states of a Hamiltonian which are embedded in and interact with many continua is considered. Exact continuum eigenstates are found by a direct extension of Fano's formalism, without use of scattering theory. Our procedure reduces the computation time when many continua are involved. The conditions for the existence of exact discrete states are discussed. The line shapes in ${\epsilon }_1\mathrm{}\left(E\right)\mathrm{}$, ${\epsilon }_2\mathrm{}\left(E\right)\mathrm{}$, $\mathrm{Im}[-\frac{1}{\epsilon \mathrm{}\left(E\right)\mathrm{}}]$, and the absorption coefficient are compared.

• Received 9 July 1976

DOI:https://doi.org/10.1103/PhysRevB.15.2961