Numerical methods for finding continua

In the GTAW code, the matrix elements $\overline{E}_{21}^{m' m}$ and $\overline{E}_{22}^{m' m}$ are multiplied by $\overline{\omega}^2$ (check whether this will make $\overline{\omega}^2 = 0$ a root of $\ensuremath{\operatorname{Det}} (\overline{E} (\overline{\omega})) = 0$?). After this, the matrix elements $\overline{E}$ can be written in the following form

$$\overline{E} = \overline{E}_a + \overline{\omega}^2 \overline{E}_b,$$ (212)

where $\overline{E}_a$ and $\overline{E}_b$ are $2 L \times 2 L$ matrix which are both independent of $\overline{\omega}$.

The continua are determined by the condition that $\ensuremath{\operatorname{Det}} (\overline{E} (\overline{\omega})) = 0$, which is the condition that the matrix equation $\overline{E} X = 0$ has nonzero solutions. Using Eq. (212), the matrix equation $\overline{E} X = 0$ can be written

$$\overline{E}_a X = - \overline{\omega}^2 \overline{E}_b X.$$ (213)

Thus finding $\overline{\omega}$ that can make $\overline{E} X = 0$ have nonzero solution reduces to finding the eigenvalues of the generalized eigenvalue problem in Eq. (213). In my code, the generalized eigenvalue problem in Eq. (213) is solved numerically by using the zggev subroutine in Lapack library. The numerical results of the continuous spectrum are given in Sec. 8.2.