=0$). Further increasing of $l_{\rm max}$ and $j_{\rm max}$ does not alter the number and location of complex eigenfrequencies in the $\omega$-plane. This shows that unstable modes do not constitute a continuous family.

Figure \[pic:full-spectra\] displays the eigenfrequency spectra for the azimuthal wavenumbers $0\le m\le 5$. Eigenfrequencies on the real axis are oscillatory van Kampen modes. Their calculation requires evaluation of Cauchy’s principal value [@V03] if one uses Kalnajs’s first order theory. In the present formalism, van Kampen modes are found together with growing modes without any special treatment. More van Kampen modes are obtainable by increasing the truncation limit $l_{\rm max}$ of Fourier terms in the $\theta_R$-direction. Toomre’s $Q$ is marginally greater than 1 for the model (Figure 7[*b*]{} in JH), and therefore, one could expect that the disk is stable for $m=0$ excitations (see top-left panel in Figure \[pic:full-spectra\]). The model is highly unstable for $m>0$ excitations although the average growth rate of unstable modes decreases for larger wavenumbers. It is evident that either unstable modes are isolated or they are grouped in [*discrete families*]{}. Depending on the wavenumber, there may be one or more discrete families. The most prominent family bifurcates from van Kampen modes. Members of 