Connection between the two branches of the quantum two-stream instability across the k space

The stability of two quantum counter-streaming electron beams is investigated within the quantum plasma fluid equations for arbitrarily oriented wave vectors. The analysis reveals that the two quantum two-stream unstable branches are indeed connected by a continuum of unstable modes with oblique wave vectors. Using the longitudinal approximation, the stability domain for any k is analytically explained, together with the growth rate.