Abstract
The dispersion relations for electrostatic oscillations of an inhomogeneous beam with a longitudinal velocity spread and of two counter-streaming inhomogeneous beams are investigated. In both cases the effect of the inhomogeneity is to extinguish the discrete spectrum entirely and to make the dispersion function a multivalued analytic function: a pair of branch points is created corresponding to each zero of the dispersion function of the associated homogeneous problem. The zeros themselves do not vanish but move on other sheets of the Riemannian surface of the dispersion function. The result is, physically, that the Landau damping is enhanced and the aperiodic two-beam instability becomes an over-stability.
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