Normal Oscillating Modes of a One-Dimensional Maxwellian Beam in the Linear Space Charge Approximation

Following a method used in plasma physics, a Fourier-Hermite transform is attempted to solve the linearized Vlasov-Poisson equation for small fluctuations of a stationary beam. This expansion method is well fitted to the case of a maxwellian beam confined by linear external forces. As a first approximation, if one retains only the linear part of the space charge forces, the eigenmodes for the small beam oscillations are found in a straightforward way. The associated eigenfrequencies are the roots of a 3-diagonal determinant. The spectrum involves only real and discrete eigenfrequencies, and is similar to the spectrum of a beam with uniform density in real space or in phase space.