Vlasov equation of plasma in magnetic field

The linearized Vlasov equation for a plasma system in a constant external magnetic field and the corresponding linear Vlasov operator are studied. The solution of the Vlasov equation is found by the resolvent method. The spectrum and eigenfunctions of the Vlasov operator are also found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and complex. Interestingly, the real eigenvalues are uncountably infinitely degenerate, which causes difficulty in solving this initial value problem by using the conventional eigenfunction expansion method. It also breaks the natural relation between the eigenfunctions and the resolvent solution in which the eigenfunctions can normally be considered as the coefficients of e-(i omega t) in the Laplace (or resolvent) solution.