Application of the method of spectral deformation to the Vlasov-poisson system

The Vlasov equation is studied using the method of spectral deformation, a technique developed in the theory of Schrödinger operators. The method associates the linearized Vlasov operator L with a family of operators L(θ) which depend analytically on the complex parameter θ. The analysis of L(θ) leads to a novel solution to the linear initial value problem, a new eigenfunction expansion which can be applied to nonlinear problems, and a deeper understanding of the relation between the spectrum of L and Landau damping. For example, it follows from the theory of L(θ) that the embedded eigenvalues of L, which characteristically occur at the threshold of a linear instability, are generically simple. Extending the deformation construction to include the nonlinear terms leads to a family of evolution equations depending on the parameter θ. This family is shown to be time reversal invariant in a suitably generalized sense. The analyticity properties required of Vlasov solutions if they are to correspond to solutions of the new equations for complex θ are described. The eigenfunction expansion for L(θ) is applied to derive equations for the nonlinear evolution of electrostatic waves. This derivation and the resulting amplitude equations are unusual in that the standard assumptions of weak nonlinearity and separated time scales are not used. When these assumptions are made, the familiar form of the amplitude equations is recovered.