## Abstract

This paper presents a mathematical description of the linearized Vlasov-Poisson operator L_{k} acting on a family of Banach spaces X _{p}, related to L ^{p}(ℝ), and the application of the method of spectral deformation to this model. It is shown that a type-A analytic family of operators L_{k} (θ), θ∈ℂ, L_{k} (0) = L_{k} can be associated with L_{k}. By means of this family, the Landau damped modes of the plasma are identified as the spectral resonances of L_{k}. Existence and uniqueness of solutions to the initial-value problem for the evolution equation ∂_{ν}g = L _{k} (θ)g is proven. An expansion of any solution to the initial-value problem (with sufficiently smooth initial data) is obtained in terms of the eigenfunctions of L_{k} (θ) and a spectral integral over the essential spectrum. This is applied to derive an expansion for solutions to the Vlasov equation in which the Landau damped portions of the distribution function are manifestly exhibited. A self-contained discussion of the spectral deformation method and an extension of it to certain closed operators on Banach spaces is also given.

Original language | English |
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Pages (from-to) | 2819-2837 |

Number of pages | 19 |

Journal | Journal of Mathematical Physics |

Volume | 30 |

Issue number | 12 |

DOIs | |

State | Published - 1989 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics