TL;DR
This paper analyzes the spectral properties of the linearized Vlasov equation in a magnetic field, revealing complex eigenvalue structures and solving the equation via the resolvent method.
Contribution
It provides a detailed spectral analysis of the Vlasov operator in a magnetic field and introduces a resolvent method for solving the equation.
Findings
Spectrum has continuous real and discrete complex parts.
Real eigenvalues are infinitely degenerate.
Vlasov equation solved using the resolvent method.
Abstract
The linearized Vlasov equation for a plasma system in a uniform magnetic field and the corresponding linear Vlasov operator are studied. The spectrum and the corresponding eigenfunctions of the Vlasov operator are 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 infinitely degenerate, which causes difficulty solving this initial value problem by using the conventional eigenfunction expansion method. Finally, the Vlasov equation is solved by the resolvent method.
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