A New Method of Analyzing Eigenvalues and Eigenfunctions of Linearized Rosenbluth Collision Operator

TL;DR

This paper introduces a numerical approach to analyze the eigenvalues and eigenfunctions of the linearized Rosenbluth collision operator, providing new insights into its spectrum and mathematical properties.

Contribution

It offers a novel numerical method for eigenvalue analysis and rigorously proves the completeness and orthogonality of the eigenfunctions.

Findings

Minimum eigenvalues calculated numerically

Proved completeness and orthogonality of eigenfunctions

Indicated the continuum spectrum of the operator

Abstract

Eigenvalue spectrum has been a long term unsolved problem for plasma physicists. In this paper, some numerical calculations are conducted about the minimum eigenvalues of the linearized Rosenbluth collision operator and the differential part of the whole operator. Similarities of the two operator lead to further discussions about the differential operator. Completeness and orthogonality are rigorously proved. Moreover, the continuum of the whole operator is indicated.