The Vlasov?Maxwell Operators in Self-adjoint Form

Abstract

A systematic method for obtaining the scalar product with respect to which a given Vlasov-Maxwell operator is self-adjoint is illustrated by considering the operator introduced by Shure (1964) to describe one-dimensional longitudinal oscillations in a bounded plasma. By separation of variables the problem is reduced to the solution of a singular integral equation for the weight function of the scalar product. The solution of this equation is not explicit but is in the form of a simple linear Fredholm integral equation of the second kind which is easily solved. The method should be applicable to similar operators in this field.