Variational nature of eigenvalue problems with a parameter

Abstract

The following theorem is proved. Let D (ω) be an operator with eigenvalues and eigenfunctions {d_k(ω), ν_k(ω) }, where ω is a complex parameter. Given a complex number d_k0, let ω₀ be such that d_k(ω₀) =〈$Ṽ$ _k(ω₀) ‖D (ω₀) ‖ν_k(ω₀) 〉=d_k0, where $Ṽ$ _k(ω₀) is the dual eigenfunction to ν_k(ω₀). Suppose ψ and $Ṽ$ approximate ν_k(ω₀) and $Ṽ$ _k(ω₀), respectively, to order ε. Then, if D (ω) is analytic in ω in the neighborhood of ω₀, and if ω′ is such that 〈$Ṽ$ ‖D (ω′) ‖ψ〉=d_k0, ω′ usually will approximate ω₀ to order ε². By applying this theorem it is shown that roots of the inhomogeneous plasma dispersion relation usually will be accurate to second order if the associated normal modes and their duals are known merely to first order. The theorem can also be applied to solutions of the dispersion relation in a truncated function space.