Large mode number eigenvalues of the prolate spheroidal differential equation

摘要

We show that the eigenvalues of the prolate spheroidal differential equation of zeroth order, χn(c) where c is the so-called “bandwidth” or “ellipticity” parameter, are well-approximated for large mode number n by a single function Ω in the form χn∼c*2Ω(c/c*) where c*≡(π/2)(n+1/2). Ω is defined implicitly as the root of an algebraic equation, E(min(1,m−1/2),m1/2)=1/Ω where E is the usual incomplete integral of the second kind and m=γ2/Ω with γ=c/c*. Ω has a weak singularity at γ=1 proportional to (γ−1)log(γ−1) plus iterated logarithms. We give Chebyshev series for Ω accurate for γ∈[0,∞].