Solution of a Characteristic Value Problem from the Theory of Chain Molecules

Abstract

An eigenvalue problem encountered in the dynamical theory of chain molecules is ∫ −11α′′(s)(|r−s|)−12ds=−λα(r),α′(±1)=0.This is solved by three methods: use of a Fourier series for α, expansion of α in associated Legendre polynomials Pm2, and by a variation method. The eight smallest eigenvalues are calculated explicitly and an approximate formula is found for the remaining ones. Formulas are found also for the eigenfunctions.