Asymptotics and Spectral Theory for High Order Ordinary Differential Equations with Power Coefficients

Abstract

This chapter discusses the asymptotic solutions for large values of the independent variable x of the linear ordinary differential equation of arbitrary order n with power coefficients. Despite the successes of operator theory and matrix methods in the spectral theory of the n th order equation, useful information can be obtained by classical analysis of a model scalar equation. The deficiency index problem is sometimes approached by the operator theory methods. These tend to yield big theorems giving bounds on (N + ,N - ).The matrix methods have the advantage of working for general coefficients, subject only to the restrictions of the various asymptotic perturbation theorems, but the repeated diagonalisation of matrices is nontrivial. The direct methods can be applied generally only to the equations whose coefficients are powers or at best polynomials in x . They provide an explicit solution as integral representation from which a full asymptotic expansion may be constructed. This contrasts with the matrix methods that, in many cases, provide information only as an order term.