On Asymptotics of Solutions to Some Linear Differential Equations

Abstract

In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation l_2n[y] = λy of order 2n, where l_2n is the product of second-order linear differential expressions and λ is a fixed complex number. We assume that the coefficients of these differential expressions are not necessarily smooth but have a prescribed power growth at infinity. The asymptotic formulas obtained are applied for the problem on the defect index of differential operators in the case where l_2n is a symmetric (formally self-adjoint) differential expression.