On asymptotics of solutions to some linear differential equations

In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation $l_{2n}[y]=\lambda y$ of order $2n$, where $l_{2n}$ is the product of second-order linear differential expressions and $\lambda$ 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 asumptotic 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.