The Leading Term of the Asymptotics of Solutions of Linear Differential Equations with First-Order Distribution Coefficients

Abstract

Let a₁,a₂, …,a_n, and λ be complex numbers, and let p₁,p₂, …,p_n be measurable complex-valued functions on ℝ₊ (:= [0, + ∞)) such that

\(\left| {{p_1}} \right| + \left( {1 + \left| {{p_2} - {p_1}} \right|} \right)\sum\limits_{j = 2}^n {\left| {{p_j}} \right|} \; \in \;L_{{\rm{loc}}}^1\left( {{\mathbb{R}_ + }} \right).\)

A construction is proposed which makes it possible to well define the differential equation

\({y^{\left( n \right)}} + \left( {{a_1} + {p_1}\left( x \right)} \right){y^{\left( {n - 1} \right)}} + \left( {{a_2} + p_{2}^{\prime} \left( x \right)} \right){y^{\left( {n - 2} \right)}} + \cdots + \left( {{a_n} + p_{n}^{\prime}\left( x \right)} \right)y = \lambda y\)

under this condition, where all derivatives are understood in the sense of distributions. This construction is used to show that the leading term of the asymptotics as x → + ∞ of a fundamental system of solutions of this equation and of their derivatives can be determined, as in the classical case, from the roots of the polynomial

\(Q\left( z \right) = {z^n} + {a_1}{z^{n - 1}} + \cdots + {a_n} - \lambda ,\)

provided that the functions p₁,p₂, …,p_n satisfy certain conditions of integral decay at infinity. The case where a₁ = … = a_n = λ = 0 is considered separately and in more detail.