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

Let a(1), a(2), ... , a(n), and lambda be complex numbers, and let p(1), p(2), ... , p(n) be measurable complex-valued functions on R+(:= [0,+infinity)) such that vertical bar p(1)vertical bar + (1+|vertical bar p(2) - p(1)vertical bar)Sigma(n)(j=2)vertical bar p(j)vertical bar is an element of L-loc(1)(R+). A construction is proposed which makes it possible to well define the differential equation y((n)) + (a(1) + p(1)(x))y((n-1)) + (a(2) + p(2)'(x))y((n-2)) + center dot center dot center dot + (a(n) + p(n)'(x)) 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 -> +infinity 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(z) = z(n) + a(1)z(n-1) + center dot center dot center dot + a(n) - lambda, provided that the functions p(1), p(2), ... , p(n) satisfy certain conditions of integral decay at infinity. The case where a(1) = center dot center dot center dot = a(n) = lambda = 0 is considered separately and in more detail.