Polynomial Growth Solutions Of Sturm-Liouville Equations On A Half-Line And Their Zero Distribution

For α ∈ [0, 2π], consider the Sturm-Liouville equation on the half line y″(x) + (λ - q(x))y(x) = 0, 0 ≤ x \u3c ∞, with y(0) = sin α, y′(0) = -cos α. For each λ \u3e 0, denote by φ(x, λ) the solution of the above initial-value problem. It is known that the condition xq(x) ∈ L1 (ℝ1) is sufficient for φ(x, λ) to be uniformly bounded by a linear function in x for all x, λ ≥ 0; however, this condition is not necessary as the Bessel differential equation demonstrates. In this paper we extend this result to the borderline case in which q(x) = O(1/x2) as x → ∞. We show that if q(x) is continuously differentiable and q(x) - O(1/x2) as x → ∞, that is, xq(x) may not be integrable on ℝ+ then there exists a polynomial p(x) such that |(x, λ)| ≤ p(x) for any x ∈ [0, ∞) and λ ∈ [0, ∞). As a particular example, we consider the perturbed Bessel equation v″(x) +[1 - ν2 - 1/4/x2 + h(x)] v(x) = 0, where ν ∈ ℝ and h(x) = o(1/x2) as x → ∞. The technique, developed in the paper, allows us to find upper and lower bounds on the distance between consecutive zeroes xn, xn+1 + 1 of the solution v(x) of the perturbed Bessel equation, as well as the asymptotics of xn + xn-1 as n → ∞. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim