Polynomial growth solutions of Sturm-Liouville equations on a half-line and their zero distribution
✍ Scribed by Chang Eon Shin; Alexander Tovbis; Ahmed I. Zayed
- Publisher
- John Wiley and Sons
- Year
- 2004
- Tongue
- English
- Weight
- 184 KB
- Volume
- 263-264
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
For α ∈ [0, 2__π__], consider the Sturm‐Liouville equation on the half line
y″(x) + (λ − q(x))y(x) = 0, 0 ≤ x < ∞,
with
y(0) = sin__α__, y′(0) = −cos__α__.
For each λ > 0, denote by ϕ(x, λ) the solution of the above initial‐value problem. It is known that the condition xq(x) ∈ L^1^(ℝ^+^) 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/x^2^) as x → ∞. We show that if q(x) is continuously differentiable and q(x) = O(1/x^2^) as x → ∞, that is, xq(x) may not be integrable on ℝ^+^, then there exists a polynomial p(x) such that
|ϕ(x, λ)| ≤ p(x) for any x ∈ [0,∞) and λ ∈ [0,∞).
As a particular example, we consider the perturbed Bessel equation
where ν ∈ ℝ and h(x) = o(1/x^2^) as x → ∞. The technique, developed in the paper, allows us to find upper and lower bounds on the distance between consecutive zeroes x~n~, x~n+1~ of the solution v(x) of the perturbed Bessel equation, as well as the asymptotics of x~n+1~ − x~n~ as n → ∞. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)