Polynomial growth solutions of Sturm‐Liouville equations on a half‐line and their zero distribution

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$$ v '' (x) + \left[ 1 - {{\nu ^{2} - 1/4} \over {x^{2}}} + h(x) \right] v(x) = 0 \,, $$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)