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

For alpha is an element of [0, 2pi], consider the Sturm-Liouville equation on the half line y (x) + (lambda - q(x))y(x) 0, 0 less than or equal to x \u3c infinity, with y(0) = sin alpha, y\u27(0) -cos alpha. For each lambda \u3e 0, denote by phi(x, lambda) the solution of the above initial-value problem. It is known that the condition xq(x) is an element of L-1(R+) is sufficient for phi(x, lambda) to be uniformly bounded by a linear function in x for all x, lambda greater than or equal to 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 -- \u3e infinity. We show that if q(x) is continuously differentiable and q(x) = O(1/x(2)) as x -- \u3e infinity, that is, xq(x) may not be integrable on R+, then there exists a polynomial p(x) such that \phi(x, lambda)\ less than or equal to p(x) for any x is an element of [0, infinity) and lambda is an element of [0, infinity). As a particular example, we consider the perturbed Bessel equation v (x) + [1 - (nu(2) - 1/4)/x(2) + h(x)] v(x) = 0, where nu is an element of R and h(x) = o(1/x(2)) as x -- \u3e infinity. 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 -- \u3e infinity. (C) 2004 WILEY-VCH Verlag GmbH Co. KGa