Asymptotics of the Titchmarsh-Weyl m-coefficient for non-integrable potentials

Asymptotic formulae for the Titchmarsh-Weyl m-coefficient on rays in the co mplex lambda-plane for the equation -y" + qy = lambda y when the potential is limit circle and non-oscillatory at x = 0 are obtained under assumptions slightly more general than xq(x) is an element of L-1 (0, c). The behaviou r of q at the right end-point is arbitrary and may fall in either the limit -point or limit-circle case. A method of regularization of the equation is given that can be made to depend either on a solution of the equation for l ambda = 0 or more directly on an approximation to the solution in terms of q. This enables equivalent definitions of the m-coeffcient to be given for the singular Sturm-Liouville problem associated with a singular limit-circl e boundary condition, and its associated regular Sturm-Liouville problem. A s a consequence, it becomes possible to apply asymptotic results obtained b y Atkinson for the regular problem in order to give asymptotic results for the singular problem. Potentials of the form q(x) = C/x(j), 1 less than or equal to j < 2, are included. In the case j = 1, an independent calculation of the limit-point m-coefficient over the range (0, infinity), relying on Whittaker functions, verifies the main result.