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

Abstract

Asymptotic formulae for the Titchmarsh—Weyl m-coefficient on rays in the complex λ-plane for the equation − y ″ + qy = λy whenthe potential is limit circle and non-oscillatory at x = 0 are obtained under assumptions slightly more general than xq(x) ∈ L1(0,c). The behaviour 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 λ = 0 or more directly on an approximation to the solution in terms of q. This enables equivalent definitions of the m-coefficient to be given for the singular Sturm—Liouville problem associated with a singular limit-circle boundary condition, and its associatedregular Sturm—Liouville problem. As a consequence, it becomes possible to apply asymptotic results obtained by Atkinson for the regular problem in order to give asymptoticresults for the singular problem. Potentials of the form q(x) = C/xj, 1 ≤ j < 2, are included. In the case j = 1, an independent calculation of the limit-point m-coefficient over the range (0,∞), relying on Whittaker functions, verifies the main result.