caa a22 4500 388109084 CHVBK 20180307125348.0 cr unu---uuuuu 161130s1999 xx s 000 0 eng 10.1017/S0308210500013068 doi S0308210500013068 pii (NATIONALLICENCE)cambridge-10.1017/S0308210500013068 Asymptotics of the Titchmarsh—Weyl m-coefficient for non-integrable potentials [Elektronische Daten] 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. Copyright © Royal Society of Edinburgh 1999 Atkinson F. V. University of Toronto, Department of Mathematics, Toronto, Ontario, Canada M5S 1A1 Fulton C. T. Florida Institute of Technology, Applied Mathematics Program, Melbourne, FL 32901, USA Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/4(1999), 663-683 0308-2105 129:4<663 1999 129 PRM https://doi.org/10.1017/S0308210500013068 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0308210500013068 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Atkinson F. V. University of Toronto, Department of Mathematics, Toronto, Ontario, Canada M5S 1A1 NATIONALLICENCE 700 1- Fulton C. T. Florida Institute of Technology, Applied Mathematics Program, Melbourne, FL 32901, USA NATIONALLICENCE 773 0- Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/4(1999), 663-683 0308-2105 129:4<663 1999 129 PRM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge