Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I*

F. V. Atkinsona1 and C. T. Fultona2

a1 Department of Mathematics, University of Toronto, Toronto, Canada

a2 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, U.S.A.

Synopsis

Asymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

(Received February 14 1984)

Footnotes

* This research was sponsored by the National Science Foundation under grant #MCS-7902025.