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

Published online by Cambridge University Press:  14 November 2011

F. V. Atkinson  and
C. T. Fulton
F. V. Atkinson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada
C. T. Fulton
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, U.S.A.
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Extract

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 1-2 , 1984 , pp. 51 - 70
Copyright
Copyright © Royal Society of Edinburgh 1984

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