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On the L2 nature of solutions of nth order symmetric differential equations and McLeod's conjecture

Published online by Cambridge University Press:  14 November 2011

R. B. Paris  and
A. D. Wood
R. B. Paris
Affiliation:
Association Euratom—C.E.A., Centre d'Etudes Nucléaires, 92260 Fontenay-aux-Roses, France
A. D. Wood
Affiliation:
Department of Mathematics, Cranfield Institute of Technology, Cranfield, Bedford MK43 OAL, England
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Synopsis

We investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2m, whose coefficients involve a real multiple ɑr of certain positive real powers β of the independent variable x. Information on the L2 nature is obtained by variation of parameters from Meijer function solutions of an associated homogeneous equation of hypergeometric type. When the coefficients of the differential expressions are positive, it is possible, by a suitable choice of ɑr, β and m, to obtain between m and 2m —1 linearly independent solutions in L2(0, ∞). This proves a conjecture of J. B. McLeod that the deficiency index can take values between m and 2m —1 for such operators.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 90 , Issue 3-4 , 1981 , pp. 209 - 236
Copyright
Copyright © Royal Society of Edinburgh 1981

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