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Mutually conjugate solutions of formally self-adjoint differential equations

Published online by Cambridge University Press:  14 November 2011

J. M. Hill  and
R. V. Nillsen
J. M. Hill
Affiliation:
Department of Mathematics, The University of Wollongong, N. S. W., Australia
R. V. Nillsen
Affiliation:
Department of Mathematics, The University of Wollongong, N. S. W., Australia
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Synopsis

Let L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 1-2 , 1979 , pp. 93 - 101
Copyright
Copyright © Royal Society of Edinburgh 1979

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