Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T07:13:22.944Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Coppel, W. A.. Disconjugacy. Lecture Notes in Mathematics 220 (Berlin: Springer, 1971).Google Scholar
2Miller, K. S.. Linear differential équations in the real domain (New York: Norton, 1963).Google Scholar
3Pólya, G.. On the mean value theorem corresponding to a given linear homogeneous differential equation. Trans. Amer. Math. Soc. 24 (1922), 312–324.Google Scholar
4Ristroph, R.. Polya's property W and factorization—a short proof. Proc. Amer. Math. Soc. 31 (1972), 631–632.Google Scholar
5Zettl, A.. General theory of the factorization of ordinary differential operators. Trans. Amer. Math. Soc. 197 (1974), 341–352.CrossRefGoogle Scholar