Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T02:11:07.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Commuting linear differential expressions

Published online by Cambridge University Press:  14 November 2011

M. Giertz ,
M. K. Kwong  and
A. Zettl
M. Giertz
Affiliation:
Department of Mathematics, The Royal Institute of Technology, 10040 Stockholm 70, Sweden
M. K. Kwong
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
A. Zettl
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider the question: When do two ordinary linear differential expressions commute? It turns out that the set of all expressions which commute with a given one form a commutative ring. Here we study the algebraic structure of these rings. As an application a complete characterization of normal differential expressions is obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 3-4 , 1981 , pp. 331 - 347
Copyright
Copyright © Royal Society of Edinburgh 1981

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

1Burchnall, J. L. and Chaundy, T. W.. Commutative ordinary differential operators. Proc. London Math. Soc. 24 (1923), 420–440.CrossRefGoogle Scholar
2Burchnall, J. L. and Chaundy, T. W.. Commutative ordinary differential operators. Proc. Roy. Soc. London Ser. A 118 (1928), 557–583.Google Scholar
3Burchnall, J. L. and Chaundy, T. W.. Commutative ordinary differential operators II.—The identity P nxs = Q m. Proc. Roy. Soc. London Ser. A 134 (1931), 471–485.Google Scholar
4Dubrovin, B. A., Matveev, V. B., and Novikov, S. P.. Nonlinear equations of the Korteweg-de Vries type, finite-zone linear operators, and Abelian manifolds. Uspehi. Mat. Nauk. 31 (1976), 55–136.Google Scholar
5Krichever, I. M.. Integration of non-linear equations by the method of algebraic geometry. Functional Anal. Appl. 11 (1977), 15–31.Google Scholar
6Krichever, I. M.. Algebrogeometric construction of the Zakhasov-Shabat equations and their periodic solutions. Dokl. Akad. Nauk. SSSR 227 (1976), 291–294.Google Scholar
7Krichever, I. M.. Algebraic curves and commutative matrix differential operators. Functional Anal. Appl. 10 (1976), 144–146.CrossRefGoogle Scholar
8Niven, I. and Zuckerman, H. S.. An Introduction to the Theory of Numbers, 3rd edn (New York: Wiley, 1972).Google Scholar
9Ore, O.. Formale Theorie der linearen Differentialgleichungen, Erster Teil. J. Reine Angew. Math. 167 (1932), 221–234.CrossRefGoogle Scholar
10Ore, O.. Formale Theorie der linearen Differentialgleichungen, Zweiter Teil. J. Reine Angew. Math. 168 (1932), 233–252.Google Scholar
11Amitsur, S. A.. Commutative linear differential operators. Pacific J. Math. 8 (1958), 1–10.Google Scholar
12Carlson, R. C. and Goodearl, K. R.. Commutants of ordinary differential operators, pre-print.Google Scholar