Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T17:27:28.019Z Has data issue: false hasContentIssue false
  • English
  • Français

On the commutativity of certain quasi-differential expressions II

Published online by Cambridge University Press:  14 November 2011

H. Frentzen ,
D. Race  and
A. Zettl
H. Frentzen
Affiliation:
Fachbereich 6-Mathematik und Informatik, Universität Essen, Universitätsstr.3, W-4300 Essen 1, Germany
D. Race
Affiliation:
Mathematics Department, University of Surrey, Guildford, Surrey GU2 5XH, U.K.
A. Zettl
Affiliation:
Department of Mathematics, 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, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar, J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be an nth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 1 , 1993 , pp. 27 - 43
Copyright
Copyright © Royal Society of Edinburgh 1993

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

1Amitsur, S. A.. Commutative linear differential operators. Pacific J. Math. 8 (1958), 1–10.CrossRefGoogle Scholar
2Burchnall, J. L. and Chaundy, T. W.. Commutative ordinary differential operators. Proc. London Math. Soc. 21 (1923), 420–440.CrossRefGoogle Scholar
3Burchnall, J. L. and Chaundy, T. W.. Commutative ordinary differential operators. Proc. Roy. Soc. London Ser. A 118 (1928), 557–583.Google Scholar
4Burchnall, J. L. and Chaundy, T. W.. Commutative ordinary differential operators. II. - The identity P n = Qm. Proc. Roy. Soc. London Ser. A 134 (1931), 471–485.Google Scholar
5Carlson, R. C. and Goodearl, K. R.. Commutants of ordinary differential operators. J. Differential Equations 35 (1980), 339–365.CrossRefGoogle Scholar
6Frentzen, H.. Equivalence, adjoints and symmetry of quasi-differential expressions with matrixvalued coefficients and polynomials in them. Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), 123–146.CrossRefGoogle Scholar
7Frentzen, H.. On J-symmetric quasi-differential expressions with matrix-valued coefficients. Quaestiones Math. 10 (1986), 153–164.CrossRefGoogle Scholar
8Giertz, M., Kwong, M. K. and Zettl, A.. Commuting linear differential expressions. Proc. Roy. Soc. Edinburgh Sect. A 87 (1981), 331–347.CrossRefGoogle Scholar
9Kato, T.. Perturbation theory for linear operators, 2nd edn (Berlin: Springer, 1976).Google Scholar
10Krichever, I. M.. Integration of nonlinear equations by the methods of algebraic geometry. Functional Anal. Appl. 11 (1977), 12–26.CrossRefGoogle Scholar
11Ore, O.. Formale Theorie der linearen Differentialgleichungen. (Erster Teil). J. Reine Angew. Math. 167 (1932), 221–234.CrossRefGoogle Scholar
12Ore, O.. Formale Theorie der linearen Differentialgleichungen. (Zweiter Teil). J. Reine Angew. Math. 168 (1932), 233–252.CrossRefGoogle Scholar
13Race, D. and Zettl, A.. On the commutativity of certain quasi-differential expressions I. J. London Math. Soc. (2) 42 (1990), 489–504CrossRefGoogle Scholar
14Zettl, A.. Formally self-adjoint quasi-differential operators. Rocky Mountain J. Math. 5 (1975), 453–474.CrossRefGoogle Scholar