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Transformations of second order ordinary and partial difierential operators

Published online by Cambridge University Press:  14 November 2011

Calvin D. Ahlbrandt ,
Don B. Hinton  and
Roger T. Lewis
Calvin D. Ahlbrandt
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211, U.S.A.
Don B. Hinton
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916, U.S.A.
Roger T. Lewis
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, Birmingham, Alabama 35294, U.S.A.
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Synopsis

Liouville type transformations are given for symmetric linear ordinary and partial differential operators of second order. Explicit formulas are given for the coefficients of the transformed operators. As a corollary to the general theory we obtain an “Atkinson form” for certain first order vector partial differential operators. This leads to a generalization of the concept of “g-unitary” transformations. Applications to oscillation and spectral theories are included.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 1-2 , 1982 , pp. 31 - 49
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

1Ahlbrandt, C. D.. Principal and antiprincipal solutions of self-adjoint differential systems and their reciprocals. Rocky Mountain J. Math. 2 (1972), 169–182.CrossRefGoogle Scholar
2Ahlbrandt, C. D.. Equivalence of differential operators. Lecture Notes in Mathematics 846, 1–16 (Berlin: Springer 1981).Google Scholar
3Ahlbrandt, C. D., Hinton, D. B., and Lewis, R. T.. The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory. J. Math. Anal. Appl. 81 (1981), 234–277.CrossRefGoogle Scholar
4Allegretto, W.. On the equivalence of two types of oscillation for elliptic operators. Pacific J. Math. 55 (1974), 319–328.CrossRefGoogle Scholar
5Atkinson, F. V.. Discrete and Continuous Boundary Value Problems (New York: Academic Press, 1964).Google Scholar
6Barrett, J. H.. Matrix systems of second order differential equations. Portugal. Math. 14 (1955), 79–89.Google Scholar
7Bliss, G. A.. A note on the problem of Lagrange in the calculus of variations. Bull. Amer. Math. Soc. 22 (1916), 220–225.CrossRefGoogle Scholar
8Bliss, G. A.. The transformation of Clebsch in the calculus of variations. In Proceedings of the International Congress 1 (1924) 589–603 (Toronto: University of Toronto Press, 1928).Google Scholar
9Bliss, G. A.. The problem of Lagrange in the calculus of variations. Amer. J. Math. 52 (1930), 673–744.CrossRefGoogle Scholar
10Bolza, O.. Vorlesungen über Variationsrechnung (New York: Chelsea, reprint of 1909 edition).Google Scholar
11Carathéodory, C.. Calculus of Variations and Partial Differential Equations of the First Order. Part I: Partial Differential Equations of the First Order (San Francisco: Holden Day, 1965).Google Scholar
12Clebsch, A.. Über die zweite Variation vielfacher Integrale. J. Reine Angew. Math. 56 (1859), 122–148.Google Scholar
13Coppel, W. A.. Disconjugacy. Lecture Notes in Mathematics 220 (New York: Springer, 1971).Google Scholar
14Devinatz, A.. Essential self-adjointness of Schrödinger-type operators. J. Functional Anal. 25 (1977), 58–59.CrossRefGoogle Scholar
15Etgen, G. J. and Lewis, R. T.. The oscillation of elliptic systems. Math. Nachr. 94 (1980), 43–50.CrossRefGoogle Scholar
16Everitt, W. N.. On the transformation theory of ordinary second-order linear symmetric differential equations. Czechoslovak Math. J. to appear.Google Scholar
17Everitt, W. N. and Zettl, A.. Generalized symmetric ordinary differential expressions I: the general theory. Nieuw Arch. Wisk. 27 (1979), 363–397.Google Scholar
18Glazman, I. M.. Direct methods of qualitative spectral analysis of singular differential operators (Israel program for scientific translation, Jerusalem, 1965).Google Scholar
19Hartman, P.. Self-adjoint, non-oscillatory systems of ordinary, second order, linear differential equations. Duke Math. J. 24 (1957), 25–35.CrossRefGoogle Scholar
20Hinton, D. B.. Limit point criteria for differential equations, II. Canad. J. Math. 26 (1974), 340–351.CrossRefGoogle Scholar
21Hinton, D. B.. Limit point-limit circle criteria for (py′)′ + qv = Afcy. In Lecture Notes in Mathematics 415, 173–183 (Berlin: Springer, 1974).Google Scholar
22Kalf, H.. On the characterization of the Friedrichs extension of ordinary or elliptic differential operators with a strongly singular potential. J. Functional Anal. 10 (1972), 230–250.CrossRefGoogle Scholar
23Kalf, H. and Walter, J.. Strongly singular potentials and essential self-adjointness of singular elliptic operators in (Rm\{0}). J. Functional Anal. 10 (1972), 114–130.CrossRefGoogle Scholar
24Knowles, I.. On essential self-adjointness for singular elliptic differential operators. Math. Ann. 227 (1977), 155–172.CrossRefGoogle Scholar
25Kreith, K.. Oscillation Theory. Lecture Notes in Mathematics 324 (New York: Springer, 1973).Google Scholar
26Kreith, K. and Travis, C.. Oscillation criteria for self-adjoint elliptic systems. Pacific J. Math. 41 (1972), 743–753.CrossRefGoogle Scholar
27Lewis, R. T.. The existence of conjugate points for self-adjoint differential equations of even order. Proc. Amer. Math. Soc. 56 (1976), 162–166.CrossRefGoogle Scholar
28Radon, J.. Zum Problem von Lagrange. Abh. Math. Sem. Univ. Hamburg 6 (1928), 273–299.CrossRefGoogle Scholar
29Reid, W. T.. Oscillation criteria for linear differential systems with complex coefficients. Pacific J. Math. 6 (1956), 733–751.CrossRefGoogle Scholar
30Reid, W. T.. Principal solutions of non-oscillatory self-adjoint linear differential systems. Pacific J. Math. 8 (1958), 147–169.CrossRefGoogle Scholar
31Reid, W. T.. Ordinary Differential Equations (New York: Wiley, 1971).Google Scholar
32Reid, W. T.. Sturmian Theory for Ordinary Differential Equations. Applied Mathematical Sciences 31 (New York: Springer, 1980).Google Scholar
33Churchill, R. V., Brown, J. W. and Verhey, R. F.. Complex Variables and Applications, 3rd edn (New York: McGraw-Hill, 1976).Google Scholar
34Fleming, W. H.. Functions of Several Variables (Reading, Mass.: Addison-Wesley: 1965).Google Scholar
35Hellwig, B.. Ein Kriterium für die Selbstadjungiertheit singular elliptischer Differentialoperatoren im Gebiet G. Math. Z. 89 (1965), 333–344.CrossRefGoogle Scholar
36Loewner, C. and Nirenberg, L.. Partial differential equations invariant under conformal or projective transformations. In Contributions to Analysis, 245–272 ed. Ahlfors, , Kra, , Maskit, , and Nirenberg, (New York, Academic Press: 1974).CrossRefGoogle Scholar
37Moss, W. F. and Piepenbrink, J.. Positive solutions of elliptic equations. Pacific J. Math. 75 (1978), 219–226.CrossRefGoogle Scholar
38Nehari, Z.. Conformal Mapping (New York: Dover, 1952).Google Scholar